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14130:-dimensional topological vector space, up to isomorphism. It follows from this that any finite-dimensional subspace of a TVS is closed. A characterization of finite dimensionality is that a Hausdorff TVS is locally compact if and only if it is finite-dimensional (therefore isomorphic to some Euclidean space).
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25342:" and column "Absorbing" is colored green. But since the arbitrary intersection of absorbing sets need not be absorbing, the cell in row "Arbitrary intersections (of at least 1 set)" and column "Absorbing" is colored red. If a cell is not colored then that information has yet to be filled in.
16754:
25313:(indicated by the column name, "convex" for instance) is preserved under the set operator (indicated by the row's name, "closure" for instance). If in every TVS, a property is preserved under the indicated set operator then that cell will be colored green; otherwise, it will be colored red.
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Every relatively compact set is totally bounded and the closure of a totally bounded set is totally bounded. The image of a totally bounded set under a uniformly continuous map (such as a continuous linear map for instance) is totally bounded. If
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Depending on the application additional constraints are usually enforced on the topological structure of the space. In fact, several principal results in functional analysis fail to hold in general for topological vector spaces: the
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The closure of a vector subspace of a TVS is a vector subspace. Every finite dimensional vector subspace of a
Hausdorff TVS is closed. The sum of a closed vector subspace and a finite-dimensional vector subspace is closed. If
14248:. This may not be the only natural topology on the dual space; for instance, the dual of a normed space has a natural norm defined on it. However, it is very important in applications because of its compactness properties (see
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13520:: these are complete locally convex spaces where the topology comes from a translation-invariant metric, or equivalently: from a countable family of seminorms. Many interesting spaces of functions fall into this class --
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16620:{\displaystyle \operatorname {Int} _{X}S~=~\operatorname {Int} _{X}\left(\operatorname {cl} _{X}S\right)~{\text{ and }}~\operatorname {cl} _{X}S~=~\operatorname {cl} _{X}\left(\operatorname {Int} _{X}S\right)}
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In general, the set of all balanced and absorbing subsets of a vector space does not satisfy the conditions of this theorem and does not form a neighborhood basis at the origin for any vector topology.
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A family of neighborhoods of the origin with the above two properties determines uniquely a topological vector space. The system of neighborhoods of any other point in the vector space is obtained by
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it can be shown that a space is locally convex if and only if its topology can be defined by a family of seminorms. Local convexity is the minimum requirement for "geometrical" arguments like the
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21479:
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8633:). But if the TVS is not Hausdorff then there exist compact subsets that are not closed. However, the closure of a compact subset of a non-Hausdorff TVS is again compact (so compact subsets are
32727:
has the trivial topology, so does each of its subsets, which makes them all compact. It is known that a subset of any uniform space is compact if and only if it is complete and totally bounded.
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The closure of any convex (respectively, any balanced, any absorbing) subset of any TVS has this same property. In particular, the closure of any convex, balanced, and absorbing subset is a
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22071:{\displaystyle \operatorname {cl} _{X}(R)+\operatorname {cl} _{X}(S)~\subseteq ~\operatorname {cl} _{X}(R+S)~{\text{ and }}~\operatorname {cl} _{X}\left~=~\operatorname {cl} _{X}(R+S)}
16925:
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3541:{\displaystyle \mathbb {S} (u):=\left\{n_{\bullet }=\left(n_{1},\ldots ,n_{k}\right)~:~k\geq 1,n_{i}\geq 0{\text{ for all }}i,{\text{ and }}u\in U_{n_{1}}+\cdots +U_{n_{k}}\right\}.}
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24400:{\displaystyle a(R+S)=aR+aS,~{\text{ and }}~\operatorname {co} (R+S)=\operatorname {co} R+\operatorname {co} S,~{\text{ and }}~\operatorname {co} (aS)=a\operatorname {co} S.}
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and every
Hausdorff TVS has a Hausdorff completion. Every TVS (even those that are Hausdorff and/or complete) has infinitely many non-isomorphic non-Hausdorff completions.
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Every Cauchy sequence is bounded, although Cauchy nets and Cauchy filters may not be bounded. A topological vector space where every Cauchy sequence converges is called
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13828:: Banach spaces naturally isomorphic to their double dual (see below), which ensures that some geometrical arguments can be carried out. An important example which is
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is a real vector space (where addition and scalar multiplication are defined pointwise, as usual) that can be identified with (and indeed, is often defined to be) the
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if its topology can be induced by a norm. A topological vector space is normable if and only if it is
Hausdorff and has a convex bounded neighborhood of the origin.
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with respect to the operation of addition, and in a topological vector space the inverse operation is always continuous (since it is the same as multiplication by
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This shows, in particular, that it will often suffice to consider nets indexed by a neighborhood basis of the origin rather than nets on arbitrary directed sets.
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23053:{\displaystyle \operatorname {cl} _{X}(\operatorname {co} (R+S))~=~\operatorname {cl} _{X}(\operatorname {co} R)+\operatorname {cl} _{X}(\operatorname {co} S).}
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8617:, etc., which are always assumed to be with respect to this uniformity (unless indicated other). This implies that every Hausdorff topological vector space is
13927:; even though these spaces may be infinite-dimensional, most geometrical reasoning familiar from finite dimensions can be carried out in them. These include
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35081:
25059:
22225:
15626:{\displaystyle \operatorname {Int} _{X}K~\subseteq ~\{x\in X:p(x)<1\}~\subseteq ~K~\subseteq ~\{x\in X:p(x)\leq 1\}~\subseteq ~\operatorname {cl} _{X}K}
10181:; indeed, every neighborhood of the origin in the product topology contains lines (that is, 1-dimensional vector subspaces, which are subsets of the form
9325:
always contains a TVS topology. Any vector space (including those that are infinite dimensional) endowed with the trivial topology is a compact (and thus
783:. The topological and linear algebraic structures can be tied together even more closely with additional assumptions, the most common of which are listed
456:
There are topological vector spaces whose topology is not induced by a norm, but are still of interest in analysis. Examples of such spaces are spaces of
35631:
14973:
The intersection of any family of convex sets is convex and the convex hull of a subset is equal to the intersection of all convex sets that contain it.
12947:
consist of the trivial topology, the
Hausdorff Euclidean topology, and then the infinitely many remaining non-trivial non-Euclidean vector topologies on
13097:
A linear operator between two topological vector spaces which is continuous at one point is continuous on the whole domain. Moreover, a linear operator
35484:
31951:
31209:
21732:
18313:
16749:{\displaystyle \operatorname {Int} _{X}(R)+\operatorname {Int} _{X}(S)~\subseteq ~R+\operatorname {Int} _{X}S\subseteq \operatorname {Int} _{X}(R+S).}
35039:
21302:
The sum of a compact set and a closed set is closed. However, the sum of two closed subsets may fail to be closed (see this footnote for examples).
15188:
5724:
Defining vector topologies using collections of strings is particularly useful for defining classes of TVSs that are not necessarily locally convex.
35073:
31102:
10998:
and since it plays an important role in the definition of topological vector spaces, this dichotomy plays an important role in the definition of an
7769:
is bounded if and only if every countable subset of it is bounded. A set is bounded if and only if each of its subsequences is a bounded set. Also,
17519:
35320:
18587:
18497:
34480:. Grundlehren der mathematischen Wissenschaften. Vol. 159. Translated by Garling, D.J.H. New York: Springer Science & Business Media.
32884:{\displaystyle s+\operatorname {cl} _{X}\{0\}=\operatorname {cl} _{X}(s+\{0\})=\operatorname {cl} _{X}\{s\}\subseteq \operatorname {cl} _{X}S.}
9305:
and it is the coarsest TVS topology possible. An important consequence of this is that the intersection of any collection of TVS topologies on
5019:
36141:
11439:
35147:
19325:
6887:
3831:{\displaystyle f(x):=\inf _{}\left\{2^{-n_{1}}+\cdots 2^{-n_{k}}~:~n_{\bullet }=\left(n_{1},\ldots ,n_{k}\right)\in \mathbb {S} (x)\right\}.}
22107:
21597:
21484:
14348:
4518:
24150:{\displaystyle \operatorname {bal} (\operatorname {co} S)~\subseteq ~\operatorname {cobal} S~=~\operatorname {co} (\operatorname {bal} S),}
20797:
16139:
480:. An infinite-dimensional Montel space is never normable. The existence of a norm for a given topological vector space is characterized by
24533:
24477:
18042:
13684:
4944:
4590:
4527:
8896:
but in general, it is almost never uniformly continuous. Because of this, every topological vector space can be completed and is thus a
35758:
35733:
35310:
35119:
22812:
13278:
24932:
32552:
30177: – topological group for which the underlying topology is locally compact and Hausdorff, so that the Haar measure can be defined
22758:
14339:
A topological vector space has a non-trivial continuous dual space if and only if it has a proper convex neighborhood of the origin.
3124:
2262:
17:
22708:
13697:: these are locally convex spaces with the property that every bounded map from the nuclear space to an arbitrary Banach space is a
35715:
35437:
35292:
23140:
23095:
20646:
8371:
if and only if it has a countable neighborhood basis at the origin, or equivalent, if and only if its topology is generated by an
36183:
35685:
35624:
35268:
30192:
24030:
13353:
9631:
9339:
1250:
87:
20396:
16793:
13561:
5408:
35928:
35752:
30063:
25535:
25281:
The convex hull of a balanced (resp. open) set is balanced (respectively, open). However, the convex hull of a closed set need
19106:
13508:
7673:
20163:
15792:
10853:
in this case) is
Hausdorff. The trivial topology on a vector space is Hausdorff if and only if the vector space has dimension
8651:
35089:
35055:
35021:
34991:
34934:
34882:
34848:
34821:
34781:
34677:
34647:
34620:
34590:
34556:
34485:
34455:
34428:
33961:
10184:
481:
18161:
36334:
34548:
22609:
20018:
19841:
11063:
30895:
30536:
25274:
of two compact (respectively, bounded, balanced, convex) sets has that same property. But the sum of two closed sets need
24748:
20601:
19386:
and this set is compact. Thus the closure of a compact subset of a TVS is compact (said differently, all compact sets are
5690:{\textstyle \operatorname {Knots} \mathbb {S} :=\bigcup _{U_{\bullet }\in \mathbb {S} }\operatorname {Knots} U_{\bullet }}
4752:
36193:
35690:
35660:
32600:
32212:
30581:
21441:
18713:
16852:
16356:
is a neighborhood of the origin. This is no longer guaranteed if the set is not convex (a counter-example exists even in
11349:
10174:
9638:
9333:
8352:
6193:
1254:
32092:
9875:
5966:
5534:
5494:
587:
209:
36344:
36313:
35964:
35617:
35160:
34912:
34520:
30484:
In general topology, the closure of a compact subset of a non-Hausdorff space may fail to be compact (for example, the
23366:
if and only if its closure is a neighborhood of the origin. A vector subspace of a TVS that is closed but not open is
461:
33066:
20220:
19538:
861:
634:
307:
36101:
35249:
35140:
34964:
34743:
20714:
20575:
17375:
14087:
with the topology induced by the standard inner product. As pointed out in the preceding section, for a given finite
10162:
9330:
8912:
8904:
8606:
8516:
8509:
3081:
Summative sequences of sets have the particularly nice property that they define non-negative continuous real-valued
1258:
79:
31063:
30651:
30491:
19738:
17884:
16989:
16930:
12005:
9806:
9680:
3223:
2614:
36006:
35519:
34707:
30198:
23515:
21720:{\displaystyle \left(\operatorname {cl} A\right)\left(\operatorname {cl} _{X}S\right)=\operatorname {cl} _{X}(AS).}
20275:
19255:
18816:
17417:
16100:
14748:
14211:
A topology on the dual can be defined to be the coarsest topology such that the dual pairing each point evaluation
9765:
8630:
824:
34735:
32987:
32691:
20514:
20337:
20104:
19621:
19164:
19065:
19022:
18979:
8918:
A compact subset of a TVS (not necessarily
Hausdorff) is complete. A complete subset of a Hausdorff TVS is closed.
5459:
90:. This article focuses on TVSs that are not necessarily locally convex. Other well-known examples of TVSs include
35164:
34929:. Grundlehren der mathematischen Wissenschaften. Vol. 237. New York: Springer Science & Business Media.
23846:
11910:
6157:
1072:
1030:
32345:
31915:{\displaystyle x={\tfrac {1}{s}}y\in \operatorname {cl} \left({\tfrac {1}{s}}C\right)\cap \operatorname {int} C}
31806:
31774:
31289:
14546:
14275:
3085:
functions. These functions can then be used to prove many of the basic properties of topological vector spaces.
36036:
24688:
16897:
14244:
is continuous. This turns the dual into a locally convex topological vector space. This topology is called the
13523:
11876:
672:
195:
113:
acting on topological vector spaces, and the topology is often defined so as to capture a particular notion of
33865:
31573:
22755:
The closed balanced hull of a set is equal to the closure of the balanced hull of that set; that is, equal to
21794:
20764:
19785:
14840:
14513:
11782:
9536:
36168:
35770:
35747:
35315:
34769:
34582:
33875:
31764:{\displaystyle \operatorname {cl} _{X}\left({\tfrac {1}{s}}C\right)={\tfrac {1}{s}}\operatorname {cl} _{X}C.}
31357:
30183:
14892:
14637:
12226:
11577:
11249:
8069:
30406:
14609:
9082:
347:
36219:
35598:
35371:
35305:
35133:
34900:
32271:
31925:
22701:
In a locally convex space, convex hulls of bounded sets are bounded. This is not true for TVSs in general.
17086:
14581:
13466:
12547:
8229:
32377:
23889:
22705:
The closed convex hull of a set is equal to the closure of the convex hull of that set; that is, equal to
17048:
14669:
13759:
7376:
6043:
4795:
3986:
3863:
1304:
36339:
36040:
35335:
33870:
30049:
25521:
18233:
15766:
2085:
2055:
1991:
1337:
24442:
17145:
16359:
14214:
12132:
9256:
7623:
then it also has a neighborhood basis consisting of closed convex balanced neighborhoods of the origin.
1674:
is additive. This statement remains true if the word "neighborhood" is replaced by "open neighborhood."
114:
62:
with the property that the vector space operations (vector addition and scalar multiplication) are also
36276:
35813:
35728:
35723:
35665:
35580:
35534:
35458:
35340:
34983:
30835:
30779:
30727:
30250:
18032:{\displaystyle [x,y):=\{tx+(1-t)y:0<t\leq 1\}\subseteq \operatorname {Int} _{X}{\text{ if }}x\neq y}
16248:
15925:
14249:
13476:
12750:
that are induced by linear functionals with distinct kernels will induce distinct vector topologies on
12512:
10581:
9150:
796:
473:
24164:
22551:{\displaystyle \operatorname {cl} _{X}U~\supseteq ~\bigcap \{U:S\subseteq U,U{\text{ is open in }}X\}}
20450:
14061:
14032:
12655:
10613:
10526:
10473:
8779:
8452:
6581:
1751:
1571:
256:
36072:
35882:
35575:
35391:
34907:. Addison-Wesley series in mathematics. Vol. 1. Reading, MA: Addison-Wesley Publishing Company.
34547:. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY:
30485:
30381:
In fact, this is true for topological group, since the proof does not use the scalar multiplications.
30203:
23607:
21086:
11663:
6327:
3632:
2949:
2115:
1367:
32507:
25293:
The following table, the color of each cell indicates whether or not a given property of subsets of
24410:
24196:
23939:
23063:
22863:
22561:
22434:
21830:
16422:
16224:
15995:
12404:
7832:
7500:
7426:
5600:
4438:
4172:
2153:
2031:
1959:
1888:
1832:
1539:
1276:
35845:
35840:
35833:
35828:
35427:
35325:
33553:
30954:
30867:
22329:
22196:
19400:
18561:
18288:
17223:
16042:
14189:
12798:
12730:
and seminorms with distinct kernels induce distinct topologies so that in particular, seminorms on
12591:
12380:
12254:
12191:
12077:
11542:
11198:
10644:
10139:
9849:
7720:
7304:
6858:
6272:
6074:
5700:
2546:
2485:
1678:
All of the above conditions are consequently a necessity for a topology to form a vector topology.
983:
905:
155:
33117:
32472:
32180:
32021:
31428:
30998:
30932:
23977:
23793:
23581:
21551:
21308:
19660:
19492:
19299:
17858:
17611:
17325:{\displaystyle t\operatorname {Int} C+(1-t)\operatorname {cl} C~\subseteq ~\operatorname {Int} C.}
16986:
to also be convex (in addition to being balanced and having non-empty interior).; The conclusion
15113:
15001:
14356:
13891:
13431:
11618:
11500:
11407:
11223:
11041:
11019:
10979:
10937:
10724:
10399:
10353:
10331:
10309:
10287:
9716:
8426:
8393:
6797:
6103:
5994:
5924:
5815:
5773:
5574:
5386:
5348:
5303:
5290:{\displaystyle \ U_{\bullet }\cap V_{\bullet }:=\left(U_{i}\cap V_{i}\right)_{i\in \mathbb {N} }.}
4725:
4696:
4391:
3097:
2849:
2784:
2576:
2386:
2351:
939:
911:
802:
553:
394:
130:
36106:
36087:
35763:
35743:
35524:
35300:
34808:
31177:
23686:
23319:{\displaystyle \operatorname {cl} _{X}(\operatorname {co} (R\cup S))~=~\operatorname {co} \left.}
14991:
13864:
13676:
13480:
13369:
12892:
12278:
11737:
10278:
7007:
6713:
6624:
6519:
3314:
1689:
1639:
31020:
24872:
23729:
19132:
In fact, a vector subspace of a TVS is bounded if and only if it is contained in the closure of
15958:
11539:
will be translation invariant and invariant under non-zero scalar multiplication, and for every
11297:
7333:
7254:
5841:
4098:
3190:
36349:
36295:
36285:
36269:
35969:
35918:
35818:
35803:
35555:
35499:
35463:
34866:
17800:
10692:
10421:
9603:
9509:
9388:
9349:
8645:
8523:
8360:
8322:
8192:
8079:
6125:
5741:
4239:
1645:
1615:
1509:
508:
436:
is (jointly) continuous. This follows from the triangle inequality and homogeneity of the norm.
106:
32060:
25208:) rather than not just a subset or else these equalities are no longer guaranteed; similarly,
15893:
15419:
13986:
13402:
13035:
a TVS topology because despite making addition and negation continuous (which makes it into a
12837:
10878:
10784:
10074:
8851:
3928:
3288:
2998:
998:
36264:
35951:
35933:
35898:
35738:
35262:
31657:
31631:
30241:
30174:
30169:
26458:
26265:
25808:
25600:
25319:
25033:
24904:
24720:
23359:
22806:
21216:
20758:
20710:
20269:
19730:
16763:
16071:
15393:
14814:
14414:
14183:
14154:
14144:
13494:
13328:
11132:
10816:
10270:
10238:
9843:
8881:
8825:
8601:
Every TVS is assumed to be endowed with this canonical uniformity, which makes all TVSs into
7465:
7459:
7205:
7150:
6661:
4929:{\displaystyle \ \operatorname {Knots} U_{\bullet }:=\left\{U_{i}:i\in \mathbb {N} \right\}.}
4275:
4032:
3553:
3075:
2923:
2738:
1197:
935:
121:
35258:
33364:
32957:{\displaystyle S\subseteq S+\operatorname {cl} _{X}\{0\}\subseteq \operatorname {cl} _{X}S,}
32739:
25316:
So for instance, since the union of two absorbing sets is again absorbing, the cell in row "
25231:
25165:
24694:
19135:
17743:{\textstyle \operatorname {Int} _{X}N\subseteq B_{1}N=\bigcup _{0<|a|<1}aN\subseteq N}
15298:
13284:
Below are some common topological vector spaces, roughly in order of increasing "niceness."
11038:) is straightforward so only an outline with the important observations is given. As usual,
11016:
The proof of this dichotomy (i.e. that a vector topology is either trivial or isomorphic to
9956:
8275:
7179:
6957:
6832:
6771:
6687:
6555:
6421:
3597:
2745:
1725:
36280:
36224:
36203:
35538:
35009:
34495:
33332:
31325:
30321:
28613:
28457:
26682:
26662:
26508:
26488:
26067:
26047:
25858:
25838:
23971:
20075:
19201:
18767:
18664:
18487:{\displaystyle x\in \operatorname {Int} N{\text{ and }}r:=\sup\{r>0:[0,r)x\subseteq N\}}
17753:
16759:
15703:
15450:
15322:
14140:
13957:
13930:
13836:
13732:
13375:
13365:
13299:
13274:
12346:
9982:
8559:
8115:
7532:
5877:
4324:
4145:
4067:
2515:
2435:
1466:
1430:
457:
35125:
24843:
24661:
15723:
13120:
10110:
7596:. The origin even has a neighborhood basis consisting of closed balanced neighborhoods of
6742:
5188:{\displaystyle \ U_{\bullet }+V_{\bullet }:=\left(U_{i}+V_{i}\right)_{i\in \mathbb {N} }.}
8:
36163:
36158:
36116:
35695:
35504:
35442:
35156:
34959:. Reading, Mass.–London–Don Mills, Ont.: Addison-Wesley Publishing Co., Inc.
34773:
31563:{\displaystyle s~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~{\tfrac {r}{r-1}}<0}
30154:
27946:
27795:
26993:
26845:
23394:
22081:
20551:
19944:
15139:
15067:
14314:
13726:
11003:
10006:
9032:
8610:
6983:
6449:
6373:
6301:
5594:
3960:
1460:
298:
191:
75:
63:
51:
32659:
30696:
30488:
on an infinite set). This result shows that this does not happen in non-Hausdorff TVSs.
25379:
25357:
24617:
24594:
24007:
22809:
hull of a set is equal to the closure of the disked hull of that set; that is, equal to
22674:
22458:
22283:
21282:
21126:
21043:
19992:
19686:
19112:
18862:
18793:
18690:
17496:
16281:
16019:
15743:
15347:
14956:
14721:
14466:
14382:
14160:
14090:
13794:
13070:
12869:
12753:
12489:
12323:
11853:
11694:
10856:
10650:
10052:
9446:
9415:
9183:
9057:
8736:
8579:
8481:
8298:
8168:
8040:
7977:
7928:
7599:
6477:
6211:
6020:
5325:
4835:
4494:
2876:
2679:
2413:
2328:
2205:
1936:
1486:
1143:
1116:
698:
416:
86:(although this article does not). One of the most widely studied categories of TVSs are
36148:
36091:
36025:
36010:
35877:
35867:
35529:
35396:
34871:
34813:
34574:
33143:
33046:
33026:
32967:
32639:
32452:
32251:
31481:
30815:
30759:
30631:
30458:
30347:
30301:
30272:
30227:
30086:
30039:
29928:
29725:
27788:
27617:
25558:
25511:
25402:
25296:
25211:
25191:
24228:
23773:
23666:
23642:
23495:
23467:
23447:
23424:
23400:
23376:
22915:
22895:
22654:
22589:
22306:
22176:
21577:
21421:
21334:
21262:
21242:
21196:
21176:
21156:
21136:
21066:
21023:
21003:
20983:
20963:
20940:
20920:
20900:
20880:
20860:
20740:
20720:
20581:
20557:
20494:
20376:
20317:
20143:
20084:
19972:
19924:
19904:
19821:
19712:
19597:
19518:
19387:
19235:
19207:
18959:
18644:
18268:
18141:
18121:
18097:
17838:
17780:
17637:
17476:
17456:
17355:
17335:
17203:
17028:
16969:
16832:
16769:
16454:
16394:
16339:
16307:
16204:
15873:
15853:
15680:
15660:
15640:
15456:
15399:
15372:
15327:
15168:
15093:
15047:
15027:
14742:
14714:
14695:
14493:
14446:
14426:
14255:
14113:
13729:. Most of functional analysis is formulated for Banach spaces. This class includes the
13712:
13472:
13239:
13219:
13199:
13176:
13149:
13100:
13046:
13010:
12986:
12950:
12930:
12778:
12733:
12713:
12469:
12442:
12171:
12112:
12053:
11985:
11965:
11945:
11833:
11813:
11717:
11522:
11329:
11277:
11178:
11158:
10959:
10928:
10910:
10756:
10672:
10555:
10506:
10500:
10453:
10379:
10373:
10032:
10012:
9936:
9742:
9736:
9660:
9579:
9489:
9469:
9308:
9288:
9219:
9130:
9037:
9011:
9005:
8987:
8964:
8944:
8924:
8759:
8716:
8634:
8614:
8446:
8255:
8235:
8162:
8141:
8020:
8000:
7957:
7908:
7888:
7864:
7812:
7792:
7772:
7752:
7700:
7680:
7654:
7634:
7575:
7472:
7234:
7130:
7101:
7081:
7053:
7033:
7013:
6401:
6353:
6333:
6249:
6236:
6014:
5946:
5904:
5835:
5795:
4673:
4653:
4418:
4371:
4351:
4304:
4219:
3843:
3057:
3036:
2929:
2902:
2829:
2463:
2242:
2185:
1916:
1868:
1812:
1792:
1406:
1196:
topological homomorphism. Equivalently, a TVS-embedding is a linear map that is also a
739:
526:
34922:
34473:
25155:{\displaystyle s\{x\in X:P(x)\}=\left\{x\in X:P\left({\tfrac {1}{s}}x\right)\right\}.}
19200:
also carries the trivial topology and so is itself a compact, and thus also complete,
8885:; in general, it may not be complete (in the sense that all Cauchy filters converge).
6504:
One of the most used properties of vector topologies is that every vector topology is
297:
is (jointly) continuous with respect to this topology. This follows directly from the
35860:
35786:
35509:
35095:
35085:
35061:
35051:
35027:
35017:
34997:
34987:
34960:
34940:
34930:
34908:
34888:
34878:
34854:
34844:
34827:
34817:
34803:
34787:
34777:
34749:
34739:
34683:
34673:
34653:
34643:
34626:
34616:
34596:
34586:
34562:
34552:
34542:
34526:
34516:
34499:
34481:
34461:
34451:
34434:
34424:
30221:
30212:
30078:
25550:
19391:
19016:
15185:(not necessarily Hausdorff or locally convex) are exactly those that are of the form
14995:
14148:
13815:
13194:
13040:
13036:
13004:
9930:
9760:
9650:
8893:
8641:
6350:
is closed. This permits the following construction: given a topological vector space
6267:
6232:
1910:
1137:
721:
581:
548:
488:
59:
6658:
Scalar multiplication by a non-zero scalar is a TVS-isomorphism. This means that if
36253:
35823:
35808:
35609:
35514:
35432:
35401:
35381:
35366:
35361:
35356:
34727:
34585:. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer.
30232:
30073:
25545:
19059:
16766:
is not empty if and only if this interior contains the origin. More generally, if
13708:
13698:
13486:
13257:
13028:
10846:
10640:
9800:
9654:
9336:
9246:
8222:
1533:
676:
36136:
35675:
35193:
34877:. Translated by Chaljub, Orlando. New York: Gordon and Breach Science Publishers.
13680:
13517:
10273:, a Hausdorff topological vector space is finite-dimensional if and only if it is
503:
36228:
36076:
35376:
35330:
35278:
35273:
35244:
34665:
34491:
34420:
32011:{\displaystyle c_{0}\in \left({\tfrac {1}{s}}C\right)\cap \operatorname {int} C,}
31060:
This is because every non-empty balanced set must contain the origin and because
30111:
30068:
30022:
25583:
25540:
25494:
25264:
23567:
23338:
16329:
16325:
14987:
14026:
13825:
13490:
13462:
10574:
10274:
10166:
9343:
9326:
8900:
8626:
8618:
8420:
8416:
8108:
is a topological vector space then the following four conditions are equivalent:
7951:
2732:
780:
768:
764:
492:
83:
35203:
31279:{\displaystyle w_{0}~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~rx+(1-r)y}
21784:{\displaystyle S\subseteq X{\text{ and }}S+S\subseteq 2\operatorname {cl} _{X}S}
19482:{\displaystyle S+\operatorname {cl} _{X}\{0\}\subseteq \operatorname {cl} _{X}S}
18399:{\displaystyle \operatorname {Int} N=[0,1)\operatorname {Int} N=(-1,1)N=B_{1}N,}
13719:. In normed spaces a linear operator is continuous if and only if it is bounded.
36259:
36208:
35923:
35565:
35417:
35218:
34799:
34761:
34608:
33322:{\displaystyle z+\{x\in X:P(x)\}=\{z+x:x\in X,P(x)\}=\{z+x:x\in X,P((z+x)-z)\}}
30139:
30116:
30044:
25588:
25516:
21411:{\displaystyle a\operatorname {cl} _{X}S\subseteq \operatorname {cl} _{X}(aS),}
18949:{\displaystyle \operatorname {cl} _{X}\{0\}=\bigcap _{N\in {\mathcal {N}}(0)}N}
13292:
13281:, and the fact that the dual space of the space separates points in the space.
12707:
12704:
10170:
8413:
8372:
8368:
7620:
5812:
that is directed downward and such that the set of all knots of all strings in
1686:
Since every vector topology is translation invariant (which means that for all
1262:
1243:
990:
577:
465:
202:
and the metric induces a topology. This is a topological vector space because:
125:
35099:
31159:{\displaystyle \operatorname {Int} _{X}S=\{0\}\cup \operatorname {Int} _{X}S.}
22424:{\displaystyle \operatorname {cl} _{X}S~=~\bigcap _{N\in {\mathcal {N}}}(S+N)}
20794:
is totally bounded, if and only if its image under the canonical quotient map
13493:, where the dual space is endowed with the topology of uniform convergence on
11346:
contains an "unbounded sequence", by which it is meant a sequence of the form
36328:
36243:
36153:
36096:
36056:
35984:
35959:
35903:
35855:
35791:
35570:
35494:
35223:
35208:
35198:
35065:
35031:
34944:
34738:. Translated by Eggleston, H.G.; Madan, S. Berlin New York: Springer-Verlag.
34687:
34630:
34600:
34530:
34515:. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press.
34503:
34438:
30890:
30148:
30010:
29967:
29952:
28282:
28111:
28104:
27953:
25482:
25439:
25424:
25271:
24158:
24001:
23489:
23367:
23363:
21548:
In particular, every non-zero scalar multiple of a closed set is closed. If
19105:(even if its dimension is non-zero or even infinite) and consequently also a
19102:
18262:
17599:{\displaystyle \{tx+(1-t)y:0<t<1\}\subseteq \operatorname {int} _{X}C.}
16333:
15390:
14245:
13981:
13924:
13920:
13694:
13688:
13295:
topological vector spaces with a translation-invariant metric. These include
12371:
10999:
8622:
8602:
8553:
7565:
7493:
7123:
6618:
6205:
4092:
2920:
2726:
2708:
2023:
1862:
1788:
1239:
1193:
469:
445:
99:
95:
71:
34858:
34831:
34791:
34753:
34657:
34566:
33541:{\displaystyle \{y:y-z\in X,P(y-z)\}=\{y:y\in X,P(y-z)\}=\{y\in X:P(y-z)\}.}
23341:) set has that same property. The convex hull of a finite union of compact
6474:
is then a
Hausdorff topological vector space that can be studied instead of
36290:
36238:
36198:
36188:
36066:
35913:
35908:
35705:
35655:
35560:
35213:
35183:
34538:
34465:
30160:
30133:
29957:
25429:
20013:
18115:
16787:
14410:
13722:
13704:
13500:
8629:(for Hausdorff TVSs, a set being totally bounded is equivalent to it being
8378:. A TVS is metrizable if and only if it is Hausdorff and pseudometrizable.
7593:
7569:
7293:
6240:
6196:
if and only if its topology can be induced by a single topological string.
2702:
2019:
1858:
1186:
1066:
733:
544:
477:
441:
199:
91:
55:
35001:
34892:
18627:{\displaystyle rx\in \operatorname {cl} N\setminus \operatorname {Int} N.}
18551:{\displaystyle r>1{\text{ and }}[0,r)x\subseteq \operatorname {Int} N,}
14311:
is never continuous, no matter which vector space topology one chooses on
36248:
36233:
36126:
36020:
36015:
36000:
35979:
35943:
35850:
35670:
35489:
35479:
35386:
35188:
30056:
30029:
29258:
29097:
28450:
28289:
25528:
25501:
23418:
14664:
14487:
14020:
13512:
13063:(where a subset is open if and only if its complement is finite) is also
9597:
6826:
which is consequently a linear homeomorphism and thus a TVS-isomorphism.
5086:{\displaystyle \ sU_{\bullet }:=\left(sU_{i}\right)_{i\in \mathbb {N} }.}
3082:
1981:
573:
150:
31:
13711:: locally convex spaces where the topology can be described by a single
13039:
under addition), it fails to make scalar multiplication continuous. The
13003:
is a non-trivial vector space (that is, of non-zero dimension) then the
11490:{\displaystyle \left(a_{i}\right)_{i=1}^{\infty }\subseteq \mathbb {K} }
1681:
36061:
35974:
35938:
35798:
35680:
35422:
35254:
34952:
34040:
34038:
33932:
33930:
33928:
33824:
33822:
33820:
33818:
33816:
30215: – Algebraic structure with addition, multiplication, and division
30106:
29962:
25578:
25434:
24649:
19615:
19379:{\displaystyle \operatorname {cl} _{X}S=S+\operatorname {cl} _{X}\{0\}}
16927:
will be balanced if and only if it contains the origin. For this (i.e.
14403:
13504:
13361:
13357:
13261:
13170:
10639:
is endowed with the usual
Euclidean topology (which is the same as the
7369:
6947:{\displaystyle \operatorname {cl} _{X}(x+S)=x+\operatorname {cl} _{X}S}
1236:
993:
110:
33750:
33748:
33746:
33744:
33742:
33740:
33694:
33692:
33690:
33688:
33686:
33684:
33682:
33680:
33678:
33676:
30224: – Group that is a topological space with continuous group action
22558:
and it is possible for this containment to be proper (for example, if
22164:{\displaystyle \operatorname {cl} _{X}(R)+\operatorname {cl} _{X}(S).}
21627:{\displaystyle \operatorname {cl} S{\text{ nor }}\operatorname {cl} A}
21541:{\displaystyle \operatorname {cl} _{X}(aS)=a\operatorname {cl} _{X}S.}
8888:
The vector space operation of addition is uniformly continuous and an
36213:
36030:
34299:
34297:
34197:
34195:
34193:
34191:
34189:
34187:
33674:
33672:
33670:
33668:
33666:
33664:
33662:
33660:
33658:
33656:
26838:
Arbitrary intersections (of at least 1 set)
26689:
Arbitrary intersections (of at least 1 set)
23563:
23485:
22273:{\displaystyle \bigcap _{r>1}rS\subseteq \operatorname {cl} _{X}S}
18636:
16194:{\displaystyle s_{\bullet }=\left(s_{N}\right)_{N\in {\mathcal {N}}}}
8897:
8159:
2873:
is a string and each of its knots is a neighborhood of the origin in
1854:
124:
field of a topological vector space will be assumed to be either the
34338:
34270:
34079:
34035:
33925:
33813:
24582:{\displaystyle R\cap \operatorname {co} (S\cup \{x\})=\varnothing .}
19252:; for instance, this will be true of any non-empty proper subset of
12466:
Some of these topologies are now described: Every linear functional
7592:
so every topological vector space has a local base of absorbing and
5004:{\textstyle \ \ker U_{\bullet }:=\bigcap _{i\in \mathbb {N} }U_{i}.}
36178:
36173:
36131:
36111:
36081:
35872:
35047:
34369:
34367:
34365:
34135:
34133:
33984:
33982:
33834:
33737:
30365:
24523:{\displaystyle S\cap \operatorname {co} (R\cup \{x\})=\varnothing }
24161:
of a convex set need not be convex (counter-examples exist even in
13833:
13819:
13672:
13296:
12542:
10178:
9926:
8383:
7882:
4643:{\displaystyle V_{\bullet }=\left(V_{i}\right)_{i\in \mathbb {N} }}
4580:{\displaystyle U_{\bullet }=\left(U_{i}\right)_{i\in \mathbb {N} }}
1027:
between topological vector spaces (TVSs) such that the induced map
931:
757:
34417:
Topological Vector Spaces: The Theory
Without Convexity Conditions
34294:
34184:
33653:
30195: – A vector space with a topology defined by convex open sets
30186: – relatively new C*-algebraic approach toward quantum groups
23659:
is convex then equality holds. For an example where equality does
36121:
34230:
34228:
34226:
34224:
34222:
33023:
is a vector space, it is readily verified that the complement in
30142: – algebraic structure that is complete relative to a metric
24614:
there exist two disjoint non-empty convex subsets whose union is
22850:{\displaystyle \operatorname {cl} _{X}(\operatorname {cobal} S).}
13288:
34714:. Singapore-New Jersey-Hong Kong: Universitätsbibliothek: 35–133
34419:. Lecture Notes in Mathematics. Vol. 639. Berlin New York:
34362:
34207:
34130:
34023:
34011:
33979:
33889:
33760:
33715:
33713:
33711:
33709:
33707:
25023:{\displaystyle z+\{x\in X:\|x\|<1\}=\{x\in X:\|x-z\|<1\}.}
19062:, which in particular implies that the topological vector space
9653:
of a family of topological vector spaces, when endowed with the
35114:
34326:
33548:
32590:{\displaystyle h\left(\operatorname {int} C\right)\subseteq C.}
22796:{\displaystyle \operatorname {cl} _{X}(\operatorname {bal} S).}
20709:
A subset of a TVS is compact if and only if it is complete and
15288:{\displaystyle z+\{x\in X:p(x)<1\}~=~\{x\in X:p(x-z)<1\}}
12375:
9657:, is a topological vector space. Consider for instance the set
3180:{\displaystyle U_{\bullet }=\left(U_{i}\right)_{i=0}^{\infty }}
2318:{\displaystyle U_{\bullet }=\left(U_{i}\right)_{i=1}^{\infty }}
580:
numbers with their standard topologies) that is endowed with a
34245:
34243:
34219:
34162:
34160:
33901:
33846:
26258:
Arbitrary unions (of at least 1 set)
26074:
Arbitrary unions (of at least 1 set)
23337:
to be compact. The balanced hull of a compact (respectively,
23333:
In a general TVS, the closed convex hull of a compact set may
22746:{\displaystyle \operatorname {cl} _{X}(\operatorname {co} S).}
12775:
However, while there are infinitely many vector topologies on
12074:
contain any "unbounded sequence". This implies that for every
10773:
does not have a unique vector topology, it does have a unique
8419:
topological field, for example the real or complex numbers. A
8359:
By the
Birkhoff–Kakutani theorem, it follows that there is an
33704:
33612:
23175:{\displaystyle \operatorname {cl} _{X}(\operatorname {co} S)}
23130:{\displaystyle \operatorname {cl} _{X}(\operatorname {co} R)}
14272:
is a non-normable locally convex space, then the pairing map
34316:
34314:
34312:
33624:
31425:
if necessary, we may assume without loss of generality that
25188:
of these sets must range over a vector space (that is, over
22323:
is a convex neighborhood of the origin then equality holds.
20717:, a closed and totally bounded subset is compact. A subset
20690:{\displaystyle X\cong H\times \operatorname {cl} _{X}\{0\}.}
9840:
becomes a topological vector space whose topology is called
2202:(rather than a neighborhood basis) for a vector topology on
799:
of topological vector spaces over a given topological field
35155:
34260:
34258:
34240:
34157:
34067:
33913:
33725:
33643:
33641:
33639:
24061:{\displaystyle \operatorname {co} (\operatorname {bal} S).}
22280:
where the left hand side is independent of the topology on
18956:
so that in particular, every neighborhood of the origin in
13662:{\textstyle \|f\|_{k,\ell }=\sup _{x\in }|f^{(\ell )}(x)|.}
13399:
spaces are locally convex (in fact, Banach spaces) for all
7881:
is locally convex, the boundedness can be characterized by
779:
if it is Hausdorff; importantly, "separated" does not mean
34145:
34001:
33999:
33997:
33791:
33789:
33787:
32730:
20440:{\displaystyle H\times \operatorname {cl} _{X}\{0\}\to X,}
16822:{\displaystyle \operatorname {Int} _{X}S\neq \varnothing }
7789:
is bounded if and only if for every balanced neighborhood
6370:(that is probably not Hausdorff), form the quotient space
5449:{\displaystyle U_{\bullet },V_{\bullet }\in \mathbb {S} ,}
34309:
23092:
each have a closed convex hull that is compact (that is,
14349:
Locally convex topological vector space § Properties
4415:
forms a basis of balanced neighborhoods of the origin in
34350:
34255:
33636:
33583:
25257:
20210:{\displaystyle \{0\}=H\cap \operatorname {cl} _{X}\{0\}}
19224:
is not Hausdorff then there exist subsets that are both
15843:{\displaystyle x_{\bullet }=\left(x_{i}\right)_{i\in I}}
8702:{\displaystyle x_{\bullet }=\left(x_{i}\right)_{i\in I}}
8605:. This allows one to talk about related notions such as
8381:
More strongly: a topological vector space is said to be
4517:
A proof of the above theorem is given in the article on
34812:. Pure and applied mathematics. Vol. 1. New York:
33994:
33942:
33784:
30269:
The topological properties of course also require that
30217:
Pages displaying short descriptions of redirect targets
30165:
Pages displaying short descriptions of redirect targets
19390:), which is not guaranteed for arbitrary non-Hausdorff
11907:
at the origin guarantees the existence of an open ball
10643:). This Hausdorff vector topology is also the (unique)
10228:{\displaystyle \mathbb {R} f:=\{rf:r\in \mathbb {R} \}}
6330:
is a Hausdorff topological vector space if and only if
6231:). Hence, every topological vector space is an abelian
3187:
be a collection of subsets of a vector space such that
1791:), to define a vector topology it suffices to define a
675:(where the domains of these functions are endowed with
34282:
34120:
34118:
34103:
34091:
34057:
34055:
34053:
33772:
32380:
32274:
31974:
31881:
31852:
31731:
31708:
31584:
31535:
31520:
31235:
30533:
is compact because it is the image of the compact set
30409:
25125:
23518:
23488:, which happens if and only if there does not exist a
18223:{\displaystyle B_{1}:=\{a\in \mathbb {K} :|a|<1\},}
17660:
13564:
9211:
9085:
5770:
is a topological vector space then there exists a set
5631:
4947:
3989:
3074:
has countable dimension then every string contains an
34172:
33602:
33600:
33598:
33556:
33402:
33367:
33335:
33174:
33146:
33120:
33069:
33049:
33029:
32990:
32970:
32897:
32768:
32742:
32694:
32662:
32642:
32603:
32555:
32510:
32475:
32455:
32348:
32254:
32215:
32183:
32095:
32063:
32024:
31954:
31928:
31844:
31809:
31777:
31688:
31660:
31634:
31576:
31504:
31484:
31431:
31360:
31328:
31292:
31212:
31180:
31105:
31066:
31023:
31001:
30957:
30935:
30898:
30870:
30838:
30818:
30782:
30762:
30730:
30699:
30654:
30634:
30584:
30539:
30494:
30461:
30350:
30324:
30304:
30275:
30089:
29931:
29728:
26996:
26848:
26665:
26491:
26461:
26268:
26050:
25841:
25811:
25603:
25561:
25405:
25382:
25360:
25322:
25299:
25285:
be closed. And the convex hull of a bounded set need
25234:
25214:
25194:
25168:
25062:
25036:
24935:
24907:
24875:
24846:
24751:
24723:
24697:
24664:
24620:
24597:
24536:
24480:
24445:
24413:
24251:
24231:
24199:
24167:
24074:
24033:
24010:
23980:
23942:
23892:
23849:
23796:
23776:
23732:
23689:
23669:
23645:
23610:
23584:
23498:
23470:
23450:
23427:
23403:
23379:
23188:
23143:
23098:
23066:
22938:
22918:
22898:
22866:
22815:
22761:
22711:
22677:
22657:
22644:{\displaystyle \operatorname {cl} _{X}U\subseteq U+U}
22612:
22592:
22564:
22484:
22461:
22437:
22360:
22332:
22309:
22286:
22228:
22199:
22179:
22110:
22084:
21865:
21833:
21797:
21735:
21640:
21600:
21580:
21554:
21487:
21444:
21424:
21357:
21337:
21311:
21285:
21265:
21245:
21219:
21199:
21179:
21159:
21139:
21089:
21069:
21046:
21026:
21006:
20986:
20966:
20943:
20923:
20903:
20883:
20863:
20846:{\displaystyle X\to X/\operatorname {cl} _{X}(\{0\})}
20800:
20767:
20743:
20723:
20649:
20604:
20584:
20560:
20517:
20497:
20453:
20399:
20379:
20340:
20320:
20278:
20223:
20166:
20146:
20107:
20087:
20067:{\displaystyle q:X\to X/\operatorname {cl} _{X}\{0\}}
20021:
19995:
19975:
19947:
19927:
19907:
19890:{\displaystyle X\to X/\operatorname {cl} _{X}(\{0\})}
19844:
19824:
19788:
19741:
19715:
19689:
19663:
19624:
19600:
19541:
19521:
19495:
19431:
19403:
19328:
19302:
19258:
19238:
19210:
19167:
19138:
19115:
19068:
19025:
18982:
18962:
18888:
18865:
18819:
18796:
18770:
18716:
18693:
18667:
18647:
18590:
18564:
18500:
18412:
18316:
18291:
18271:
18236:
18164:
18144:
18124:
18100:
18045:
17929:
17887:
17861:
17841:
17803:
17783:
17756:
17640:
17614:
17522:
17499:
17479:
17459:
17420:
17378:
17358:
17338:
17261:
17226:
17206:
17148:
17089:
17051:
17031:
16992:
16972:
16933:
16900:
16855:
16835:
16796:
16772:
16633:
16477:
16457:
16425:
16397:
16362:
16342:
16310:
16284:
16251:
16227:
16207:
16142:
16103:
16074:
16045:
16022:
15998:
15961:
15928:
15896:
15876:
15856:
15795:
15769:
15746:
15726:
15706:
15683:
15663:
15643:
15479:
15459:
15422:
15402:
15375:
15350:
15330:
15301:
15191:
15171:
15142:
15116:
15096:
15070:
15050:
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11122:{\displaystyle B_{r}:=\{a\in \mathbb {K} :|a|<r\}}
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is assumed have the (normed) Euclidean topology. Let
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The definition of boundedness can be weakened a bit;
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Defining topologies using neighborhoods of the origin
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Every topological vector space is also a commutative
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35014:
Topological Vector Spaces, Distributions and Kernels
34708:"An Introduction to Locally Convex Inductive Limits"
34415:
Adasch, Norbert; Ernst, Bruno; Keim, Dieter (1978).
34391:
30922:{\displaystyle \mathbb {Z} +{\sqrt {2}}\mathbb {Z} }
30628:
Recall also that the sum of a compact set (that is,
30571:{\displaystyle S\times \operatorname {cl} _{X}\{0\}}
30246:
Pages displaying wikidata descriptions as a fallback
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Pages displaying wikidata descriptions as a fallback
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Pages displaying wikidata descriptions as a fallback
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Pages displaying wikidata descriptions as a fallback
24833:{\displaystyle z+\{x\in X:P(x)\}=\{x\in X:P(x-z)\}.}
20636:{\displaystyle C\times \operatorname {cl} _{X}\{0\}}
13146:
is bounded (as defined below) for some neighborhood
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is bounded if and only if every continuous seminorm
4785:{\displaystyle \ U_{\bullet }\subseteq V_{\bullet }}
2234:
34115:
34050:
32984:is closed then equality holds. Using the fact that
32629:{\displaystyle h\left(\operatorname {int} C\right)}
32241:{\displaystyle h\left(\operatorname {int} C\right)}
30621:{\displaystyle \cdot \,+\,\cdot \;:X\times X\to X.}
30163: – Vector space on which a distance is defined
23574:
Important algebraic facts and common misconceptions
21474:{\displaystyle a\neq 0,{\text{ or }}S=\varnothing }
18757:{\displaystyle \{0\}=\operatorname {cl} _{X}\{0\}.}
18087:{\displaystyle [x,x)=\varnothing {\text{ if }}x=y.}
16887:{\displaystyle \{0\}\cup \operatorname {Int} _{X}S}
11397:{\displaystyle \left(a_{i}x\right)_{i=1}^{\infty }}
6654:
then it is not linear and so not a TVS-isomorphism.
35485:Spectral theory of ordinary differential equations
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32170:{\displaystyle x\mapsto rx+(1-r)sc_{0}=rx-rc_{0},}
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18637:Non-Hausdorff spaces and the closure of the origin
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10578:vector topology, which makes it TVS-isomorphic to
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9918:{\displaystyle \left(f_{n}\right)_{n=1}^{\infty }}
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7010:(resp. open neighborhood, closed neighborhood) of
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5984:{\displaystyle \operatorname {Knots} \mathbb {S} }
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5564:{\displaystyle W_{\bullet }\subseteq V_{\bullet }}
5563:
5524:{\displaystyle W_{\bullet }\subseteq U_{\bullet }}
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1289:
1249:Many properties of TVSs that are studied, such as
1155:
1128:
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1057:
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922:
896:
850:
813:
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624:{\displaystyle \cdot \,+\,\cdot \;:X\times X\to X}
623:
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535:
428:
405:
383:
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289:
246:{\displaystyle \cdot \,+\,\cdot \;:X\times X\to X}
245:
169:
141:
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34373:
34344:
34332:
34303:
34276:
34213:
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34139:
34085:
34044:
33988:
33962:"A quick application of the closed graph theorem"
33936:
33828:
33754:
33698:
30235: – ring where ring operations are continuous
13489:: a locally convex space satisfying a variant of
11002:and has consequences that reverberate throughout
8941:is a complete subset of a TVS then any subset of
4650:are two collections of subsets of a vector space
50:) is one of the basic structures investigated in
36326:
34974:Robertson, Alex P.; Robertson, Wendy J. (1980).
33107:{\displaystyle S+\operatorname {cl} _{X}\{0\}=S}
30206: – topological group whose group is abelian
23441:has no balanced absorbing nowhere dense subset.
20261:{\displaystyle X=H+\operatorname {cl} _{X}\{0\}}
19579:{\displaystyle S+\operatorname {cl} _{X}\{0\}=S}
18439:
14417:, closed convex, closed balanced, closed disked'
13591:
10277:, which happens if and only if it has a compact
3685:
897:{\displaystyle \mathrm {TVect} _{\mathbb {K} }.}
664:{\displaystyle \cdot :\mathbb {K} \times X\to X}
337:{\displaystyle \cdot :\mathbb {K} \times X\to X}
82:. Some authors also require that the space is a
34798:
34414:
33766:
33719:
17407:{\displaystyle y\in \operatorname {int} _{X}C,}
17372:(not necessarily Hausdorff or locally convex),
14182:of all continuous linear functionals, that is,
12320:contains an open neighborhood of the origin in
9576:into another TVS is necessarily continuous. If
9466:that is finer than every other TVS-topology on
6122:is the set of all topological strings in a TVS
2178:a filter base then it will form a neighborhood
491:is a topological vector space over each of its
34672:. Mineola, New York: Dover Publications, Inc.
34511:Narici, Lawrence; Beckenstein, Edward (2011).
31092:{\displaystyle 0\in \operatorname {Int} _{X}S}
31017:denotes the set of all topological strings in
30686:{\displaystyle S+\operatorname {cl} _{X}\{0\}}
30526:{\displaystyle S+\operatorname {cl} _{X}\{0\}}
25263:The balanced hull of a compact (respectively,
24157:where the inclusion might be strict since the
23555:{\textstyle X=\bigcup _{n\in \mathbb {N} }nD.}
22892:and the closed convex hull of one of the sets
19773:{\displaystyle S+\operatorname {cl} _{X}\{0\}}
19204:(see footnote for a proof). In particular, if
18230:then by considering intersections of the form
17916:{\displaystyle y\in \operatorname {cl} _{X}S,}
17634:is any balanced neighborhood of the origin in
17018:{\displaystyle 0\in \operatorname {Int} _{X}S}
16959:{\displaystyle 0\in \operatorname {Int} _{X}S}
14976:
12043:{\displaystyle B_{r}\cdot S\subseteq W\neq X,}
9872:The reason for this name is the following: if
9833:{\displaystyle X:=\mathbb {R} ^{\mathbb {R} }}
9706:{\displaystyle f:\mathbb {R} \to \mathbb {R} }
3278:{\displaystyle U_{i+1}+U_{i+1}\subseteq U_{i}}
2669:{\displaystyle U_{i+1}+U_{i+1}\subseteq U_{i}}
1403:Characterization of continuity of addition at
736:), but not this page, require the topology on
35625:
35141:
34573:
34249:
34234:
34166:
34073:
33919:
33907:
33618:
32656:that contains the origin and is contained in
20307:{\displaystyle \operatorname {cl} _{X}\{0\}.}
19287:{\displaystyle \operatorname {cl} _{X}\{0\}.}
18848:{\displaystyle \operatorname {cl} _{X}\{0\},}
17446:{\displaystyle x\in \operatorname {cl} _{X}C}
16129:{\displaystyle x\in \operatorname {cl} _{X}S}
15677:was continuous (which happens if and only if
14804:{\displaystyle t_{1}s_{1}+\cdots +t_{n}s_{n}}
13236:has either dense or closed kernel. Moreover,
11873:then the continuity of scalar multiplication
11519:(in the usual sense). Any vector topology on
9792:{\displaystyle \mathbb {R} ^{\mathbb {R} },,}
9285:is always a TVS topology on any vector space
851:{\displaystyle \mathrm {TVS} _{\mathbb {K} }}
105:Many topological vector spaces are spaces of
34865:
34841:Functional Analysis: Theory and Applications
33630:
33532:
33499:
33493:
33454:
33448:
33403:
33316:
33259:
33253:
33214:
33208:
33181:
33095:
33089:
33016:{\displaystyle \operatorname {cl} _{X}\{0\}}
33010:
33004:
32929:
32923:
32856:
32850:
32828:
32822:
32794:
32788:
32720:{\displaystyle \operatorname {cl} _{X}\{0\}}
32714:
32708:
31131:
31125:
30680:
30674:
30565:
30559:
30520:
30514:
30331:
30325:
30136: – Normed vector space that is complete
25093:
25066:
25014:
25005:
24993:
24978:
24972:
24963:
24957:
24942:
24882:
24876:
24824:
24791:
24785:
24758:
24564:
24558:
24508:
24502:
23754:
23739:
23711:
23696:
22545:
22513:
22455:is any neighborhood basis at the origin for
20837:
20831:
20681:
20675:
20630:
20624:
20543:{\displaystyle \operatorname {cl} _{X}\{0\}}
20537:
20531:
20425:
20419:
20366:{\displaystyle \operatorname {cl} _{X}\{0\}}
20360:
20354:
20298:
20292:
20255:
20249:
20204:
20198:
20173:
20167:
20133:{\displaystyle \operatorname {cl} _{X}\{0\}}
20127:
20121:
20061:
20055:
19881:
19875:
19767:
19761:
19650:{\displaystyle \operatorname {cl} _{X}\{0\}}
19644:
19638:
19567:
19561:
19457:
19451:
19373:
19367:
19278:
19272:
19193:{\displaystyle \operatorname {cl} _{X}\{0\}}
19187:
19181:
19145:
19139:
19094:{\displaystyle \operatorname {cl} _{X}\{0\}}
19088:
19082:
19051:{\displaystyle \operatorname {cl} _{X}\{0\}}
19045:
19039:
19008:{\displaystyle \operatorname {cl} _{X}\{0\}}
19002:
18996:
18908:
18902:
18839:
18833:
18777:
18771:
18748:
18742:
18723:
18717:
18674:
18668:
18481:
18442:
18214:
18178:
17996:
17948:
17571:
17523:
17182:
17149:
17129:
17096:
17083:the interior of the closed and balanced set
16894:will necessarily be balanced; consequently,
16862:
16856:
15595:
15562:
15538:
15505:
15282:
15243:
15231:
15198:
13669:A locally convex F-space is a Fréchet space.
13572:
13565:
13479:to any locally convex space are exactly the
11116:
11080:
10832:
10826:
10689:has a unique vector topology if and only if
10499:(explicitly, this means that there exists a
10264:
10222:
10199:
9272:
9260:
8125:
8119:
6431:
6425:
5484:{\displaystyle W_{\bullet }\in \mathbb {S} }
4491:is a metric defining the vector topology on
34905:Topological Vector Spaces and Distributions
34670:Modern Methods in Topological Vector Spaces
30244: – semigroup with continuous operation
29717:Image under a continuous linear surjection
29598:Image under a continuous linear surjection
25228:must belong to this vector space (that is,
23879:{\displaystyle s>0{\text{ and }}t>0,}
12188:does not carry the trivial topology and if
11935:{\displaystyle B_{r}\subseteq \mathbb {K} }
10976:-dimensional topological vector space over
6185:{\displaystyle \tau _{\mathbb {S} }=\tau .}
5873:Such a collection of strings is said to be
1269:A necessary condition for a vector topology
1106:{\displaystyle \operatorname {Im} u:=u(X),}
1058:{\displaystyle u:X\to \operatorname {Im} u}
771:. A topological vector space is said to be
448:are examples of topological vector spaces.
35632:
35618:
35148:
35134:
35080:. Compact Textbooks in Mathematics. Cham:
35046:. Vol. 67. Amsterdam New York, N.Y.:
32367:{\displaystyle c\in \operatorname {int} C}
31831:{\displaystyle y\in \operatorname {cl} C,}
31796:{\displaystyle x\in \operatorname {int} C}
31315:{\displaystyle \operatorname {int} _{X}C.}
30596:
30475:if the sequence of partial sums converges.
22606:is the rational numbers). It follows that
21193:is a closed neighborhood of the origin in
20699:
14994:and any connected open subset of a TVS is
14572:{\displaystyle \operatorname {Int} _{X}S,}
14304:{\displaystyle X'\times X\to \mathbb {K} }
11850:of the origin that is not equal to all of
11660:is a continuous linear bijection. Because
10931:and the (non-Hausdorff) trivial topology.
6040:In this case, this topology is denoted by
5697:be the set of all knots of all strings in
2174:satisfies the above two conditions but is
602:
224:
34772:. Vol. 96 (2nd ed.). New York:
34702:
34607:
33731:
31003:
30959:
30937:
30915:
30900:
30872:
30841:
30733:
30592:
30588:
30157: – Concept in condensed mathematics
24644:Every TVS topology can be generated by a
24170:
23537:
22572:
18293:
18244:
18188:
17060:
16920:{\displaystyle \operatorname {Int} _{X}S}
16411:is not of the second category in itself.
16371:
16016:be a neighborhood basis of the origin in
14982:Properties of neighborhoods and open sets
14297:
14230:
14194:
14067:
14038:
13888:but is strictly contained in the dual of
13551:{\displaystyle C^{\infty }(\mathbb {R} )}
13541:
12574:
12518:
12259:
12237:
11928:
11900:{\displaystyle \mathbb {K} \times X\to X}
11881:
11793:
11674:
11595:
11505:
11483:
11260:
11203:
11090:
11046:
11024:
10984:
10942:
10619:
10587:
10532:
10479:
10404:
10358:
10336:
10314:
10292:
10218:
10189:
10144:
9854:
9824:
9818:
9777:
9771:
9721:
9699:
9691:
8458:
8431:
8398:
6167:
6108:
6079:
6053:
5999:
5977:
5929:
5820:
5778:
5705:
5665:
5639:
5608:
5604:
5579:
5477:
5439:
5391:
5353:
5308:
5278:
5176:
5074:
4982:
4914:
4634:
4571:
4321:is a topological vector space and if all
3807:
3355:
3102:
1984:that satisfies the following conditions:
944:
916:
885:
842:
807:
645:
598:
594:
558:
399:
318:
220:
216:
160:
135:
70:and every topological vector space has a
35438:Group algebra of a locally compact group
35038:
34726:
34664:
34613:Handbook of Analysis and Its Foundations
34320:
34288:
34109:
34097:
33858:
33778:
33647:
31621:{\displaystyle y={\tfrac {r}{r-1}}x=sx.}
30151: – Type of topological vector space
29429:Pre-image under a continuous linear map
29265:Pre-image under a continuous linear map
24691:(a true or false statement dependent on
21817:{\displaystyle \operatorname {cl} _{X}S}
20787:{\displaystyle \operatorname {cl} _{X}S}
19808:{\displaystyle \operatorname {cl} _{X}S}
19618:whose vertical side is the vector space
17855:belongs to the interior of a convex set
15657:had any topological properties nor that
14882:{\displaystyle s_{1},\ldots ,s_{n}\in S}
14536:{\displaystyle \operatorname {cl} _{X}S}
13354:Locally convex topological vector spaces
12973:
11803:{\displaystyle F\subseteq \mathbb {K} .}
10552:). This finite-dimensional vector space
9569:{\displaystyle \left(X,\tau _{f}\right)}
8445:is locally compact if and only if it is
6199:
5963:that is directed downward, then the set
5322:is a collection sequences of subsets of
1265:, are invariant under TVS isomorphisms.
502:
88:locally convex topological vector spaces
34899:
34838:
34732:Topological Vector Spaces: Chapters 1–5
34445:
34356:
34264:
34151:
31418:{\displaystyle C-w_{0},x-w_{0},y-w_{0}}
30444:{\textstyle \sum _{i=1}^{\infty }x_{i}}
30193:Locally convex topological vector space
24439:are convex non-empty disjoint sets and
14946:{\displaystyle t_{1},\ldots ,t_{n}\in }
14653:{\displaystyle \operatorname {cobal} S}
14490:, balanced hull, disked hull) of a set
13558:is a Fréchet space under the seminorms
12244:{\displaystyle B\subseteq \mathbb {K} }
11608:{\displaystyle M_{x}:\mathbb {K} \to X}
11267:{\displaystyle B\subseteq \mathbb {K} }
9120:{\textstyle \sum _{i=1}^{\infty }x_{i}}
7564:Every neighborhood of the origin is an
6266:be a topological vector space. Given a
6017:at the origin for a vector topology on
1297:of subsets of a vector space is called
908:are the topological vector spaces over
14:
36327:
35771:Uniform boundedness (Banach–Steinhaus)
35008:
34760:
34640:An introduction to Functional Analysis
34637:
34397:
34178:
33948:
33114:must also satisfy this equality (when
32335:{\textstyle h(c_{0})=rc_{0}-rc_{0}=0.}
25346:Properties preserved by set operators
21594:is a set of scalars such that neither
20078:onto the (necessarily) Hausdorff TVS.
14628:{\displaystyle \operatorname {bal} S,}
12967:are all TVS-isomorphic to one another.
8713:if and only if for every neighborhood
2808:is summative, absorbing, and balanced.
1829:is a real or complex vector space. If
384:{\displaystyle (s,x)\mapsto s\cdot x,}
27:Vector space with a notion of nearness
35613:
35129:
35078:A Course on Topological Vector Spaces
35072:
35016:. Mineola, N.Y.: Dover Publications.
34921:
34537:
34472:
34385:
34124:
34061:
34029:
34017:
34005:
33895:
33852:
33840:
33807:
33795:
33606:
33589:
32442:{\textstyle h(c)=rc+(1-r)sc_{0}\in C}
31941:{\displaystyle \operatorname {int} C}
31498:is a neighborhood of the origin. Let
25258:Properties preserved by set operators
17135:{\displaystyle S:=\{(x,y):xy\geq 0\}}
14600:{\displaystyle \operatorname {co} S,}
14486:The closure (respectively, interior,
14153:Every topological vector space has a
12581:{\displaystyle |f|:X\to \mathbb {R} }
12168:From this, it can be deduced that if
10927:has two vector topologies: the usual
10845:has exactly one vector topology: the
10049:if and only if for every real number
9644:
8984:A Cauchy sequence in a Hausdorff TVS
4348:are neighborhoods of the origin then
4022:{\textstyle \bigcap _{i\geq 0}U_{i};}
3116:-valued function induced by a string)
967:topological vector space homomorphism
34951:
34712:Functional Analysis and Applications
34549:McGraw-Hill Science/Engineering/Math
30318:is Hausdorff if and only if the set
29593:Image under a continuous linear map
29434:Image under a continuous linear map
26681: decreasing nonempty
26507: decreasing nonempty
26066: increasing nonempty
25857: increasing nonempty
23929:{\displaystyle tC+(1-t)C\subseteq C}
17076:{\displaystyle X:=\mathbb {R} ^{2},}
17045:is not also convex; for example, in
14685:{\displaystyle \operatorname {co} S}
14463:that has this property and contains
13784:{\displaystyle 1\leq p\leq \infty ,}
12509:which is vector space isomorphic to
8552:is the unique translation-invariant
7416:{\displaystyle tE+(1-t)E\subseteq E}
6062:{\displaystyle \tau _{\mathbb {S} }}
4825:{\displaystyle U_{i}\subseteq V_{i}}
4519:metrizable topological vector spaces
4368:is continuous, where if in addition
3918:{\displaystyle f(x+y)\leq f(x)+f(y)}
1795:(or subbasis) for it at the origin.
1327:{\displaystyle N\in {\mathcal {N}},}
1207:topological vector space isomorphism
763:; it then follows that the space is
451:
34982:. Vol. 53. Cambridge England:
29917:
29908:
29901:
29880:
29841:
29707:
29687:
29672:
29665:
29658:
29635:
29628:
29621:
29614:
29607:
29600:
29583:
29563:
29548:
29541:
29534:
29471:
29464:
29457:
29450:
29443:
29358:
29351:
29344:
29337:
29330:
29323:
29316:
29309:
29302:
29295:
29288:
29281:
29274:
29267:
29232:
29225:
29211:
29172:
29165:
29150:
29143:
29136:
29129:
29122:
29115:
29108:
29101:
29066:
29059:
29031:
29024:
29017:
29010:
29003:
28996:
28989:
28975:
28968:
28961:
28954:
28947:
28940:
28907:
28900:
28872:
28865:
28858:
28851:
28844:
28837:
28830:
28816:
28809:
28802:
28795:
28788:
28781:
28748:
28741:
28734:
28713:
28706:
28699:
28692:
28685:
28678:
28671:
28657:
28650:
28643:
28629:
28622:
28587:
28580:
28566:
28547:
28540:
28533:
28524:
28517:
28510:
28503:
28496:
28489:
28482:
28475:
28468:
28461:
28424:
28417:
28403:
28390:
28379:
28372:
28365:
28358:
28342:
28335:
28328:
28321:
28314:
28307:
28300:
28293:
28256:
28249:
28242:
28235:
28215:
28208:
28201:
28194:
28187:
28178:
28164:
28157:
28150:
28143:
28136:
28122:
28115:
28092:
28085:
28074:
28055:
28048:
28041:
28034:
28027:
28020:
28013:
28006:
27999:
27992:
27985:
27978:
27971:
27964:
27957:
27927:
27916:
27843:
27836:
27820:
27813:
27774:
27767:
27760:
27753:
27740:
27733:
27726:
27719:
27712:
27705:
27698:
27691:
27684:
27677:
27670:
27656:
27649:
27642:
27635:
27628:
27621:
27604:
27597:
27590:
27583:
27576:
27569:
27562:
27553:
27546:
27537:
27530:
27523:
27516:
27509:
27502:
27495:
27488:
27481:
27474:
27467:
27460:
27453:
27446:
27439:
27432:
27425:
27410:
27403:
27396:
27389:
27382:
27375:
27368:
27361:
27354:
27347:
27340:
27333:
27326:
27319:
27312:
27305:
27298:
27291:
27284:
27277:
27270:
27263:
27256:
27249:
27242:
27235:
27228:
27221:
27206:
27199:
27192:
27185:
27164:
27155:
27148:
27141:
27134:
27127:
27120:
27057:
27050:
27043:
27036:
27029:
26971:
26956:
26949:
26918:
26911:
26904:
26897:
26890:
26883:
26876:
26869:
26830:
26819:
26808:
26789:
26782:
26768:
26761:
26754:
26747:
26733:
26719:
26712:
26705:
26698:
26651:
26640:
26629:
26610:
26603:
26589:
26582:
26575:
26568:
26547:
26540:
26533:
26526:
26519:
26448:
26439:
26432:
26421:
26412:
26401:
26394:
26387:
26380:
26373:
26366:
26359:
26352:
26345:
26338:
26331:
26324:
26317:
26310:
26303:
26296:
26289:
26232:
26225:
26125:
26118:
26097:
26083:
26076:
26018:
26011:
25911:
25904:
25897:
25890:
25883:
25876:
25869:
25862:
25798:
25791:
25784:
25777:
25770:
25763:
25756:
25747:
25738:
25724:
25717:
25687:
25680:
25673:
25666:
25645:
25631:
25624:
25267:, open) set has that same property.
24000:is equal to the convex hull of the
23348:
20101:that is an algebraic complement of
18254:{\displaystyle N\cap \mathbb {R} x}
17453:then the open line segment joining
15782:{\displaystyle \tau \subseteq \nu }
14186:from the space into the base field
13475:: a locally convex space where the
12109:there exists some positive integer
9236:be a real or complex vector space.
9212:Finest and coarsest vector topology
9031:If a Cauchy filter in a TVS has an
8640:With respect to this uniformity, a
8353:metrizable topological vector space
7050:if and only if the same is true of
2105:{\displaystyle U\in {\mathcal {B}}}
2075:{\displaystyle B\in {\mathcal {B}}}
2011:{\displaystyle B\in {\mathcal {B}}}
1804:(Neighborhood filter of the origin)
1357:{\displaystyle U\in {\mathcal {N}}}
1231:isomorphism in the category of TVSs
476:on them. These are all examples of
462:infinitely differentiable functions
24:
34696:
30578:under the continuous addition map
30426:
29894:
29887:
29871:
29864:
29855:
29848:
29834:
29827:
29820:
29813:
29806:
29799:
29792:
29785:
29778:
29771:
29764:
29757:
29750:
29743:
29698:
29574:
29527:
29520:
29513:
29506:
29499:
29492:
29485:
29478:
29436:
29421:
29414:
29399:
29386:
29379:
29372:
29365:
29246:
29239:
29218:
29200:
29193:
29186:
29179:
29080:
29073:
29052:
28982:
28921:
28914:
28893:
28823:
28762:
28755:
28664:
28636:
28601:
28594:
28573:
28438:
28431:
28410:
28349:
28270:
28263:
28222:
28171:
28129:
27934:
27901:
27894:
27885:
27878:
27871:
27864:
27857:
27850:
27829:
27806:
27799:
27663:
27178:
27171:
27113:
27106:
27099:
27092:
27085:
27078:
27071:
27064:
27022:
26934:
26925:
26775:
26740:
26726:
26691:
26596:
26561:
26554:
26512:
26246:
26239:
26218:
26209:
26202:
26195:
26188:
26181:
26174:
26167:
26160:
26153:
26146:
26139:
26132:
26111:
26104:
26090:
26032:
26025:
26004:
25995:
25988:
25981:
25974:
25967:
25960:
25953:
25946:
25939:
25932:
25925:
25918:
25731:
25710:
25701:
25694:
25659:
25652:
25638:
24467:{\displaystyle x\not \in R\cup S,}
23345:sets is again compact and convex.
22440:
22399:
18927:
18571:
17191:{\displaystyle \{(x,y):xy>0\}.}
16384:{\displaystyle X=\mathbb {R} ^{2}}
16230:
16184:
16136:if and only if there exists a net
16001:
15697:is a neighborhood of the origin).
14237:{\displaystyle X'\to \mathbb {K} }
13900:
13873:
13775:
13532:
13465:: locally convex spaces where the
12161:{\displaystyle S\subseteq B_{n}x.}
11474:
11389:
9910:
9278:{\displaystyle \{X,\varnothing \}}
9102:
8625:if and only if it is complete and
8504:Completeness and uniform structure
5571:(said differently, if and only if
3172:
2310:
2159:
2097:
2067:
2037:
2003:
1965:
1956:That is, the assumptions are that
1894:
1838:
1349:
1316:
1282:
1169:topological vector space embedding
879:
876:
873:
870:
867:
836:
833:
830:
482:Kolmogorov's normability criterion
413:is the underlying scalar field of
177:unless clearly stated otherwise.
54:. A topological vector space is a
25:
36361:
35107:
35042:(1982). Nachbin, Leopoldo (ed.).
34615:. San Diego, CA: Academic Press.
33124:
32177:which is a homeomorphism because
30857:{\displaystyle \mathbb {R} ^{2}.}
30805:{\displaystyle y={\frac {1}{x}},}
30749:{\displaystyle \mathbb {R} ^{2},}
24573:
24517:
23393:is a TVS that does not carry the
21468:
21063:then there exists a neighborhood
20715:complete topological vector space
19838:under the canonical quotient map
18609:
18064:
16816:
16271:{\displaystyle s_{\bullet }\to x}
15948:{\displaystyle x_{\bullet }\to 0}
15165:The open convex subsets of a TVS
15162:is a neighborhood of the origin.
13256:is continuous if and only if its
12534:{\displaystyle \mathbb {K} ^{n},}
10603:{\displaystyle \mathbb {K} ^{n},}
9269:
9173:{\displaystyle x_{\bullet }\to 0}
8905:complete topological vector space
8510:Complete topological vector space
4690:is a scalar, then by definition:
2742:, etc.) if this is true of every
2235:Defining topologies using strings
1612:) is continuous at the origin of
36309:
36308:
35594:
35593:
35520:Topological quantum field theory
35113:
34843:. New York: Dover Publications.
30648:) and a closed set is closed so
30199:Ordered topological vector space
29918:
29909:
29902:
29895:
29888:
29881:
29872:
29865:
29856:
29849:
29842:
29835:
29828:
29821:
29814:
29807:
29800:
29793:
29786:
29779:
29772:
29765:
29758:
29751:
29744:
29708:
29699:
29688:
29673:
29666:
29659:
29636:
29629:
29622:
29615:
29608:
29601:
29584:
29575:
29564:
29549:
29542:
29535:
29528:
29521:
29514:
29507:
29500:
29493:
29486:
29479:
29472:
29465:
29458:
29451:
29444:
29437:
29422:
29415:
29400:
29387:
29380:
29373:
29366:
29359:
29352:
29345:
29338:
29331:
29324:
29317:
29310:
29303:
29296:
29289:
29282:
29275:
29268:
29247:
29240:
29233:
29226:
29219:
29212:
29201:
29194:
29187:
29180:
29173:
29166:
29151:
29144:
29137:
29130:
29123:
29116:
29109:
29102:
29081:
29074:
29067:
29060:
29053:
29032:
29025:
29018:
29011:
29004:
28997:
28990:
28983:
28976:
28969:
28962:
28955:
28948:
28941:
28922:
28915:
28908:
28901:
28894:
28873:
28866:
28859:
28852:
28845:
28838:
28831:
28824:
28817:
28810:
28803:
28796:
28789:
28782:
28763:
28756:
28749:
28742:
28735:
28714:
28707:
28700:
28693:
28686:
28679:
28672:
28665:
28658:
28651:
28644:
28637:
28630:
28623:
28602:
28595:
28588:
28581:
28574:
28567:
28548:
28541:
28534:
28525:
28518:
28511:
28504:
28497:
28490:
28483:
28476:
28469:
28462:
28439:
28432:
28425:
28418:
28411:
28404:
28391:
28380:
28373:
28366:
28359:
28350:
28343:
28336:
28329:
28322:
28315:
28308:
28301:
28294:
28271:
28264:
28257:
28250:
28243:
28236:
28223:
28216:
28209:
28202:
28195:
28188:
28179:
28172:
28165:
28158:
28151:
28144:
28137:
28130:
28123:
28116:
28093:
28086:
28075:
28056:
28049:
28042:
28035:
28028:
28021:
28014:
28007:
28000:
27993:
27986:
27979:
27972:
27965:
27958:
27935:
27928:
27917:
27902:
27895:
27886:
27879:
27872:
27865:
27858:
27851:
27844:
27837:
27830:
27821:
27814:
27807:
27800:
27775:
27768:
27761:
27754:
27741:
27734:
27727:
27720:
27713:
27706:
27699:
27692:
27685:
27678:
27671:
27664:
27657:
27650:
27643:
27636:
27629:
27622:
27605:
27598:
27591:
27584:
27577:
27570:
27563:
27554:
27547:
27538:
27531:
27524:
27517:
27510:
27503:
27496:
27489:
27482:
27475:
27468:
27461:
27454:
27447:
27440:
27433:
27426:
27411:
27404:
27397:
27390:
27383:
27376:
27369:
27362:
27355:
27348:
27341:
27334:
27327:
27320:
27313:
27306:
27299:
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27285:
27278:
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27250:
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27229:
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27207:
27200:
27193:
27186:
27179:
27172:
27165:
27156:
27149:
27142:
27135:
27128:
27121:
27114:
27107:
27100:
27093:
27086:
27079:
27072:
27065:
27058:
27051:
27044:
27037:
27030:
27023:
26972:
26957:
26950:
26935:
26926:
26919:
26912:
26905:
26898:
26891:
26884:
26877:
26870:
26831:
26820:
26809:
26790:
26783:
26776:
26769:
26762:
26755:
26748:
26741:
26734:
26727:
26720:
26713:
26706:
26699:
26692:
26652:
26641:
26630:
26611:
26604:
26597:
26590:
26583:
26576:
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26562:
26555:
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26402:
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26388:
26381:
26374:
26367:
26360:
26353:
26346:
26339:
26332:
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26318:
26311:
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26297:
26290:
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26175:
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26147:
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26126:
26119:
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26105:
26098:
26091:
26084:
26077:
26033:
26026:
26019:
26012:
26005:
25996:
25989:
25982:
25975:
25968:
25961:
25954:
25947:
25940:
25933:
25926:
25919:
25912:
25905:
25898:
25891:
25884:
25877:
25870:
25863:
25799:
25792:
25785:
25778:
25771:
25764:
25757:
25748:
25739:
25732:
25725:
25718:
25711:
25702:
25695:
25688:
25681:
25674:
25667:
25660:
25653:
25646:
25639:
25632:
25625:
24591:In any non-trivial vector space
24183:{\displaystyle \mathbb {R} ^{2}}
23886:or equivalently, if and only if
23464:is a Baire space if and only if
23354:Meager, nowhere dense, and Baire
20705:Compact and totally bounded sets
20484:{\displaystyle (h,n)\mapsto h+n}
18813:the same is true of its closure
18710:or equivalently, if and only if
17923:then the half-open line segment
14080:{\displaystyle \mathbb {C} ^{n}}
14051:{\displaystyle \mathbb {R} ^{n}}
13503:: a barrelled space where every
12979:Discrete and cofinite topologies
12696:{\displaystyle \ker f=\ker |f|.}
10632:{\displaystyle \mathbb {K} ^{n}}
10545:{\displaystyle \mathbb {K} ^{n}}
10492:{\displaystyle \mathbb {K} ^{n}}
10372:with its usual Hausdorff normed
9342:topological vector space. It is
8815:{\displaystyle x_{i}-x_{j}\in V}
8471:{\displaystyle \mathbb {K} ^{n}}
8062:
8037:if and only if it is bounded in
7073:
6610:{\displaystyle x\mapsto x_{0}+x}
5597:with respect to the containment
1780:{\displaystyle x\mapsto x_{0}+x}
1605:{\displaystyle (x,y)\mapsto x+y}
290:{\displaystyle (x,y)\mapsto x+y}
185:
36296:With the approximation property
35044:Topics in Locally Convex Spaces
34980:Cambridge Tracts in Mathematics
34766:A Course in Functional Analysis
34407:
33954:
33163:
32682:
31628:Since scalar multiplication by
31478:and so it remains to show that
31168:
31054:
30995:This condition is satisfied if
30989:
30929:is a countable dense subset of
30812:which is the complement of the
30718:
30478:
30397:
30384:
30375:
30292:
30263:
23632:{\displaystyle 2S\subseteq S+S}
21111:{\displaystyle K+N\subseteq U.}
17332:Explicitly, this means that if
11810:And if this vector topology on
11684:{\displaystyle X=\mathbb {K} x}
11195:-dimensional vector space over
8363:that is translation-invariant.
8295:which is the given topology on
6500:Invariance of vector topologies
5991:of all knots of all strings in
3655:{\displaystyle x\not \in U_{0}}
2988:{\displaystyle U_{i}:=2^{1-i}U}
2140:{\displaystyle U+U\subseteq B,}
1392:{\displaystyle U+U\subseteq N.}
1113:which is the range or image of
679:). Such a topology is called a
35759:Open mapping (Banach–Schauder)
33529:
33517:
33490:
33478:
33445:
33433:
33313:
33304:
33292:
33289:
33250:
33244:
33205:
33199:
32831:
32813:
32542:{\displaystyle sc_{0},c\in C,}
32417:
32405:
32390:
32384:
32291:
32278:
32123:
32111:
32099:
32073:
31664:
31453:
31441:
31270:
31258:
31036:
31024:
30609:
25090:
25084:
24856:
24850:
24821:
24809:
24782:
24776:
24674:
24668:
24567:
24549:
24511:
24493:
24432:{\displaystyle R,S\subseteq X}
24376:
24367:
24320:
24308:
24267:
24255:
24218:{\displaystyle R,S\subseteq X}
24141:
24129:
24093:
24081:
24052:
24040:
23961:{\displaystyle 0\leq t\leq 1.}
23914:
23902:
23809:
23797:
23305:
23293:
23274:
23262:
23226:
23223:
23211:
23202:
23169:
23157:
23124:
23112:
23085:{\displaystyle R,S\subseteq X}
23044:
23032:
23013:
23001:
22976:
22973:
22961:
22952:
22885:{\displaystyle R,S\subseteq X}
22841:
22829:
22787:
22775:
22737:
22725:
22579:{\displaystyle X=\mathbb {R} }
22448:{\displaystyle {\mathcal {N}}}
22418:
22406:
22155:
22149:
22130:
22124:
22065:
22053:
22023:
22017:
21998:
21992:
21947:
21935:
21910:
21904:
21885:
21879:
21852:{\displaystyle R,S\subseteq X}
21711:
21702:
21510:
21501:
21402:
21393:
20840:
20828:
20804:
20469:
20466:
20454:
20428:
20334:is an algebraic complement of
20031:
19921:is a vector subspace of a TVS
19884:
19872:
19848:
19706:the following are equivalent:
18938:
18932:
18527:
18515:
18469:
18457:
18371:
18356:
18341:
18329:
18204:
18196:
18058:
18046:
17972:
17960:
17942:
17930:
17813:
17805:
17716:
17708:
17547:
17535:
17289:
17277:
17164:
17152:
17111:
17099:
16966:) to be true, it suffices for
16740:
16728:
16678:
16672:
16653:
16647:
16444:{\displaystyle R,S\subseteq X}
16262:
16238:{\displaystyle {\mathcal {N}}}
16009:{\displaystyle {\mathcal {N}}}
15974:
15962:
15939:
15909:
15897:
15789:if and only if whenever a net
15586:
15580:
15529:
15523:
15273:
15261:
15222:
15216:
14940:
14928:
14293:
14226:
13816:functions of bounded variation
13652:
13648:
13642:
13637:
13631:
13622:
13616:
13601:
13545:
13537:
13216:on a topological vector space
13173:in a topological vector space
13133:
13127:
13092:
12927:then the vector topologies on
12686:
12678:
12638:
12634:
12628:
12621:
12614:
12608:
12604:
12596:
12570:
12560:
12552:
12432:{\displaystyle \dim X=n\geq 2}
12298:
12292:
11891:
11766:
11760:
11638:
11632:
11599:
11106:
11098:
10470:is vector space isomorphic to
10123:
10117:
10094:
10088:
9803:. With this product topology,
9695:
9486:(that is, any TVS-topology on
9164:
8539:
8527:
8423:topological vector space over
8338:
8326:
8208:
8196:
8095:
8083:
7974:is a vector subspace of a TVS
7854:{\displaystyle E\subseteq tV.}
7651:of a topological vector space
7522:{\displaystyle -E\subseteq E,}
7448:{\displaystyle 0\leq t\leq 1.}
7401:
7389:
7346:
7338:
7267:
7259:
6916:
6904:
6804:
6778:
6720:
6694:
6588:
6562:
6141:
6129:
5943:is a collection of strings in
5857:
5845:
5757:
5745:
5612:{\displaystyle \,\subseteq \,}
4484:{\displaystyle d(x,y):=f(x-y)}
4478:
4466:
4457:
4445:
4252:
4244:
4209:{\displaystyle f(sx)\leq f(x)}
4203:
4197:
4188:
4179:
4129:
4123:
4114:
4105:
4045:
4039:
3912:
3906:
3897:
3891:
3882:
3870:
3817:
3811:
3678:
3672:
3610:
3604:
3581:
3569:
3566:
3365:
3359:
2995:forms a string beginning with
2167:{\displaystyle {\mathcal {B}}}
2045:{\displaystyle {\mathcal {B}}}
1973:{\displaystyle {\mathcal {B}}}
1902:{\displaystyle {\mathcal {B}}}
1846:{\displaystyle {\mathcal {B}}}
1758:
1732:
1661:
1649:
1590:
1587:
1575:
1561:{\displaystyle X\times X\to X}
1552:
1446:
1434:
1290:{\displaystyle {\mathcal {N}}}
1223:topological vector isomorphism
1097:
1091:
1043:
1011:
655:
615:
366:
363:
351:
328:
304:The scalar multiplication map
275:
272:
260:
237:
66:. Such a topology is called a
13:
1:
35316:Uniform boundedness principle
34770:Graduate Texts in Mathematics
34374:Narici & Beckenstein 2011
34345:Narici & Beckenstein 2011
34333:Narici & Beckenstein 2011
34304:Narici & Beckenstein 2011
34277:Narici & Beckenstein 2011
34214:Narici & Beckenstein 2011
34202:Narici & Beckenstein 2011
34140:Narici & Beckenstein 2011
34086:Narici & Beckenstein 2011
34045:Narici & Beckenstein 2011
33989:Narici & Beckenstein 2011
33937:Narici & Beckenstein 2011
33843:, p. 27-28 Theorem 1.37.
33829:Narici & Beckenstein 2011
33767:Adasch, Ernst & Keim 1978
33755:Narici & Beckenstein 2011
33720:Adasch, Ernst & Keim 1978
33699:Narici & Beckenstein 2011
33563:{\displaystyle \blacksquare }
30969:{\displaystyle \mathbb {R} .}
30882:{\displaystyle \mathbb {R} ,}
30184:Locally compact quantum group
22348:{\displaystyle S\subseteq X,}
22215:{\displaystyle S\subseteq X,}
20980:is a compact subset of a TVS
20511:is necessarily Hausdorff and
19419:{\displaystyle S\subseteq X,}
18577:{\displaystyle r\neq \infty }
18303:{\displaystyle \mathbb {R} x}
17248:{\displaystyle 0<t\leq 1,}
16471:has non-empty interior then
16061:{\displaystyle S\subseteq X,}
15321:and some positive continuous
14342:
14204:{\displaystyle \mathbb {K} .}
14134:
13193:is either dense or closed. A
12823:{\displaystyle \dim X\geq 2,}
12645:{\displaystyle |f|(x)=|f(x)|}
12390:{\displaystyle \blacksquare }
12269:{\displaystyle \mathbb {K} ,}
12216:{\displaystyle 0\neq x\in X,}
12102:{\displaystyle 0\neq x\in X,}
11567:{\displaystyle 0\neq x\in X,}
11497:is unbounded in normed space
11213:{\displaystyle \mathbb {K} .}
10154:{\displaystyle \mathbb {R} .}
9864:{\displaystyle \mathbb {R} .}
7739:{\displaystyle E\subseteq tV}
7323:{\displaystyle tE\subseteq E}
6877:{\displaystyle S\subseteq X,}
6291:{\displaystyle M\subseteq X,}
6089:{\displaystyle \mathbb {S} .}
5733:(Topology induced by strings)
5715:{\displaystyle \mathbb {S} .}
2946:then the sequence defined by
2566:{\displaystyle U_{\bullet }.}
2505:{\displaystyle U_{\bullet }.}
498:
460:on an open domain, spaces of
180:
170:{\displaystyle \mathbb {R} ,}
72:uniform topological structure
34927:Topological Vector Spaces II
34577:; Wolff, Manfred P. (1999).
33576:
33133:{\displaystyle X\setminus S}
32497:{\displaystyle 0<r<1,}
32202:{\displaystyle 0<r<1.}
32050:{\displaystyle sc_{0}\in C.}
31471:{\displaystyle rx+(1-r)y=0,}
31010:{\displaystyle \mathbb {S} }
30944:{\displaystyle \mathbb {R} }
29092:Closed convex balanced hull
28938:Closed convex balanced hull
23993:{\displaystyle S\subseteq X}
23836:{\displaystyle (s+t)C=sC+tC}
23597:{\displaystyle S\subseteq X}
21567:{\displaystyle S\subseteq X}
21324:{\displaystyle S\subseteq X}
20897:such that every sequence in
20491:is a TVS-isomorphism, where
20140:(that is, a vector subspace
19969:is Hausdorff if and only if
19676:{\displaystyle S\subseteq X}
19508:{\displaystyle S\subseteq X}
19315:{\displaystyle S\subseteq X}
18882:This vector space satisfies
18661:is Hausdorff if and only if
17874:{\displaystyle S\subseteq X}
17627:{\displaystyle N\subseteq X}
17352:is a convex subset of a TVS
16790:set with non-empty interior
15740:be two vector topologies on
15136:has non-empty interior then
15129:{\displaystyle S\subseteq X}
15017:{\displaystyle S\subseteq X}
14372:{\displaystyle S\subseteq X}
13911:{\displaystyle L^{\infty }.}
13453:{\displaystyle 0<p<1.}
12463:distinct vector topologies:
11653:{\displaystyle M_{x}(a):=ax}
11512:{\displaystyle \mathbb {K} }
11429:{\displaystyle 0\neq x\in X}
11239:{\displaystyle S\subseteq X}
11053:{\displaystyle \mathbb {K} }
11031:{\displaystyle \mathbb {K} }
10991:{\displaystyle \mathbb {K} }
10949:{\displaystyle \mathbb {K} }
10746:{\displaystyle \dim X\neq 0}
10411:{\displaystyle \mathbb {K} }
10365:{\displaystyle \mathbb {K} }
10343:{\displaystyle \mathbb {C} }
10321:{\displaystyle \mathbb {R} }
10299:{\displaystyle \mathbb {K} }
9728:{\displaystyle \mathbb {R} }
9385:There exists a TVS topology
9028:is not necessarily compact).
8438:{\displaystyle \mathbb {K} }
8405:{\displaystyle \mathbb {K} }
8230:translation-invariant metric
7809:of the origin, there exists
6819:{\displaystyle x\mapsto -x,}
6115:{\displaystyle \mathbb {S} }
6006:{\displaystyle \mathbb {S} }
5936:{\displaystyle \mathbb {S} }
5827:{\displaystyle \mathbb {S} }
5785:{\displaystyle \mathbb {S} }
5586:{\displaystyle \mathbb {S} }
5398:{\displaystyle \mathbb {S} }
5360:{\displaystyle \mathbb {S} }
5315:{\displaystyle \mathbb {S} }
4742:{\displaystyle U_{\bullet }}
4713:{\displaystyle V_{\bullet }}
4408:{\displaystyle U_{\bullet }}
3109:{\displaystyle \mathbb {R} }
3054:Moreover, if a vector space
2866:{\displaystyle U_{\bullet }}
2801:{\displaystyle U_{\bullet }}
2593:{\displaystyle U_{\bullet }}
2403:{\displaystyle U_{\bullet }}
2368:{\displaystyle U_{\bullet }}
2325:be a sequence of subsets of
1463:(as all vector spaces are),
960:from one object to another.
951:{\displaystyle \mathbb {K} }
923:{\displaystyle \mathbb {K} }
814:{\displaystyle \mathbb {K} }
565:{\displaystyle \mathbb {K} }
406:{\displaystyle \mathbb {K} }
142:{\displaystyle \mathbb {C} }
7:
36335:Topology of function spaces
35980:Radially convex/Star-shaped
35965:Pre-compact/Totally bounded
34839:Edwards, Robert E. (1995).
34478:Topological Vector Spaces I
34450:. Stuttgart: B.G. Teubner.
33871:Encyclopedia of Mathematics
31948:is open, there exists some
31206:so it remains to show that
31199:{\displaystyle 0<r<1}
30127:
29919:
29910:
29903:
29882:
29843:
29709:
29689:
29674:
29667:
29660:
29637:
29630:
29623:
29616:
29609:
29602:
29585:
29565:
29550:
29543:
29536:
29473:
29466:
29459:
29452:
29445:
29360:
29353:
29346:
29339:
29332:
29325:
29318:
29311:
29304:
29297:
29290:
29283:
29276:
29269:
29234:
29227:
29213:
29174:
29167:
29152:
29145:
29138:
29131:
29124:
29117:
29110:
29103:
29068:
29061:
29033:
29026:
29019:
29012:
29005:
28998:
28991:
28977:
28970:
28963:
28956:
28949:
28942:
28909:
28902:
28874:
28867:
28860:
28853:
28846:
28839:
28832:
28818:
28811:
28804:
28797:
28790:
28783:
28750:
28743:
28736:
28715:
28708:
28701:
28694:
28687:
28680:
28673:
28659:
28652:
28645:
28631:
28624:
28589:
28582:
28568:
28549:
28542:
28535:
28526:
28519:
28512:
28505:
28498:
28491:
28484:
28477:
28470:
28463:
28426:
28419:
28405:
28392:
28381:
28374:
28367:
28360:
28344:
28337:
28330:
28323:
28316:
28309:
28302:
28295:
28258:
28251:
28244:
28237:
28217:
28210:
28203:
28196:
28189:
28180:
28166:
28159:
28152:
28145:
28138:
28124:
28117:
28094:
28087:
28076:
28057:
28050:
28043:
28036:
28029:
28022:
28015:
28008:
28001:
27994:
27987:
27980:
27973:
27966:
27959:
27929:
27918:
27845:
27838:
27822:
27815:
27776:
27769:
27762:
27755:
27742:
27735:
27728:
27721:
27714:
27707:
27700:
27693:
27686:
27679:
27672:
27658:
27651:
27644:
27637:
27630:
27623:
27606:
27599:
27592:
27585:
27578:
27571:
27564:
27555:
27548:
27539:
27532:
27525:
27518:
27511:
27504:
27497:
27490:
27483:
27476:
27469:
27462:
27455:
27448:
27441:
27434:
27427:
27412:
27405:
27398:
27391:
27384:
27377:
27370:
27363:
27356:
27349:
27342:
27335:
27328:
27321:
27314:
27307:
27300:
27293:
27286:
27279:
27272:
27265:
27258:
27251:
27244:
27237:
27230:
27223:
27208:
27201:
27194:
27187:
27166:
27157:
27150:
27143:
27136:
27129:
27122:
27059:
27052:
27045:
27038:
27031:
26973:
26958:
26951:
26920:
26913:
26906:
26899:
26892:
26885:
26878:
26871:
26832:
26821:
26810:
26791:
26784:
26770:
26763:
26756:
26749:
26735:
26721:
26714:
26707:
26700:
26653:
26642:
26631:
26612:
26605:
26591:
26584:
26577:
26570:
26549:
26542:
26535:
26528:
26521:
26450:
26441:
26434:
26423:
26414:
26403:
26396:
26389:
26382:
26375:
26368:
26361:
26354:
26347:
26340:
26333:
26326:
26319:
26312:
26305:
26298:
26291:
26234:
26227:
26127:
26120:
26099:
26085:
26078:
26020:
26013:
25913:
25906:
25899:
25892:
25885:
25878:
25871:
25864:
25800:
25793:
25786:
25779:
25772:
25765:
25758:
25749:
25740:
25726:
25719:
25689:
25682:
25675:
25668:
25647:
25633:
25626:
25353:
23720:{\displaystyle S=\{-x,x\};}
18641:A topological vector space
17493:belongs to the interior of
14977:Neighborhoods and open sets
14712:is equal to the set of all
13881:{\displaystyle L^{\infty }}
13477:continuous linear operators
12920:{\displaystyle n:=\dim X=2}
12313:{\displaystyle M_{x}(B)=Bx}
11772:{\displaystyle Fx=M_{x}(F)}
9506:is necessarily a subset of
9206:
8892:. Scalar multiplication is
7697:of the origin there exists
6732:{\displaystyle x\mapsto sx}
6647:{\displaystyle x_{0}\neq 0}
6545:{\displaystyle x_{0}\in X,}
5792:of neighborhood strings in
3340:{\displaystyle u\in U_{0},}
1715:{\displaystyle x_{0}\in X,}
732:Many authors (for example,
117:of sequences of functions.
10:
36366:
35666:Continuous linear operator
35459:Invariant subspace problem
34984:Cambridge University Press
34032:, p. 62-68 §3.8-3.14.
34020:, p. 27 Theorem 1.36.
33898:, p. 17 Theorem 1.22.
33866:"Topological vector space"
32248:is thus an open subset of
31654:is a linear homeomorphism
31045:{\displaystyle (X,\tau ).}
30251:Topological vector lattice
29896:
29889:
29873:
29866:
29857:
29850:
29836:
29829:
29822:
29815:
29808:
29801:
29794:
29787:
29780:
29773:
29766:
29759:
29752:
29745:
29700:
29576:
29529:
29522:
29515:
29508:
29501:
29494:
29487:
29480:
29438:
29423:
29416:
29401:
29388:
29381:
29374:
29367:
29248:
29241:
29220:
29202:
29195:
29188:
29181:
29082:
29075:
29054:
28984:
28923:
28916:
28895:
28825:
28764:
28757:
28666:
28638:
28603:
28596:
28575:
28440:
28433:
28412:
28351:
28272:
28265:
28224:
28173:
28131:
27936:
27903:
27896:
27887:
27880:
27873:
27866:
27859:
27852:
27831:
27808:
27801:
27665:
27180:
27173:
27115:
27108:
27101:
27094:
27087:
27080:
27073:
27066:
27024:
26936:
26927:
26777:
26742:
26728:
26693:
26598:
26563:
26556:
26514:
26248:
26241:
26220:
26211:
26204:
26197:
26190:
26183:
26176:
26169:
26162:
26155:
26148:
26141:
26134:
26113:
26106:
26092:
26034:
26027:
26006:
25997:
25990:
25983:
25976:
25969:
25962:
25955:
25948:
25941:
25934:
25927:
25920:
25733:
25712:
25703:
25696:
25661:
25654:
25640:
24894:{\displaystyle \|x\|<1}
23760:{\displaystyle S=\{x,2x\}}
18976:contains the vector space
17777:is the set of all scalars
15983:{\displaystyle (X,\tau ).}
15633:where importantly, it was
14443:is the smallest subset of
14346:
14138:
13364:. By a technique known as
12703:Every seminorm induces a (
11319:{\displaystyle B\cdot S=X}
10849:, which in this case (and
10503:between the vector spaces
9799:which carries the natural
8556:that induces the topology
8507:
7677:if for every neighborhood
7359:{\displaystyle |t|\leq 1.}
7283:{\displaystyle |c|\leq r.}
6768:produces the negation map
6739:is a homeomorphism. Using
5866:{\displaystyle (X,\tau ).}
4135:{\displaystyle f(-x)=f(x)}
3213:{\displaystyle 0\in U_{i}}
2259:be a vector space and let
1638:if and only if the set of
784:
631:and scalar multiplication
584:such that vector addition
36345:Topological vector spaces
36304:
36049:
36011:Algebraic interior (core)
35993:
35891:
35779:
35753:Vector-valued Hahn–Banach
35714:
35648:
35641:Topological vector spaces
35589:
35548:
35472:
35451:
35410:
35349:
35291:
35237:
35179:
35172:
35120:Topological vector spaces
34976:Topological Vector Spaces
34873:Topological Vector Spaces
34579:Topological Vector Spaces
34513:Topological Vector Spaces
34250:Schaefer & Wolff 1999
34235:Schaefer & Wolff 1999
34167:Schaefer & Wolff 1999
34074:Schaefer & Wolff 1999
33920:Schaefer & Wolff 1999
33908:Schaefer & Wolff 1999
33619:Schaefer & Wolff 1999
30982:
30486:particular point topology
30204:Topological abelian group
27612:Positive scalar multiple
27423:Positive scalar multiple
25399:and any other subsets of
25350:
23790:is convex if and only if
20081:Every vector subspace of
18857:the closure of the origin
17826:{\displaystyle |a|<1.}
16332:) then any closed convex
11962:and an open neighborhood
10714:{\displaystyle \dim X=0.}
10443:{\displaystyle n:=\dim X}
10265:Finite-dimensional spaces
9620:{\displaystyle \tau _{f}}
9533:). Every linear map from
9526:{\displaystyle \tau _{f}}
9405:{\displaystyle \tau _{f}}
9371:{\displaystyle \dim X=0.}
9008:(that is, its closure in
8545:{\displaystyle (X,\tau )}
8449:, that is, isomorphic to
8344:{\displaystyle (X,\tau )}
8225:(as a topological space).
8214:{\displaystyle (X,\tau )}
8101:{\displaystyle (X,\tau )}
8070:Birkhoff–Kakutani theorem
6147:{\displaystyle (X,\tau )}
5921:is a vector space and if
5763:{\displaystyle (X,\tau )}
5405:is not empty and for all
4265:{\displaystyle |s|\leq 1}
2348:Each set in the sequence
1933:for a vector topology on
1667:{\displaystyle (X,\tau )}
1631:{\displaystyle X\times X}
1525:{\displaystyle X\times X}
42:and commonly abbreviated
18:Topological vector spaces
35841:Topological homomorphism
35701:Topological vector space
35428:Spectrum of a C*-algebra
34736:Éléments de mathématique
34638:Swartz, Charles (1992).
33063:satisfying the equality
32082:{\displaystyle h:X\to X}
30256:
24027:that is, it is equal to
23566:locally convex TVS is a
22078:and so consequently, if
21153:is a vector subspace of
18855:which is referred to as
18790:is a vector subspace of
16324:is a TVS that is of the
15915:{\displaystyle (X,\nu )}
15442:{\displaystyle p:=p_{K}}
14510:is sometimes denoted by
14012:{\displaystyle W^{2,k},}
13481:bounded linear operators
13467:Banach–Steinhaus theorem
13421:{\displaystyle p\geq 1,}
13356:: here each point has a
13267:
12859:{\displaystyle 1+\dim X}
10900:{\displaystyle \dim X=1}
10806:{\displaystyle \dim X=0}
10100:{\displaystyle f_{n}(x)}
8870:{\displaystyle j\geq n.}
8756:there exists some index
8478:for some natural number
3950:{\displaystyle x,y\in X}
3860:is subadditive (meaning
3304:{\displaystyle i\geq 0.}
3024:{\displaystyle U_{1}=U.}
2739:absolutely convex/disked
1536:, then the addition map
1242:. Equivalently, it is a
1020:{\displaystyle u:X\to Y}
984:topological homomorphism
517:topological vector space
206:The vector addition map
40:linear topological space
36:topological vector space
35525:Noncommutative geometry
34867:Grothendieck, Alexander
34704:Bierstedt, Klaus-Dieter
34642:. New York: M. Dekker.
32268:that moreover contains
31676:{\displaystyle X\to X,}
31647:{\displaystyle s\neq 0}
30776:-axis and the graph of
26474:{\displaystyle R\cap S}
26281:{\displaystyle R\cap S}
25824:{\displaystyle R\cup S}
25616:{\displaystyle R\cup S}
25335:{\displaystyle R\cup S}
25049:{\displaystyle s\neq 0}
24923:{\displaystyle z\in X,}
24739:{\displaystyle z\in X,}
22651:for every neighborhood
21232:{\displaystyle U\cap N}
20917:has a cluster point in
20700:Closed and compact sets
16090:{\displaystyle x\in X.}
14830:{\displaystyle n\geq 1}
13826:Reflexive Banach spaces
13344:{\displaystyle p>0.}
12343:which then proves that
11779:for some unique subset
11148:{\displaystyle r>0.}
10838:{\displaystyle X=\{0\}}
10396:be a vector space over
10254:{\displaystyle f\neq 0}
8841:{\displaystyle i\geq n}
8621:. A subset of a TVS is
7489:is convex and balanced.
7224:{\displaystyle cx\in E}
7169:{\displaystyle x\in X,}
6677:{\displaystyle s\neq 0}
4294:{\displaystyle x\in X.}
4057:{\displaystyle f(0)=0.}
3587:{\displaystyle f:X\to }
2052:is additive: For every
1857:additive collection of
74:, allowing a notion of
35899:Absolutely convex/disk
35581:Tomita–Takesaki theory
35556:Approximation property
35500:Calculus of variations
34957:Differential manifolds
34446:Jarchow, Hans (1981).
33564:
33542:
33390:
33389:{\displaystyle z+X=X,}
33355:
33323:
33154:
33134:
33108:
33057:
33037:
33017:
32978:
32958:
32885:
32756:
32755:{\displaystyle s\in S}
32721:
32673:
32650:
32630:
32591:
32543:
32498:
32463:
32443:
32368:
32336:
32262:
32242:
32203:
32171:
32083:
32051:
32012:
31942:
31916:
31832:
31797:
31765:
31677:
31648:
31622:
31564:
31492:
31472:
31419:
31348:
31316:
31280:
31200:
31160:
31093:
31046:
31011:
30970:
30945:
30923:
30883:
30858:
30826:
30806:
30770:
30750:
30710:
30687:
30642:
30622:
30572:
30527:
30469:
30445:
30430:
30358:
30338:
30312:
30283:
30097:
29939:
29736:
27418:Non-0 scalar multiple
27219:Non-0 scalar multiple
27010:
26862:
26673:
26499:
26475:
26282:
26058:
25849:
25825:
25617:
25569:
25413:
25393:
25371:
25336:
25307:
25248:
25247:{\displaystyle z\in X}
25222:
25202:
25182:
25181:{\displaystyle x\in X}
25156:
25050:
25024:
24924:
24895:
24863:
24834:
24740:
24711:
24710:{\displaystyle x\in X}
24681:
24631:
24608:
24583:
24524:
24468:
24433:
24401:
24239:
24219:
24184:
24151:
24062:
24021:
23994:
23962:
23930:
23880:
23843:for all positive real
23837:
23784:
23761:
23721:
23677:
23653:
23633:
23598:
23556:
23506:
23478:
23458:
23435:
23411:
23387:
23320:
23176:
23131:
23086:
23054:
22926:
22906:
22886:
22851:
22797:
22747:
22688:
22665:
22645:
22600:
22580:
22552:
22539: is open in
22472:
22449:
22425:
22349:
22317:
22297:
22274:
22216:
22187:
22165:
22098:
22072:
21853:
21818:
21785:
21721:
21628:
21588:
21568:
21542:
21475:
21432:
21412:
21345:
21325:
21296:
21273:
21253:
21233:
21207:
21187:
21167:
21147:
21121:Closure and closed set
21112:
21077:
21057:
21034:
21014:
20994:
20974:
20951:
20931:
20911:
20891:
20871:
20847:
20788:
20751:
20731:
20691:
20637:
20592:
20568:
20544:
20505:
20485:
20441:
20393:then the addition map
20387:
20367:
20328:
20308:
20270:topological complement
20262:
20211:
20154:
20134:
20095:
20068:
20006:
19983:
19963:
19935:
19915:
19891:
19832:
19809:
19774:
19723:
19700:
19677:
19651:
19614:can be described as a
19608:
19580:
19529:
19509:
19483:
19420:
19380:
19316:
19288:
19246:
19218:
19194:
19155:
19154:{\displaystyle \{0\}.}
19126:
19095:
19052:
19009:
18970:
18950:
18876:
18849:
18807:
18784:
18758:
18704:
18687:is a closed subset of
18681:
18655:
18628:
18578:
18552:
18488:
18400:
18304:
18279:
18255:
18224:
18152:
18132:
18108:
18088:
18033:
17917:
17875:
17849:
17827:
17791:
17771:
17744:
17648:
17628:
17600:
17510:
17487:
17467:
17447:
17408:
17366:
17346:
17326:
17249:
17214:
17192:
17136:
17077:
17039:
17019:
16980:
16960:
16921:
16888:
16843:
16823:
16780:
16750:
16621:
16465:
16445:
16405:
16385:
16350:
16328:in itself (that is, a
16318:
16295:
16272:
16239:
16215:
16195:
16130:
16091:
16062:
16033:
16010:
15984:
15949:
15916:
15884:
15864:
15844:
15783:
15757:
15734:
15714:
15691:
15671:
15651:
15627:
15467:
15443:
15410:
15383:
15361:
15338:
15315:
15314:{\displaystyle z\in X}
15289:
15179:
15156:
15130:
15104:
15084:
15058:
15038:
15018:
14967:
14947:
14883:
14831:
14805:
14735:
14706:
14686:
14654:
14629:
14601:
14573:
14537:
14504:
14480:
14457:
14437:
14396:
14373:
14333:
14305:
14266:
14250:Banach–Alaoglu theorem
14238:
14205:
14184:continuous linear maps
14176:
14124:
14104:
14081:
14052:
14013:
13975:
13948:
13912:
13882:
13854:
13808:
13785:
13750:
13663:
13552:
13454:
13422:
13393:
13345:
13317:
13250:
13230:
13210:
13187:
13160:
13140:
13111:
13084:
13057:
13021:
12997:
12961:
12941:
12921:
12883:
12860:
12824:
12789:
12767:
12744:
12724:
12697:
12646:
12582:
12535:
12503:
12480:
12453:
12433:
12391:
12364:
12337:
12314:
12270:
12245:
12217:
12182:
12162:
12123:
12103:
12064:
12044:
11996:
11976:
11956:
11936:
11901:
11867:
11844:
11824:
11804:
11773:
11728:
11708:
11685:
11654:
11609:
11568:
11533:
11513:
11491:
11430:
11398:
11340:
11320:
11288:
11274:is a ball centered at
11268:
11240:
11214:
11189:
11169:
11149:
11123:
11054:
11032:
10992:
10970:
10950:
10921:
10901:
10867:
10839:
10807:
10767:
10747:
10715:
10683:
10664:
10633:
10604:
10566:
10546:
10517:
10493:
10464:
10444:
10412:
10390:
10366:
10344:
10322:
10300:
10255:
10229:
10155:
10130:
10101:
10066:
10043:
10023:
10000:
9973:
9972:{\displaystyle f\in X}
9947:
9929:(or more generally, a
9919:
9865:
9834:
9793:
9753:
9729:
9707:
9671:
9621:
9590:
9570:
9527:
9500:
9480:
9460:
9439:finest vector topology
9429:
9406:
9381:Finest vector topology
9372:
9319:
9299:
9279:
9230:
9197:
9174:
9141:
9121:
9106:
9071:
9048:
9022:
8998:
8975:
8955:
8935:
8871:
8842:
8816:
8770:
8750:
8727:
8703:
8593:
8570:
8546:
8495:
8472:
8439:
8406:
8345:
8312:
8289:
8288:{\displaystyle \tau ,}
8266:
8246:
8215:
8182:
8163:basis of neighborhoods
8152:
8132:
8102:
8054:
8031:
8011:
7991:
7968:
7942:
7919:
7899:
7875:
7855:
7823:
7803:
7783:
7763:
7740:
7711:
7691:
7665:
7645:
7613:
7586:
7555:
7523:
7483:
7449:
7417:
7360:
7324:
7284:
7245:
7225:
7196:
7195:{\displaystyle r>0}
7170:
7141:
7112:
7092:
7064:
7044:
7024:
7000:
6974:
6973:{\displaystyle 0\in S}
6948:
6878:
6849:
6848:{\displaystyle x\in X}
6820:
6788:
6787:{\displaystyle X\to X}
6762:
6733:
6704:
6703:{\displaystyle X\to X}
6678:
6648:
6611:
6572:
6571:{\displaystyle X\to X}
6546:
6491:
6468:
6441:
6440:{\displaystyle \{0\}.}
6412:
6392:
6364:
6344:
6320:
6292:
6260:
6239:but a TVS need not be
6225:
6186:
6148:
6116:
6090:
6071:topology generated by
6063:
6034:
6007:
5985:
5957:
5937:
5915:
5888:
5867:
5828:
5806:
5786:
5764:
5716:
5691:
5613:
5587:
5565:
5525:
5485:
5450:
5399:
5361:
5339:
5316:
5291:
5189:
5087:
5005:
4930:
4849:
4826:
4786:
4743:
4714:
4684:
4664:
4644:
4581:
4508:
4485:
4429:
4409:
4382:
4362:
4342:
4315:
4295:
4266:
4230:
4210:
4163:
4136:
4085:
4058:
4023:
3977:
3951:
3919:
3854:
3832:
3656:
3623:
3622:{\displaystyle f(x)=1}
3588:
3542:
3341:
3305:
3279:
3214:
3181:
3110:
3068:
3047:
3025:
2989:
2940:
2913:
2890:
2867:
2840:
2802:
2766:
2765:{\displaystyle U_{i}.}
2693:
2670:
2594:
2567:
2533:
2506:
2474:
2453:
2427:
2404:
2369:
2342:
2319:
2253:
2219:
2196:
2168:
2141:
2106:
2076:
2046:
2012:
1974:
1950:
1927:
1903:
1879:
1847:
1823:
1781:
1742:
1741:{\displaystyle X\to X}
1716:
1668:
1632:
1606:
1562:
1526:
1500:
1477:
1453:
1417:
1393:
1358:
1328:
1291:
1157:
1130:
1107:
1059:
1021:
952:
924:
898:
852:
815:
791:Category and morphisms
750:
712:
665:
625:
566:
537:
512:
430:
407:
385:
338:
291:
247:
171:
143:
35934:Complemented subspace
35748:hyperplane separation
35576:Banach–Mazur distance
35539:Generalized functions
34448:Locally convex spaces
33565:
33543:
33391:
33356:
33354:{\displaystyle y=z+x}
33324:
33155:
33135:
33109:
33058:
33038:
33018:
32979:
32959:
32886:
32757:
32722:
32674:
32651:
32636:is an open subset of
32631:
32592:
32544:
32499:
32464:
32444:
32369:
32337:
32263:
32243:
32204:
32172:
32084:
32052:
32013:
31943:
31917:
31833:
31798:
31766:
31678:
31649:
31623:
31565:
31493:
31473:
31420:
31349:
31347:{\displaystyle C,x,y}
31317:
31281:
31201:
31161:
31094:
31047:
31012:
30971:
30946:
30924:
30884:
30859:
30827:
30807:
30771:
30751:
30711:
30688:
30643:
30623:
30573:
30528:
30470:
30446:
30410:
30359:
30339:
30337:{\displaystyle \{0\}}
30313:
30284:
30242:Topological semigroup
30175:Locally compact group
30170:Locally compact field
30098:
29940:
29737:
28774:Closed balanced hull
28620:Closed balanced hull
27011:
26863:
26674:
26672:{\displaystyle \cap }
26500:
26498:{\displaystyle \cap }
26476:
26283:
26059:
26057:{\displaystyle \cup }
25850:
25848:{\displaystyle \cup }
25826:
25618:
25570:
25414:
25394:
25372:
25337:
25308:
25249:
25223:
25203:
25183:
25157:
25051:
25025:
24925:
24896:
24864:
24835:
24741:
24712:
24682:
24632:
24609:
24584:
24525:
24469:
24434:
24402:
24240:
24220:
24185:
24152:
24063:
24022:
23995:
23963:
23931:
23881:
23838:
23785:
23762:
23722:
23678:
23654:
23634:
23599:
23557:
23507:
23479:
23459:
23436:
23412:
23388:
23329:Hulls and compactness
23321:
23177:
23132:
23087:
23055:
22927:
22907:
22887:
22852:
22798:
22748:
22689:
22666:
22646:
22601:
22581:
22553:
22473:
22450:
22426:
22350:
22318:
22298:
22275:
22217:
22188:
22166:
22104:is closed then so is
22099:
22073:
21854:
21819:
21786:
21722:
21629:
21589:
21569:
21543:
21481:then equality holds:
21476:
21433:
21413:
21346:
21326:
21297:
21274:
21254:
21234:
21208:
21188:
21168:
21148:
21113:
21078:
21058:
21035:
21020:is an open subset of
21015:
20995:
20975:
20952:
20932:
20912:
20892:
20877:is a subset of a TVS
20872:
20848:
20789:
20752:
20732:
20692:
20638:
20593:
20569:
20545:
20506:
20486:
20442:
20388:
20368:
20329:
20309:
20263:
20212:
20155:
20135:
20096:
20069:
20007:
19984:
19964:
19936:
19916:
19892:
19833:
19810:
19775:
19724:
19701:
19678:
19652:
19609:
19581:
19530:
19515:is open or closed in
19510:
19489:and consequently, if
19484:
19421:
19381:
19317:
19289:
19247:
19219:
19195:
19156:
19127:
19096:
19053:
19010:
18971:
18951:
18877:
18850:
18808:
18785:
18783:{\displaystyle \{0\}}
18759:
18705:
18682:
18680:{\displaystyle \{0\}}
18656:
18629:
18579:
18553:
18489:
18401:
18305:
18280:
18256:
18225:
18153:
18133:
18109:
18089:
18034:
17918:
17876:
17850:
17828:
17792:
17772:
17770:{\displaystyle B_{1}}
17745:
17649:
17629:
17601:
17511:
17488:
17468:
17448:
17409:
17367:
17347:
17327:
17250:
17215:
17193:
17137:
17078:
17040:
17020:
16981:
16961:
16922:
16889:
16844:
16824:
16781:
16751:
16622:
16466:
16446:
16406:
16386:
16351:
16319:
16296:
16273:
16240:
16216:
16196:
16131:
16092:
16063:
16034:
16011:
15985:
15950:
15917:
15885:
15865:
15845:
15784:
15758:
15735:
15715:
15713:{\displaystyle \tau }
15692:
15672:
15652:
15628:
15468:
15444:
15411:
15384:
15362:
15339:
15316:
15290:
15180:
15157:
15131:
15105:
15085:
15059:
15044:is an open subset of
15039:
15019:
14968:
14948:
14884:
14832:
14806:
14736:
14707:
14687:
14655:
14630:
14602:
14574:
14538:
14505:
14481:
14458:
14438:
14397:
14374:
14334:
14306:
14267:
14252:). Caution: Whenever
14239:
14206:
14177:
14155:continuous dual space
14145:Continuous dual space
14125:
14105:
14082:
14053:
14014:
13976:
13974:{\displaystyle L^{2}}
13949:
13947:{\displaystyle L^{2}}
13913:
13883:
13855:
13853:{\displaystyle L^{1}}
13809:
13786:
13751:
13749:{\displaystyle L^{p}}
13664:
13553:
13491:reflexivity condition
13455:
13423:
13394:
13392:{\displaystyle L^{p}}
13366:Minkowski functionals
13346:
13318:
13316:{\displaystyle L^{p}}
13251:
13231:
13211:
13188:
13161:
13141:
13112:
13085:
13058:
13022:
12998:
12974:Non-vector topologies
12962:
12942:
12922:
12884:
12866:vector topologies on
12861:
12832:up to TVS-isomorphism
12825:
12790:
12768:
12745:
12725:
12710:) vector topology on
12698:
12647:
12583:
12536:
12504:
12481:
12454:
12434:
12392:
12365:
12363:{\displaystyle M_{x}}
12338:
12315:
12271:
12246:
12218:
12183:
12163:
12124:
12104:
12065:
12045:
11997:
11977:
11957:
11937:
11902:
11868:
11845:
11825:
11805:
11774:
11729:
11709:
11686:
11655:
11610:
11569:
11534:
11514:
11492:
11431:
11399:
11341:
11321:
11289:
11269:
11241:
11215:
11190:
11170:
11150:
11124:
11055:
11033:
10993:
10971:
10951:
10922:
10902:
10868:
10840:
10808:
10768:
10748:
10716:
10684:
10665:
10634:
10605:
10567:
10547:
10518:
10494:
10465:
10445:
10413:
10391:
10367:
10345:
10323:
10301:
10256:
10230:
10177:and consequently not
10156:
10131:
10102:
10067:
10044:
10024:
10001:
9999:{\displaystyle f_{n}}
9974:
9948:
9920:
9866:
9844:pointwise convergence
9835:
9794:
9754:
9730:
9708:
9672:
9622:
9591:
9571:
9528:
9501:
9481:
9461:
9430:
9407:
9373:
9320:
9300:
9280:
9231:
9198:
9175:
9142:
9122:
9086:
9072:
9054:then it converges to
9049:
9023:
8999:
8976:
8956:
8936:
8882:sequentially complete
8872:
8843:
8817:
8771:
8751:
8728:
8704:
8594:
8571:
8569:{\displaystyle \tau }
8547:
8496:
8473:
8440:
8407:
8346:
8313:
8290:
8267:
8247:
8216:
8183:
8153:
8133:
8131:{\displaystyle \{0\}}
8103:
8055:
8032:
8012:
7992:
7969:
7943:
7920:
7900:
7876:
7856:
7824:
7804:
7784:
7764:
7741:
7712:
7692:
7666:
7646:
7614:
7587:
7568:and contains an open
7556:
7554:{\displaystyle -E=E.}
7524:
7484:
7450:
7418:
7361:
7325:
7285:
7246:
7226:
7197:
7171:
7142:
7113:
7093:
7065:
7045:
7025:
7001:
6975:
6949:
6879:
6850:
6821:
6789:
6763:
6734:
6705:
6679:
6649:
6612:
6573:
6547:
6508:translation invariant
6492:
6469:
6442:
6413:
6393:
6365:
6345:
6321:
6293:
6261:
6226:
6204:A vector space is an
6200:Topological structure
6187:
6149:
6117:
6091:
6069:and it is called the
6064:
6035:
6008:
5986:
5958:
5938:
5916:
5889:
5887:{\displaystyle \tau }
5868:
5829:
5807:
5787:
5765:
5717:
5692:
5614:
5588:
5566:
5526:
5486:
5451:
5400:
5362:
5340:
5317:
5292:
5190:
5088:
5006:
4931:
4850:
4827:
4787:
4744:
4715:
4685:
4665:
4645:
4582:
4509:
4486:
4430:
4410:
4383:
4363:
4343:
4341:{\displaystyle U_{i}}
4316:
4296:
4267:
4231:
4211:
4164:
4162:{\displaystyle U_{i}}
4137:
4086:
4084:{\displaystyle U_{i}}
4059:
4024:
3978:
3952:
3920:
3855:
3833:
3657:
3624:
3589:
3543:
3342:
3306:
3280:
3215:
3182:
3111:
3069:
3048:
3026:
2990:
2941:
2914:
2891:
2868:
2841:
2803:
2767:
2694:
2671:
2595:
2568:
2534:
2532:{\displaystyle U_{1}}
2507:
2475:
2454:
2452:{\displaystyle U_{i}}
2428:
2405:
2370:
2343:
2320:
2254:
2220:
2197:
2169:
2142:
2107:
2077:
2047:
2013:
1975:
1951:
1928:
1904:
1880:
1848:
1824:
1782:
1743:
1717:
1669:
1633:
1607:
1563:
1527:
1501:
1478:
1476:{\displaystyle \tau }
1454:
1452:{\displaystyle (X,+)}
1418:
1394:
1359:
1329:
1292:
1198:topological embedding
1158:
1131:
1108:
1060:
1022:
953:
925:
899:
853:
816:
751:
713:
666:
626:
567:
538:
506:
458:holomorphic functions
431:
408:
386:
339:
292:
248:
198:: the norm induces a
196:topological structure
172:
144:
120:In this article, the
36184:Locally convex space
35734:Closed graph theorem
35686:Locally convex space
35321:Kakutani fixed-point
35306:Riesz representation
35122:at Wikimedia Commons
33554:
33400:
33365:
33333:
33172:
33144:
33118:
33067:
33047:
33027:
32988:
32968:
32895:
32766:
32740:
32692:
32660:
32640:
32601:
32553:
32508:
32473:
32453:
32378:
32346:
32272:
32252:
32213:
32181:
32093:
32061:
32022:
31952:
31926:
31842:
31807:
31775:
31686:
31658:
31632:
31574:
31502:
31482:
31429:
31358:
31326:
31290:
31210:
31178:
31103:
31064:
31021:
30999:
30955:
30933:
30896:
30868:
30836:
30816:
30780:
30760:
30728:
30697:
30652:
30632:
30582:
30537:
30492:
30459:
30407:
30348:
30344:is closed (that is,
30322:
30302:
30273:
30087:
29929:
29925:Non-empty subset of
29726:
29722:Non-empty subset of
28614:Convex balanced hull
28458:Convex balanced hull
26994:
26846:
26663:
26489:
26459:
26266:
26048:
25839:
25809:
25601:
25559:
25403:
25380:
25358:
25320:
25297:
25232:
25212:
25192:
25166:
25060:
25034:
24933:
24905:
24873:
24862:{\displaystyle P(x)}
24844:
24749:
24721:
24695:
24680:{\displaystyle P(x)}
24662:
24618:
24595:
24534:
24478:
24443:
24411:
24249:
24229:
24197:
24165:
24072:
24031:
24008:
23978:
23972:convex balanced hull
23940:
23890:
23847:
23794:
23774:
23730:
23687:
23683:be non-zero and set
23667:
23643:
23608:
23582:
23516:
23496:
23468:
23448:
23425:
23401:
23377:
23186:
23141:
23096:
23064:
22936:
22916:
22896:
22864:
22813:
22759:
22709:
22675:
22655:
22610:
22590:
22562:
22482:
22459:
22435:
22358:
22330:
22307:
22284:
22226:
22197:
22177:
22108:
22082:
21863:
21831:
21795:
21733:
21638:
21598:
21578:
21552:
21485:
21442:
21422:
21355:
21335:
21309:
21283:
21263:
21243:
21217:
21197:
21177:
21157:
21137:
21087:
21067:
21044:
21024:
21004:
20984:
20964:
20957:is totally bounded.
20941:
20921:
20901:
20881:
20861:
20853:is totally bounded.
20798:
20765:
20741:
20721:
20647:
20602:
20582:
20558:
20515:
20495:
20451:
20397:
20377:
20338:
20318:
20276:
20221:
20164:
20144:
20105:
20085:
20019:
19993:
19973:
19945:
19925:
19905:
19842:
19822:
19786:
19739:
19713:
19687:
19661:
19622:
19598:
19539:
19519:
19493:
19429:
19401:
19326:
19300:
19256:
19236:
19226:compact and complete
19208:
19165:
19136:
19113:
19066:
19023:
18980:
18960:
18886:
18863:
18817:
18794:
18768:
18714:
18691:
18665:
18645:
18588:
18562:
18498:
18410:
18406:and furthermore, if
18314:
18289:
18269:
18234:
18162:
18142:
18122:
18098:
18043:
17927:
17885:
17859:
17839:
17801:
17781:
17754:
17658:
17638:
17612:
17520:
17497:
17477:
17457:
17418:
17376:
17356:
17336:
17259:
17224:
17204:
17146:
17087:
17049:
17029:
16990:
16970:
16931:
16898:
16853:
16833:
16794:
16770:
16760:topological interior
16631:
16475:
16455:
16423:
16395:
16360:
16340:
16308:
16282:
16249:
16225:
16205:
16140:
16101:
16072:
16043:
16020:
15996:
15959:
15926:
15894:
15874:
15854:
15793:
15767:
15744:
15733:{\displaystyle \nu }
15724:
15704:
15681:
15661:
15641:
15477:
15457:
15451:Minkowski functional
15420:
15400:
15373:
15348:
15328:
15323:sublinear functional
15299:
15189:
15169:
15140:
15114:
15094:
15068:
15048:
15028:
15002:
14957:
14893:
14841:
14815:
14749:
14722:
14696:
14670:
14638:
14610:
14582:
14547:
14514:
14494:
14467:
14447:
14427:
14383:
14357:
14315:
14276:
14256:
14215:
14190:
14161:
14141:Algebraic dual space
14114:
14091:
14062:
14033:
13987:
13958:
13931:
13892:
13865:
13837:
13795:
13760:
13733:
13727:normed vector spaces
13562:
13524:
13495:totally bounded sets
13432:
13403:
13376:
13329:
13300:
13279:open mapping theorem
13275:closed graph theorem
13240:
13220:
13200:
13177:
13150:
13139:{\displaystyle f(X)}
13121:
13101:
13071:
13047:
13011:
12987:
12951:
12931:
12893:
12870:
12838:
12799:
12779:
12754:
12734:
12714:
12656:
12592:
12548:
12513:
12490:
12470:
12443:
12405:
12381:
12347:
12324:
12279:
12255:
12227:
12192:
12172:
12133:
12113:
12078:
12054:
12006:
11986:
11966:
11946:
11911:
11877:
11854:
11834:
11814:
11783:
11738:
11718:
11695:
11664:
11619:
11578:
11543:
11523:
11501:
11440:
11408:
11350:
11330:
11298:
11278:
11250:
11224:
11199:
11179:
11159:
11133:
11064:
11042:
11020:
10980:
10960:
10938:
10911:
10879:
10857:
10817:
10785:
10757:
10725:
10693:
10673:
10651:
10614:
10582:
10572:always has a unique
10556:
10527:
10507:
10474:
10454:
10422:
10418:of finite dimension
10400:
10380:
10354:
10332:
10310:
10288:
10239:
10185:
10140:
10129:{\displaystyle f(x)}
10111:
10075:
10053:
10033:
10013:
9983:
9957:
9937:
9876:
9850:
9807:
9766:
9743:
9717:
9681:
9661:
9604:
9580:
9537:
9510:
9490:
9470:
9447:
9416:
9389:
9350:
9309:
9289:
9257:
9220:
9184:
9151:
9131:
9083:
9058:
9038:
9012:
8988:
8965:
8945:
8925:
8852:
8826:
8780:
8760:
8737:
8717:
8652:
8580:
8560:
8524:
8517:canonical uniformity
8482:
8453:
8427:
8394:
8323:
8299:
8276:
8256:
8236:
8193:
8169:
8142:
8116:
8080:
8041:
8021:
8001:
7978:
7958:
7929:
7909:
7889:
7865:
7833:
7813:
7793:
7773:
7753:
7721:
7701:
7681:
7655:
7635:
7600:
7576:
7533:
7529:or equivalently, if
7501:
7473:
7427:
7377:
7334:
7305:
7255:
7235:
7206:
7180:
7176:there exists a real
7151:
7131:
7102:
7082:
7054:
7034:
7014:
6984:
6958:
6888:
6859:
6833:
6798:
6772:
6761:{\displaystyle s=-1}
6743:
6714:
6688:
6684:then the linear map
6662:
6625:
6582:
6556:
6520:
6478:
6450:
6422:
6402:
6374:
6354:
6334:
6302:
6273:
6250:
6212:
6158:
6126:
6104:
6075:
6044:
6021:
5995:
5967:
5947:
5925:
5905:
5878:
5842:
5816:
5796:
5774:
5742:
5701:
5629:
5601:
5575:
5535:
5495:
5460:
5409:
5387:
5349:
5326:
5304:
5204:
5102:
5020:
4945:
4864:
4836:
4796:
4753:
4726:
4697:
4674:
4654:
4591:
4528:
4495:
4439:
4419:
4392:
4372:
4352:
4325:
4305:
4276:
4240:
4220:
4173:
4146:
4099:
4068:
4033:
3987:
3961:
3929:
3864:
3844:
3666:
3633:
3598:
3554:
3351:
3315:
3289:
3224:
3191:
3125:
3098:
3058:
3037:
2999:
2950:
2930:
2903:
2877:
2850:
2830:
2785:
2746:
2680:
2615:
2577:
2547:
2516:
2486:
2464:
2436:
2414:
2410:and for every index
2387:
2352:
2329:
2263:
2243:
2206:
2186:
2154:
2116:
2086:
2056:
2032:
1992:
1960:
1937:
1917:
1889:
1869:
1833:
1813:
1752:
1726:
1690:
1646:
1616:
1572:
1540:
1532:is endowed with the
1510:
1487:
1467:
1431:
1407:
1368:
1338:
1305:
1277:
1144:
1117:
1073:
1031:
999:
940:
912:
862:
825:
821:is commonly denoted
803:
740:
728:Hausdorff assumption
699:
673:continuous functions
635:
588:
554:
527:
417:
395:
348:
308:
257:
210:
156:
131:
64:continuous functions
36164:Interpolation space
35696:Operator topologies
35505:Functional calculus
35464:Mahler's conjecture
35443:Von Neumann algebra
35157:Functional analysis
34575:Schaefer, Helmut H.
34544:Functional Analysis
34347:, pp. 108–109.
34279:, pp. 107–112.
34154:, pp. 101–104.
34047:, pp. 177–220.
33939:, pp. 115–154.
33831:, pp. 155–176.
33757:, pp. 371–423.
33734:, pp. 721–751.
33592:, p. 4-5 §1.3.
33140:is substituted for
30392:metric linear space
30155:Liquid vector space
28933:Closed convex hull
28779:Closed convex hull
27009:{\displaystyle R+S}
26861:{\displaystyle R+S}
25419:that is considered
25347:
24840:So for example, if
23395:indiscrete topology
22097:{\displaystyle R+S}
20643:is a completion of
20552:indiscrete topology
19962:{\displaystyle X/M}
19897:is totally bounded.
19815:is totally bounded.
19780:is totally bounded.
19657:). For any subset
18310:) it follows that:
15155:{\displaystyle S-S}
15083:{\displaystyle S+U}
14743:linear combinations
14715:convex combinations
14332:{\displaystyle X'.}
13370:Hahn–Banach theorem
12050:which implies that
11830:has a neighborhood
11478:
11393:
11004:functional analysis
10647:vector topology on
9914:
9596:has an uncountable
9252:indiscrete topology
9127:converges in a TVS
9004:is not necessarily
8613:, Cauchy nets, and
8611:uniform convergence
8074: —
7954:set is bounded. If
6999:{\displaystyle x+S}
6467:{\displaystyle X/M}
6391:{\displaystyle X/M}
6319:{\displaystyle X/M}
6298:the quotient space
6192:A Hausdorff TVS is
5736: —
3976:{\displaystyle f=0}
3467: for all
3176:
3119: —
3031:This is called the
2822:neighborhood string
2314:
1807: —
1425: —
301:obeyed by the norm.
299:triangle inequality
192:normed vector space
76:uniform convergence
52:functional analysis
36340:Topological spaces
36194:(Pseudo)Metrizable
36026:Minkowski addition
35878:Sublinear function
35530:Riemann hypothesis
35229:Topological vector
34814:Wiley-Interscience
34804:Schwartz, Jacob T.
34088:, p. 119-120.
33701:, pp. 67–113.
33560:
33538:
33386:
33361:and the fact that
33351:
33319:
33150:
33130:
33104:
33053:
33033:
33013:
32974:
32954:
32881:
32752:
32717:
32672:{\displaystyle C.}
32669:
32646:
32626:
32587:
32549:which proves that
32539:
32494:
32459:
32439:
32364:
32332:
32258:
32238:
32199:
32167:
32079:
32047:
32008:
31983:
31938:
31912:
31890:
31861:
31828:
31793:
31761:
31740:
31717:
31673:
31644:
31618:
31601:
31560:
31552:
31526:
31488:
31468:
31415:
31344:
31312:
31276:
31241:
31196:
31156:
31089:
31042:
31007:
30966:
30941:
30919:
30879:
30854:
30832:-axis, is open in
30822:
30802:
30766:
30746:
30709:{\displaystyle X.}
30706:
30683:
30638:
30618:
30568:
30523:
30465:
30441:
30354:
30334:
30308:
30279:
30228:Topological module
30093:
30040:Relatively compact
29935:
29732:
27006:
26858:
26669:
26495:
26471:
26278:
26054:
25845:
25821:
25613:
25565:
25512:Relatively compact
25409:
25392:{\displaystyle S,}
25389:
25370:{\displaystyle R,}
25367:
25345:
25332:
25303:
25244:
25218:
25198:
25178:
25152:
25134:
25046:
25020:
24920:
24891:
24859:
24830:
24736:
24707:
24677:
24630:{\displaystyle X.}
24627:
24607:{\displaystyle X,}
24604:
24579:
24520:
24464:
24429:
24397:
24235:
24215:
24180:
24147:
24058:
24020:{\displaystyle S;}
24017:
23990:
23958:
23926:
23876:
23833:
23780:
23757:
23717:
23673:
23649:
23629:
23594:
23552:
23542:
23502:
23474:
23454:
23431:
23407:
23383:
23316:
23182:are compact) then
23172:
23127:
23082:
23050:
22922:
22902:
22882:
22847:
22793:
22743:
22687:{\displaystyle X.}
22684:
22661:
22641:
22596:
22576:
22548:
22471:{\displaystyle X.}
22468:
22445:
22421:
22405:
22345:
22313:
22296:{\displaystyle X;}
22293:
22270:
22244:
22212:
22193:is a real TVS and
22183:
22161:
22094:
22068:
21849:
21814:
21781:
21717:
21634:contain zero then
21624:
21584:
21564:
21538:
21471:
21428:
21408:
21341:
21321:
21295:{\displaystyle X.}
21292:
21269:
21249:
21229:
21203:
21183:
21163:
21143:
21108:
21073:
21056:{\displaystyle K,}
21053:
21030:
21010:
20990:
20970:
20947:
20927:
20907:
20887:
20867:
20843:
20784:
20747:
20727:
20687:
20633:
20588:
20564:
20540:
20501:
20481:
20437:
20383:
20363:
20324:
20304:
20258:
20207:
20150:
20130:
20091:
20064:
20005:{\displaystyle X.}
20002:
19979:
19959:
19931:
19911:
19887:
19828:
19805:
19770:
19719:
19699:{\displaystyle X,}
19696:
19673:
19647:
19604:
19576:
19525:
19505:
19479:
19416:
19392:topological spaces
19388:relatively compact
19376:
19312:
19284:
19242:
19214:
19190:
19151:
19125:{\displaystyle X.}
19122:
19091:
19048:
19015:as a subset. The
19005:
18966:
18946:
18942:
18875:{\displaystyle X.}
18872:
18845:
18806:{\displaystyle X,}
18803:
18780:
18754:
18703:{\displaystyle X,}
18700:
18677:
18651:
18624:
18574:
18548:
18484:
18396:
18300:
18275:
18261:(which are convex
18251:
18220:
18148:
18128:
18104:
18084:
18029:
17913:
17871:
17845:
17823:
17787:
17767:
17740:
17727:
17644:
17624:
17596:
17509:{\displaystyle C;}
17506:
17483:
17463:
17443:
17404:
17362:
17342:
17322:
17245:
17210:
17188:
17132:
17073:
17035:
17025:could be false if
17015:
16976:
16956:
16917:
16884:
16839:
16819:
16776:
16746:
16617:
16461:
16441:
16401:
16381:
16346:
16314:
16294:{\displaystyle X.}
16291:
16268:
16235:
16211:
16191:
16126:
16087:
16058:
16032:{\displaystyle X,}
16029:
16006:
15980:
15945:
15912:
15880:
15860:
15840:
15779:
15756:{\displaystyle X.}
15753:
15730:
15710:
15687:
15667:
15647:
15623:
15463:
15439:
15406:
15379:
15360:{\displaystyle X.}
15357:
15334:
15311:
15285:
15175:
15152:
15126:
15100:
15090:is an open set in
15080:
15054:
15034:
15014:
14966:{\displaystyle 1.}
14963:
14943:
14879:
14827:
14801:
14734:{\displaystyle S,}
14731:
14702:
14682:
14650:
14625:
14597:
14569:
14533:
14500:
14479:{\displaystyle S.}
14476:
14453:
14433:
14395:{\displaystyle X,}
14392:
14369:
14329:
14301:
14262:
14234:
14201:
14175:{\displaystyle X'}
14172:
14120:
14110:there is only one
14103:{\displaystyle n,}
14100:
14077:
14048:
14009:
13971:
13944:
13908:
13878:
13850:
13807:{\displaystyle BV}
13804:
13781:
13746:
13691:of Hilbert spaces.
13659:
13620:
13548:
13473:Bornological space
13450:
13418:
13389:
13341:
13313:
13246:
13226:
13206:
13183:
13156:
13136:
13107:
13083:{\displaystyle X.}
13080:
13067:a TVS topology on
13053:
13017:
12993:
12957:
12937:
12917:
12882:{\displaystyle X.}
12879:
12856:
12820:
12785:
12766:{\displaystyle X.}
12763:
12740:
12720:
12693:
12642:
12578:
12531:
12502:{\displaystyle X,}
12499:
12476:
12449:
12429:
12387:
12360:
12336:{\displaystyle X,}
12333:
12310:
12266:
12241:
12223:then for any ball
12213:
12178:
12158:
12119:
12099:
12060:
12040:
11992:
11972:
11952:
11932:
11897:
11866:{\displaystyle X,}
11863:
11840:
11820:
11800:
11769:
11734:can be written as
11724:
11707:{\displaystyle x,}
11704:
11681:
11650:
11605:
11564:
11529:
11509:
11487:
11443:
11426:
11394:
11353:
11336:
11316:
11284:
11264:
11236:
11210:
11185:
11165:
11145:
11119:
11050:
11028:
11014:
10988:
10966:
10946:
10929:Euclidean topology
10917:
10897:
10866:{\displaystyle 0.}
10863:
10835:
10803:
10763:
10743:
10711:
10679:
10663:{\displaystyle X.}
10660:
10629:
10600:
10562:
10542:
10513:
10501:linear isomorphism
10489:
10460:
10440:
10408:
10386:
10374:Euclidean topology
10362:
10340:
10318:
10296:
10271:F. Riesz's theorem
10251:
10225:
10151:
10126:
10097:
10065:{\displaystyle x,}
10062:
10039:
10019:
9996:
9969:
9943:
9915:
9879:
9861:
9830:
9789:
9749:
9737:Euclidean topology
9735:carries its usual
9725:
9703:
9667:
9645:Cartesian products
9617:
9586:
9566:
9523:
9496:
9476:
9459:{\displaystyle X,}
9456:
9428:{\displaystyle X,}
9425:
9402:
9368:
9315:
9295:
9275:
9226:
9196:{\displaystyle X.}
9193:
9170:
9137:
9117:
9070:{\displaystyle x.}
9067:
9044:
9033:accumulation point
9018:
9006:relatively compact
8994:
8971:
8961:that is closed in
8951:
8931:
8867:
8838:
8812:
8766:
8749:{\displaystyle 0,}
8746:
8723:
8699:
8635:relatively compact
8615:uniform continuity
8592:{\displaystyle X.}
8589:
8566:
8542:
8494:{\displaystyle n.}
8491:
8468:
8447:finite-dimensional
8435:
8402:
8341:
8311:{\displaystyle X.}
8308:
8285:
8262:
8242:
8211:
8181:{\displaystyle X.}
8178:
8148:
8128:
8098:
8072:
8053:{\displaystyle X.}
8050:
8027:
8007:
7990:{\displaystyle X,}
7987:
7964:
7941:{\displaystyle E.}
7938:
7915:
7895:
7871:
7851:
7819:
7799:
7779:
7759:
7736:
7707:
7687:
7661:
7641:
7612:{\displaystyle 0;}
7609:
7582:
7551:
7519:
7479:
7445:
7413:
7356:
7320:
7280:
7241:
7221:
7192:
7166:
7137:
7108:
7098:of a vector space
7088:
7060:
7040:
7020:
6996:
6970:
6944:
6874:
6845:
6816:
6784:
6758:
6729:
6700:
6674:
6644:
6607:
6568:
6542:
6490:{\displaystyle X.}
6487:
6464:
6437:
6418:is the closure of
6408:
6388:
6360:
6340:
6316:
6288:
6256:
6237:completely regular
6224:{\displaystyle -1}
6221:
6182:
6144:
6112:
6086:
6059:
6033:{\displaystyle X.}
6030:
6015:neighborhood basis
6003:
5981:
5953:
5933:
5911:
5884:
5863:
5838:at the origin for
5836:neighborhood basis
5824:
5802:
5782:
5760:
5730:
5712:
5687:
5670:
5609:
5583:
5561:
5521:
5481:
5456:there exists some
5446:
5395:
5357:
5338:{\displaystyle X,}
5335:
5312:
5287:
5185:
5083:
5001:
4987:
4926:
4848:{\displaystyle i.}
4845:
4822:
4782:
4739:
4710:
4680:
4660:
4640:
4577:
4507:{\displaystyle X.}
4504:
4481:
4425:
4405:
4378:
4358:
4338:
4311:
4291:
4262:
4226:
4206:
4169:are balanced then
4159:
4132:
4081:
4054:
4029:so in particular,
4019:
4005:
3973:
3947:
3915:
3850:
3828:
3690:
3662:and otherwise let
3652:
3619:
3584:
3538:
3337:
3301:
3275:
3210:
3177:
3141:
3106:
3091:
3064:
3043:
3033:natural string of
3021:
2985:
2936:
2926:in a vector space
2909:
2889:{\displaystyle X.}
2886:
2863:
2836:
2814:Topological string
2798:
2762:
2692:{\displaystyle i.}
2689:
2666:
2590:
2563:
2529:
2502:
2470:
2449:
2426:{\displaystyle i,}
2423:
2400:
2365:
2341:{\displaystyle X.}
2338:
2315:
2279:
2249:
2218:{\displaystyle X.}
2215:
2192:
2164:
2137:
2102:
2072:
2042:
2008:
1970:
1949:{\displaystyle X.}
1946:
1923:
1899:
1875:
1843:
1819:
1801:
1793:neighborhood basis
1777:
1738:
1712:
1664:
1628:
1602:
1558:
1522:
1499:{\displaystyle X,}
1496:
1473:
1449:
1423:
1413:
1389:
1354:
1334:there exists some
1324:
1287:
1156:{\displaystyle Y.}
1153:
1129:{\displaystyle u,}
1126:
1103:
1055:
1017:
948:
920:
894:
848:
811:
746:
711:{\displaystyle X.}
708:
677:product topologies
661:
621:
562:
533:
513:
472:and the spaces of
429:{\displaystyle X,}
426:
403:
381:
334:
287:
243:
167:
139:
36322:
36321:
36041:Relative interior
35787:Bilinear operator
35671:Linear functional
35607:
35606:
35510:Integral operator
35287:
35286:
35118:Media related to
35091:978-3-030-32945-7
35057:978-0-08-087178-3
35050:Science Pub. Co.
35023:978-0-486-45352-1
34993:978-0-521-29882-7
34936:978-0-387-90400-9
34884:978-0-677-30020-7
34850:978-0-486-68143-6
34823:978-0-471-60848-6
34783:978-0-387-97245-9
34728:Bourbaki, Nicolas
34679:978-0-486-49353-4
34649:978-0-8247-8643-4
34622:978-0-12-622760-4
34592:978-1-4612-7155-0
34558:978-0-07-054236-5
34487:978-3-642-64988-2
34457:978-3-519-02224-4
34430:978-3-540-08662-8
34359:, pp. 30–32.
34323:, pp. 43–44.
34306:, pp. 19–45.
34267:, pp. 56–73.
34237:, pp. 12–35.
34204:, pp. 47–66.
34008:, p. 9 §1.8.
33951:, pp. 27–29.
33910:, pp. 12–19.
33798:, p. 6 §1.4.
33769:, pp. 10–15.
33650:, pp. 40–47.
33633:, pp. 34–36.
33631:Grothendieck 1973
33621:, pp. 74–78.
33396:this is equal to
33153:{\displaystyle S}
33056:{\displaystyle S}
33036:{\displaystyle X}
32977:{\displaystyle S}
32649:{\displaystyle X}
32462:{\displaystyle C}
32261:{\displaystyle X}
31982:
31889:
31860:
31739:
31716:
31600:
31551:
31533:
31528:
31524:
31510:
31491:{\displaystyle C}
31248:
31243:
31239:
31225:
30951:so not closed in
30912:
30825:{\displaystyle y}
30797:
30769:{\displaystyle y}
30641:{\displaystyle S}
30468:{\displaystyle X}
30357:{\displaystyle X}
30311:{\displaystyle X}
30282:{\displaystyle X}
30222:Topological group
30213:Topological field
30125:
30124:
30096:{\displaystyle X}
29938:{\displaystyle R}
29735:{\displaystyle R}
25568:{\displaystyle X}
25412:{\displaystyle X}
25306:{\displaystyle X}
25221:{\displaystyle z}
25201:{\displaystyle X}
25133:
25056:is a scalar then
24360:
24356:
24352:
24301:
24297:
24293:
24245:is a scalar then
24238:{\displaystyle a}
24122:
24116:
24104:
24098:
23862:
23783:{\displaystyle C}
23676:{\displaystyle x}
23652:{\displaystyle S}
23525:
23505:{\displaystyle D}
23477:{\displaystyle X}
23457:{\displaystyle X}
23434:{\displaystyle X}
23410:{\displaystyle X}
23386:{\displaystyle X}
23237:
23231:
22987:
22981:
22925:{\displaystyle R}
22905:{\displaystyle S}
22671:of the origin in
22664:{\displaystyle U}
22599:{\displaystyle S}
22540:
22509:
22503:
22386:
22385:
22379:
22316:{\displaystyle S}
22229:
22186:{\displaystyle X}
22039:
22033:
21960:
21956:
21952:
21921:
21915:
21748:
21613:
21587:{\displaystyle A}
21460:
21431:{\displaystyle X}
21351:is a scalar then
21344:{\displaystyle a}
21272:{\displaystyle M}
21252:{\displaystyle X}
21206:{\displaystyle X}
21186:{\displaystyle N}
21166:{\displaystyle X}
21146:{\displaystyle M}
21076:{\displaystyle N}
21033:{\displaystyle X}
21013:{\displaystyle U}
20993:{\displaystyle X}
20973:{\displaystyle K}
20950:{\displaystyle S}
20930:{\displaystyle S}
20910:{\displaystyle S}
20890:{\displaystyle X}
20870:{\displaystyle S}
20750:{\displaystyle X}
20730:{\displaystyle S}
20591:{\displaystyle H}
20567:{\displaystyle C}
20504:{\displaystyle H}
20386:{\displaystyle X}
20327:{\displaystyle H}
20314:Consequently, if
20153:{\displaystyle H}
20094:{\displaystyle X}
19982:{\displaystyle M}
19934:{\displaystyle X}
19914:{\displaystyle M}
19831:{\displaystyle S}
19722:{\displaystyle S}
19607:{\displaystyle S}
19528:{\displaystyle X}
19397:For every subset
19322:is compact, then
19245:{\displaystyle X}
19217:{\displaystyle X}
19017:subspace topology
18969:{\displaystyle X}
18914:
18654:{\displaystyle X}
18513:
18431:
18278:{\displaystyle 0}
18265:neighborhoods of
18151:{\displaystyle X}
18131:{\displaystyle 0}
18107:{\displaystyle N}
18070:
18018:
17848:{\displaystyle x}
17790:{\displaystyle a}
17696:
17647:{\displaystyle X}
17486:{\displaystyle y}
17466:{\displaystyle x}
17365:{\displaystyle X}
17345:{\displaystyle C}
17309:
17303:
17213:{\displaystyle C}
17038:{\displaystyle S}
16979:{\displaystyle S}
16842:{\displaystyle X}
16779:{\displaystyle S}
16689:
16683:
16577:
16571:
16552:
16548:
16544:
16502:
16496:
16464:{\displaystyle S}
16404:{\displaystyle X}
16349:{\displaystyle X}
16317:{\displaystyle X}
16214:{\displaystyle S}
15883:{\displaystyle 0}
15863:{\displaystyle X}
15690:{\displaystyle K}
15670:{\displaystyle p}
15650:{\displaystyle K}
15606:
15600:
15561:
15555:
15549:
15543:
15504:
15498:
15466:{\displaystyle K}
15409:{\displaystyle X}
15382:{\displaystyle K}
15337:{\displaystyle p}
15242:
15236:
15178:{\displaystyle X}
15103:{\displaystyle X}
15057:{\displaystyle X}
15037:{\displaystyle U}
14996:arcwise connected
14992:locally connected
14741:which are finite
14705:{\displaystyle S}
14503:{\displaystyle S}
14456:{\displaystyle X}
14436:{\displaystyle S}
14265:{\displaystyle X}
14149:Strong dual space
14123:{\displaystyle n}
13709:seminormed spaces
13590:
13249:{\displaystyle f}
13229:{\displaystyle X}
13209:{\displaystyle f}
13195:linear functional
13186:{\displaystyle X}
13159:{\displaystyle X}
13117:is continuous if
13110:{\displaystyle f}
13056:{\displaystyle X}
13041:cofinite topology
13037:topological group
13027:(which is always
13020:{\displaystyle X}
13005:discrete topology
12996:{\displaystyle X}
12960:{\displaystyle X}
12940:{\displaystyle X}
12889:For instance, if
12788:{\displaystyle X}
12743:{\displaystyle X}
12723:{\displaystyle X}
12479:{\displaystyle f}
12452:{\displaystyle X}
12181:{\displaystyle X}
12122:{\displaystyle n}
12063:{\displaystyle S}
11995:{\displaystyle X}
11982:of the origin in
11975:{\displaystyle S}
11955:{\displaystyle 0}
11843:{\displaystyle W}
11823:{\displaystyle X}
11727:{\displaystyle X}
11532:{\displaystyle X}
11339:{\displaystyle S}
11287:{\displaystyle 0}
11188:{\displaystyle 1}
11168:{\displaystyle X}
11012:
10969:{\displaystyle 1}
10920:{\displaystyle X}
10777:vector topology.
10766:{\displaystyle X}
10682:{\displaystyle X}
10565:{\displaystyle X}
10516:{\displaystyle X}
10463:{\displaystyle X}
10389:{\displaystyle X}
10042:{\displaystyle X}
10022:{\displaystyle f}
9946:{\displaystyle X}
9933:) of elements in
9761:Cartesian product
9752:{\displaystyle X}
9677:of all functions
9670:{\displaystyle X}
9651:Cartesian product
9589:{\displaystyle X}
9499:{\displaystyle X}
9479:{\displaystyle X}
9318:{\displaystyle X}
9298:{\displaystyle X}
9229:{\displaystyle X}
9140:{\displaystyle X}
9047:{\displaystyle x}
9021:{\displaystyle X}
8997:{\displaystyle X}
8974:{\displaystyle C}
8954:{\displaystyle C}
8934:{\displaystyle C}
8894:Cauchy continuous
8769:{\displaystyle n}
8726:{\displaystyle V}
8361:equivalent metric
8265:{\displaystyle X}
8245:{\displaystyle X}
8165:at the origin in
8151:{\displaystyle X}
8068:
8030:{\displaystyle M}
8010:{\displaystyle M}
7997:then a subset of
7967:{\displaystyle M}
7918:{\displaystyle p}
7898:{\displaystyle E}
7874:{\displaystyle X}
7822:{\displaystyle t}
7802:{\displaystyle V}
7782:{\displaystyle E}
7762:{\displaystyle E}
7710:{\displaystyle t}
7690:{\displaystyle V}
7664:{\displaystyle X}
7644:{\displaystyle E}
7585:{\displaystyle 0}
7482:{\displaystyle E}
7466:absolutely convex
7330:for every scalar
7244:{\displaystyle c}
7140:{\displaystyle X}
7111:{\displaystyle X}
7091:{\displaystyle E}
7063:{\displaystyle S}
7043:{\displaystyle X}
7023:{\displaystyle x}
6954:and moreover, if
6411:{\displaystyle M}
6363:{\displaystyle X}
6343:{\displaystyle M}
6328:quotient topology
6259:{\displaystyle X}
6233:topological group
5956:{\displaystyle X}
5914:{\displaystyle X}
5805:{\displaystyle X}
5728:
5646:
5381:directed downward
5209:
5107:
5025:
4970:
4950:
4869:
4758:
4683:{\displaystyle s}
4663:{\displaystyle X}
4428:{\displaystyle X}
4388:is Hausdorff and
4381:{\displaystyle X}
4361:{\displaystyle f}
4314:{\displaystyle X}
4229:{\displaystyle s}
3990:
3853:{\displaystyle f}
3750:
3744:
3684:
3479:
3468:
3436:
3430:
3089:
3076:absolutely convex
3067:{\displaystyle X}
3046:{\displaystyle U}
2939:{\displaystyle X}
2912:{\displaystyle U}
2839:{\displaystyle X}
2473:{\displaystyle i}
2252:{\displaystyle X}
2195:{\displaystyle 0}
1926:{\displaystyle 0}
1911:neighborhood base
1878:{\displaystyle X}
1822:{\displaystyle X}
1799:
1642:of the origin in
1483:is a topology on
1416:{\displaystyle 0}
1402:
1235:, is a bijective
1219:), also called a
1181:), also called a
1138:subspace topology
979:), also called a
749:{\displaystyle X}
722:topological group
549:topological field
536:{\displaystyle X}
489:topological field
452:Non-normed spaces
60:topological space
16:(Redirected from
36357:
36312:
36311:
36286:Uniformly smooth
35955:
35947:
35914:Balanced/Circled
35904:Absorbing/Radial
35634:
35627:
35620:
35611:
35610:
35597:
35596:
35515:Jones polynomial
35433:Operator algebra
35177:
35176:
35150:
35143:
35136:
35127:
35126:
35117:
35103:
35082:Birkhäuser Basel
35069:
35040:Valdivia, Manuel
35035:
35010:Trèves, François
35005:
34970:
34948:
34923:Köthe, Gottfried
34918:
34896:
34876:
34862:
34835:
34809:Linear Operators
34795:
34757:
34723:
34721:
34719:
34691:
34666:Wilansky, Albert
34661:
34634:
34604:
34570:
34534:
34507:
34474:Köthe, Gottfried
34469:
34442:
34401:
34395:
34389:
34383:
34377:
34371:
34360:
34354:
34348:
34342:
34336:
34330:
34324:
34318:
34307:
34301:
34292:
34286:
34280:
34274:
34268:
34262:
34253:
34247:
34238:
34232:
34217:
34211:
34205:
34199:
34182:
34176:
34170:
34164:
34155:
34149:
34143:
34137:
34128:
34122:
34113:
34107:
34101:
34095:
34089:
34083:
34077:
34071:
34065:
34059:
34048:
34042:
34033:
34027:
34021:
34015:
34009:
34003:
33992:
33986:
33977:
33976:
33974:
33973:
33958:
33952:
33946:
33940:
33934:
33923:
33917:
33911:
33905:
33899:
33893:
33887:
33886:
33885:
33883:
33862:
33856:
33855:, section 15.11.
33850:
33844:
33838:
33832:
33826:
33811:
33805:
33799:
33793:
33782:
33776:
33770:
33764:
33758:
33752:
33735:
33729:
33723:
33717:
33702:
33696:
33651:
33645:
33634:
33628:
33622:
33616:
33610:
33604:
33593:
33587:
33570:
33569:
33567:
33566:
33561:
33547:
33545:
33544:
33539:
33395:
33393:
33392:
33387:
33360:
33358:
33357:
33352:
33328:
33326:
33325:
33320:
33167:
33161:
33159:
33157:
33156:
33151:
33139:
33137:
33136:
33131:
33113:
33111:
33110:
33105:
33085:
33084:
33062:
33060:
33059:
33054:
33042:
33040:
33039:
33034:
33022:
33020:
33019:
33014:
33000:
32999:
32983:
32981:
32980:
32975:
32963:
32961:
32960:
32955:
32944:
32943:
32919:
32918:
32890:
32888:
32887:
32882:
32871:
32870:
32846:
32845:
32809:
32808:
32784:
32783:
32761:
32759:
32758:
32753:
32734:
32728:
32726:
32724:
32723:
32718:
32704:
32703:
32686:
32680:
32678:
32676:
32675:
32670:
32655:
32653:
32652:
32647:
32635:
32633:
32632:
32627:
32625:
32621:
32596:
32594:
32593:
32588:
32577:
32573:
32548:
32546:
32545:
32540:
32523:
32522:
32503:
32501:
32500:
32495:
32468:
32466:
32465:
32460:
32448:
32446:
32445:
32440:
32432:
32431:
32373:
32371:
32370:
32365:
32341:
32339:
32338:
32333:
32325:
32324:
32309:
32308:
32290:
32289:
32267:
32265:
32264:
32259:
32247:
32245:
32244:
32239:
32237:
32233:
32208:
32206:
32205:
32200:
32176:
32174:
32173:
32168:
32163:
32162:
32138:
32137:
32088:
32086:
32085:
32080:
32056:
32054:
32053:
32048:
32037:
32036:
32018:which satisfies
32017:
32015:
32014:
32009:
31992:
31988:
31984:
31975:
31964:
31963:
31947:
31945:
31944:
31939:
31921:
31919:
31918:
31913:
31899:
31895:
31891:
31882:
31862:
31853:
31838:it follows that
31837:
31835:
31834:
31829:
31802:
31800:
31799:
31794:
31770:
31768:
31767:
31762:
31751:
31750:
31741:
31732:
31726:
31722:
31718:
31709:
31698:
31697:
31682:
31680:
31679:
31674:
31653:
31651:
31650:
31645:
31627:
31625:
31624:
31619:
31602:
31599:
31585:
31569:
31567:
31566:
31561:
31553:
31550:
31536:
31531:
31530:
31529:
31527:
31525:
31522:
31518:
31513:
31508:
31497:
31495:
31494:
31489:
31477:
31475:
31474:
31469:
31424:
31422:
31421:
31416:
31414:
31413:
31395:
31394:
31376:
31375:
31353:
31351:
31350:
31345:
31321:
31319:
31318:
31313:
31302:
31301:
31285:
31283:
31282:
31277:
31246:
31245:
31244:
31242:
31240:
31237:
31233:
31228:
31223:
31222:
31221:
31205:
31203:
31202:
31197:
31172:
31166:
31165:
31163:
31162:
31157:
31146:
31145:
31115:
31114:
31098:
31096:
31095:
31090:
31082:
31081:
31058:
31052:
31051:
31049:
31048:
31043:
31016:
31014:
31013:
31008:
31006:
30993:
30976:
30975:
30973:
30972:
30967:
30962:
30950:
30948:
30947:
30942:
30940:
30928:
30926:
30925:
30920:
30918:
30913:
30908:
30903:
30888:
30886:
30885:
30880:
30875:
30863:
30861:
30860:
30855:
30850:
30849:
30844:
30831:
30829:
30828:
30823:
30811:
30809:
30808:
30803:
30798:
30790:
30775:
30773:
30772:
30767:
30755:
30753:
30752:
30747:
30742:
30741:
30736:
30722:
30716:
30715:
30713:
30712:
30707:
30692:
30690:
30689:
30684:
30670:
30669:
30647:
30645:
30644:
30639:
30627:
30625:
30624:
30619:
30577:
30575:
30574:
30569:
30555:
30554:
30532:
30530:
30529:
30524:
30510:
30509:
30482:
30476:
30474:
30472:
30471:
30466:
30450:
30448:
30447:
30442:
30440:
30439:
30429:
30424:
30401:
30395:
30388:
30382:
30379:
30373:
30363:
30361:
30360:
30355:
30343:
30341:
30340:
30335:
30317:
30315:
30314:
30309:
30296:
30290:
30288:
30286:
30285:
30280:
30267:
30247:
30238:
30233:Topological ring
30218:
30209:
30189:
30180:
30166:
30145:
30117:Pseudometrizable
30102:
30100:
30099:
30094:
30074:Infrabornivorous
29944:
29942:
29941:
29936:
29922:
29921:
29913:
29912:
29906:
29905:
29899:
29898:
29892:
29891:
29885:
29884:
29876:
29875:
29869:
29868:
29860:
29859:
29853:
29852:
29846:
29845:
29839:
29838:
29832:
29831:
29825:
29824:
29818:
29817:
29811:
29810:
29804:
29803:
29797:
29796:
29790:
29789:
29783:
29782:
29776:
29775:
29769:
29768:
29762:
29761:
29755:
29754:
29748:
29747:
29741:
29739:
29738:
29733:
29712:
29711:
29703:
29702:
29692:
29691:
29677:
29676:
29670:
29669:
29663:
29662:
29640:
29639:
29633:
29632:
29626:
29625:
29619:
29618:
29612:
29611:
29605:
29604:
29588:
29587:
29579:
29578:
29568:
29567:
29553:
29552:
29546:
29545:
29539:
29538:
29532:
29531:
29525:
29524:
29518:
29517:
29511:
29510:
29504:
29503:
29497:
29496:
29490:
29489:
29483:
29482:
29476:
29475:
29469:
29468:
29462:
29461:
29455:
29454:
29448:
29447:
29441:
29440:
29426:
29425:
29419:
29418:
29404:
29403:
29391:
29390:
29384:
29383:
29377:
29376:
29370:
29369:
29363:
29362:
29356:
29355:
29349:
29348:
29342:
29341:
29335:
29334:
29328:
29327:
29321:
29320:
29314:
29313:
29307:
29306:
29300:
29299:
29293:
29292:
29286:
29285:
29279:
29278:
29272:
29271:
29251:
29250:
29244:
29243:
29237:
29236:
29230:
29229:
29223:
29222:
29216:
29215:
29205:
29204:
29198:
29197:
29191:
29190:
29184:
29183:
29177:
29176:
29170:
29169:
29155:
29154:
29148:
29147:
29141:
29140:
29134:
29133:
29127:
29126:
29120:
29119:
29113:
29112:
29106:
29105:
29085:
29084:
29078:
29077:
29071:
29070:
29064:
29063:
29057:
29056:
29036:
29035:
29029:
29028:
29022:
29021:
29015:
29014:
29008:
29007:
29001:
29000:
28994:
28993:
28987:
28986:
28980:
28979:
28973:
28972:
28966:
28965:
28959:
28958:
28952:
28951:
28945:
28944:
28926:
28925:
28919:
28918:
28912:
28911:
28905:
28904:
28898:
28897:
28877:
28876:
28870:
28869:
28863:
28862:
28856:
28855:
28849:
28848:
28842:
28841:
28835:
28834:
28828:
28827:
28821:
28820:
28814:
28813:
28807:
28806:
28800:
28799:
28793:
28792:
28786:
28785:
28767:
28766:
28760:
28759:
28753:
28752:
28746:
28745:
28739:
28738:
28718:
28717:
28711:
28710:
28704:
28703:
28697:
28696:
28690:
28689:
28683:
28682:
28676:
28675:
28669:
28668:
28662:
28661:
28655:
28654:
28648:
28647:
28641:
28640:
28634:
28633:
28627:
28626:
28606:
28605:
28599:
28598:
28592:
28591:
28585:
28584:
28578:
28577:
28571:
28570:
28552:
28551:
28545:
28544:
28538:
28537:
28529:
28528:
28522:
28521:
28515:
28514:
28508:
28507:
28501:
28500:
28494:
28493:
28487:
28486:
28480:
28479:
28473:
28472:
28466:
28465:
28443:
28442:
28436:
28435:
28429:
28428:
28422:
28421:
28415:
28414:
28408:
28407:
28395:
28394:
28384:
28383:
28377:
28376:
28370:
28369:
28363:
28362:
28354:
28353:
28347:
28346:
28340:
28339:
28333:
28332:
28326:
28325:
28319:
28318:
28312:
28311:
28305:
28304:
28298:
28297:
28275:
28274:
28268:
28267:
28261:
28260:
28254:
28253:
28247:
28246:
28240:
28239:
28227:
28226:
28220:
28219:
28213:
28212:
28206:
28205:
28199:
28198:
28192:
28191:
28183:
28182:
28176:
28175:
28169:
28168:
28162:
28161:
28155:
28154:
28148:
28147:
28141:
28140:
28134:
28133:
28127:
28126:
28120:
28119:
28097:
28096:
28090:
28089:
28079:
28078:
28060:
28059:
28053:
28052:
28046:
28045:
28039:
28038:
28032:
28031:
28025:
28024:
28018:
28017:
28011:
28010:
28004:
28003:
27997:
27996:
27990:
27989:
27983:
27982:
27976:
27975:
27969:
27968:
27962:
27961:
27939:
27938:
27932:
27931:
27921:
27920:
27906:
27905:
27899:
27898:
27890:
27889:
27883:
27882:
27876:
27875:
27869:
27868:
27862:
27861:
27855:
27854:
27848:
27847:
27841:
27840:
27834:
27833:
27825:
27824:
27818:
27817:
27811:
27810:
27804:
27803:
27779:
27778:
27772:
27771:
27765:
27764:
27758:
27757:
27745:
27744:
27738:
27737:
27731:
27730:
27724:
27723:
27717:
27716:
27710:
27709:
27703:
27702:
27696:
27695:
27689:
27688:
27682:
27681:
27675:
27674:
27668:
27667:
27661:
27660:
27654:
27653:
27647:
27646:
27640:
27639:
27633:
27632:
27626:
27625:
27609:
27608:
27602:
27601:
27595:
27594:
27588:
27587:
27581:
27580:
27574:
27573:
27567:
27566:
27558:
27557:
27551:
27550:
27542:
27541:
27535:
27534:
27528:
27527:
27521:
27520:
27514:
27513:
27507:
27506:
27500:
27499:
27493:
27492:
27486:
27485:
27479:
27478:
27472:
27471:
27465:
27464:
27458:
27457:
27451:
27450:
27444:
27443:
27437:
27436:
27430:
27429:
27415:
27414:
27408:
27407:
27401:
27400:
27394:
27393:
27387:
27386:
27380:
27379:
27373:
27372:
27366:
27365:
27359:
27358:
27352:
27351:
27345:
27344:
27338:
27337:
27331:
27330:
27324:
27323:
27317:
27316:
27310:
27309:
27303:
27302:
27296:
27295:
27289:
27288:
27282:
27281:
27275:
27274:
27268:
27267:
27261:
27260:
27254:
27253:
27247:
27246:
27240:
27239:
27233:
27232:
27226:
27225:
27214:Scalar multiple
27211:
27210:
27204:
27203:
27197:
27196:
27190:
27189:
27183:
27182:
27176:
27175:
27169:
27168:
27160:
27159:
27153:
27152:
27146:
27145:
27139:
27138:
27132:
27131:
27125:
27124:
27118:
27117:
27111:
27110:
27104:
27103:
27097:
27096:
27090:
27089:
27083:
27082:
27076:
27075:
27069:
27068:
27062:
27061:
27055:
27054:
27048:
27047:
27041:
27040:
27034:
27033:
27027:
27026:
27020:Scalar multiple
27015:
27013:
27012:
27007:
26976:
26975:
26961:
26960:
26954:
26953:
26939:
26938:
26930:
26929:
26923:
26922:
26916:
26915:
26909:
26908:
26902:
26901:
26895:
26894:
26888:
26887:
26881:
26880:
26874:
26873:
26867:
26865:
26864:
26859:
26835:
26834:
26824:
26823:
26813:
26812:
26794:
26793:
26787:
26786:
26780:
26779:
26773:
26772:
26766:
26765:
26759:
26758:
26752:
26751:
26745:
26744:
26738:
26737:
26731:
26730:
26724:
26723:
26717:
26716:
26710:
26709:
26703:
26702:
26696:
26695:
26680:
26678:
26676:
26675:
26670:
26656:
26655:
26645:
26644:
26634:
26633:
26615:
26614:
26608:
26607:
26601:
26600:
26594:
26593:
26587:
26586:
26580:
26579:
26573:
26572:
26566:
26565:
26559:
26558:
26552:
26551:
26545:
26544:
26538:
26537:
26531:
26530:
26524:
26523:
26517:
26516:
26506:
26504:
26502:
26501:
26496:
26480:
26478:
26477:
26472:
26453:
26452:
26444:
26443:
26437:
26436:
26426:
26425:
26417:
26416:
26406:
26405:
26399:
26398:
26392:
26391:
26385:
26384:
26378:
26377:
26371:
26370:
26364:
26363:
26357:
26356:
26350:
26349:
26343:
26342:
26336:
26335:
26329:
26328:
26322:
26321:
26315:
26314:
26308:
26307:
26301:
26300:
26294:
26293:
26287:
26285:
26284:
26279:
26251:
26250:
26244:
26243:
26237:
26236:
26230:
26229:
26223:
26222:
26214:
26213:
26207:
26206:
26200:
26199:
26193:
26192:
26186:
26185:
26179:
26178:
26172:
26171:
26165:
26164:
26158:
26157:
26151:
26150:
26144:
26143:
26137:
26136:
26130:
26129:
26123:
26122:
26116:
26115:
26109:
26108:
26102:
26101:
26095:
26094:
26088:
26087:
26081:
26080:
26065:
26063:
26061:
26060:
26055:
26037:
26036:
26030:
26029:
26023:
26022:
26016:
26015:
26009:
26008:
26000:
25999:
25993:
25992:
25986:
25985:
25979:
25978:
25972:
25971:
25965:
25964:
25958:
25957:
25951:
25950:
25944:
25943:
25937:
25936:
25930:
25929:
25923:
25922:
25916:
25915:
25909:
25908:
25902:
25901:
25895:
25894:
25888:
25887:
25881:
25880:
25874:
25873:
25867:
25866:
25856:
25854:
25852:
25851:
25846:
25830:
25828:
25827:
25822:
25803:
25802:
25796:
25795:
25789:
25788:
25782:
25781:
25775:
25774:
25768:
25767:
25761:
25760:
25752:
25751:
25743:
25742:
25736:
25735:
25729:
25728:
25722:
25721:
25715:
25714:
25706:
25705:
25699:
25698:
25692:
25691:
25685:
25684:
25678:
25677:
25671:
25670:
25664:
25663:
25657:
25656:
25650:
25649:
25643:
25642:
25636:
25635:
25629:
25628:
25622:
25620:
25619:
25614:
25589:Pseudometrizable
25574:
25572:
25571:
25566:
25546:Infrabornivorous
25418:
25416:
25415:
25410:
25398:
25396:
25395:
25390:
25376:
25374:
25373:
25368:
25348:
25344:
25341:
25339:
25338:
25333:
25312:
25310:
25309:
25304:
25253:
25251:
25250:
25245:
25227:
25225:
25224:
25219:
25207:
25205:
25204:
25199:
25187:
25185:
25184:
25179:
25161:
25159:
25158:
25153:
25148:
25144:
25143:
25139:
25135:
25126:
25055:
25053:
25052:
25047:
25029:
25027:
25026:
25021:
24929:
24927:
24926:
24921:
24900:
24898:
24897:
24892:
24868:
24866:
24865:
24860:
24839:
24837:
24836:
24831:
24745:
24743:
24742:
24737:
24716:
24714:
24713:
24708:
24686:
24684:
24683:
24678:
24640:Other properties
24636:
24634:
24633:
24628:
24613:
24611:
24610:
24605:
24588:
24586:
24585:
24580:
24529:
24527:
24526:
24521:
24473:
24471:
24470:
24465:
24438:
24436:
24435:
24430:
24406:
24404:
24403:
24398:
24358:
24357:
24354:
24350:
24299:
24298:
24295:
24291:
24244:
24242:
24241:
24236:
24224:
24222:
24221:
24216:
24189:
24187:
24186:
24181:
24179:
24178:
24173:
24156:
24154:
24153:
24148:
24120:
24114:
24102:
24096:
24068:But in general,
24067:
24065:
24064:
24059:
24026:
24024:
24023:
24018:
23999:
23997:
23996:
23991:
23967:
23965:
23964:
23959:
23935:
23933:
23932:
23927:
23885:
23883:
23882:
23877:
23863:
23860:
23842:
23840:
23839:
23834:
23789:
23787:
23786:
23781:
23766:
23764:
23763:
23758:
23726:
23724:
23723:
23718:
23682:
23680:
23679:
23674:
23658:
23656:
23655:
23650:
23638:
23636:
23635:
23630:
23603:
23601:
23600:
23595:
23561:
23559:
23558:
23553:
23541:
23540:
23511:
23509:
23508:
23503:
23483:
23481:
23480:
23475:
23463:
23461:
23460:
23455:
23440:
23438:
23437:
23432:
23416:
23414:
23413:
23408:
23392:
23390:
23389:
23384:
23362:in a TVS is not
23349:Other properties
23325:
23323:
23322:
23317:
23312:
23308:
23289:
23288:
23258:
23257:
23235:
23229:
23198:
23197:
23181:
23179:
23178:
23173:
23153:
23152:
23136:
23134:
23133:
23128:
23108:
23107:
23091:
23089:
23088:
23083:
23059:
23057:
23056:
23051:
23028:
23027:
22997:
22996:
22985:
22979:
22948:
22947:
22932:is compact then
22931:
22929:
22928:
22923:
22911:
22909:
22908:
22903:
22891:
22889:
22888:
22883:
22856:
22854:
22853:
22848:
22825:
22824:
22802:
22800:
22799:
22794:
22771:
22770:
22752:
22750:
22749:
22744:
22721:
22720:
22693:
22691:
22690:
22685:
22670:
22668:
22667:
22662:
22650:
22648:
22647:
22642:
22622:
22621:
22605:
22603:
22602:
22597:
22585:
22583:
22582:
22577:
22575:
22557:
22555:
22554:
22549:
22541:
22538:
22507:
22501:
22494:
22493:
22477:
22475:
22474:
22469:
22454:
22452:
22451:
22446:
22444:
22443:
22430:
22428:
22427:
22422:
22404:
22403:
22402:
22383:
22377:
22370:
22369:
22354:
22352:
22351:
22346:
22322:
22320:
22319:
22314:
22302:
22300:
22299:
22294:
22279:
22277:
22276:
22271:
22263:
22262:
22243:
22221:
22219:
22218:
22213:
22192:
22190:
22189:
22184:
22170:
22168:
22167:
22162:
22145:
22144:
22120:
22119:
22103:
22101:
22100:
22095:
22077:
22075:
22074:
22069:
22049:
22048:
22037:
22031:
22030:
22026:
22013:
22012:
21988:
21987:
21970:
21969:
21958:
21957:
21954:
21950:
21931:
21930:
21919:
21913:
21900:
21899:
21875:
21874:
21858:
21856:
21855:
21850:
21823:
21821:
21820:
21815:
21807:
21806:
21790:
21788:
21787:
21782:
21774:
21773:
21749:
21746:
21726:
21724:
21723:
21718:
21698:
21697:
21685:
21681:
21674:
21673:
21659:
21655:
21633:
21631:
21630:
21625:
21614:
21611:
21593:
21591:
21590:
21585:
21573:
21571:
21570:
21565:
21547:
21545:
21544:
21539:
21528:
21527:
21497:
21496:
21480:
21478:
21477:
21472:
21461:
21458:
21437:
21435:
21434:
21429:
21417:
21415:
21414:
21409:
21389:
21388:
21370:
21369:
21350:
21348:
21347:
21342:
21330:
21328:
21327:
21322:
21301:
21299:
21298:
21293:
21278:
21276:
21275:
21270:
21258:
21256:
21255:
21250:
21238:
21236:
21235:
21230:
21212:
21210:
21209:
21204:
21192:
21190:
21189:
21184:
21172:
21170:
21169:
21164:
21152:
21150:
21149:
21144:
21117:
21115:
21114:
21109:
21082:
21080:
21079:
21074:
21062:
21060:
21059:
21054:
21039:
21037:
21036:
21031:
21019:
21017:
21016:
21011:
20999:
20997:
20996:
20991:
20979:
20977:
20976:
20971:
20956:
20954:
20953:
20948:
20936:
20934:
20933:
20928:
20916:
20914:
20913:
20908:
20896:
20894:
20893:
20888:
20876:
20874:
20873:
20868:
20852:
20850:
20849:
20844:
20824:
20823:
20814:
20793:
20791:
20790:
20785:
20777:
20776:
20756:
20754:
20753:
20748:
20736:
20734:
20733:
20728:
20696:
20694:
20693:
20688:
20671:
20670:
20642:
20640:
20639:
20634:
20620:
20619:
20597:
20595:
20594:
20589:
20573:
20571:
20570:
20565:
20549:
20547:
20546:
20541:
20527:
20526:
20510:
20508:
20507:
20502:
20490:
20488:
20487:
20482:
20446:
20444:
20443:
20438:
20415:
20414:
20392:
20390:
20389:
20384:
20372:
20370:
20369:
20364:
20350:
20349:
20333:
20331:
20330:
20325:
20313:
20311:
20310:
20305:
20288:
20287:
20267:
20265:
20264:
20259:
20245:
20244:
20216:
20214:
20213:
20208:
20194:
20193:
20159:
20157:
20156:
20151:
20139:
20137:
20136:
20131:
20117:
20116:
20100:
20098:
20097:
20092:
20073:
20071:
20070:
20065:
20051:
20050:
20041:
20011:
20009:
20008:
20003:
19988:
19986:
19985:
19980:
19968:
19966:
19965:
19960:
19955:
19940:
19938:
19937:
19932:
19920:
19918:
19917:
19912:
19896:
19894:
19893:
19888:
19868:
19867:
19858:
19837:
19835:
19834:
19829:
19814:
19812:
19811:
19806:
19798:
19797:
19779:
19777:
19776:
19771:
19757:
19756:
19728:
19726:
19725:
19720:
19705:
19703:
19702:
19697:
19682:
19680:
19679:
19674:
19656:
19654:
19653:
19648:
19634:
19633:
19613:
19611:
19610:
19605:
19585:
19583:
19582:
19577:
19557:
19556:
19534:
19532:
19531:
19526:
19514:
19512:
19511:
19506:
19488:
19486:
19485:
19480:
19472:
19471:
19447:
19446:
19425:
19423:
19422:
19417:
19385:
19383:
19382:
19377:
19363:
19362:
19338:
19337:
19321:
19319:
19318:
19313:
19293:
19291:
19290:
19285:
19268:
19267:
19251:
19249:
19248:
19243:
19223:
19221:
19220:
19215:
19199:
19197:
19196:
19191:
19177:
19176:
19161:Every subset of
19160:
19158:
19157:
19152:
19131:
19129:
19128:
19123:
19100:
19098:
19097:
19092:
19078:
19077:
19060:trivial topology
19057:
19055:
19054:
19049:
19035:
19034:
19014:
19012:
19011:
19006:
18992:
18991:
18975:
18973:
18972:
18967:
18955:
18953:
18952:
18947:
18941:
18931:
18930:
18898:
18897:
18881:
18879:
18878:
18873:
18854:
18852:
18851:
18846:
18829:
18828:
18812:
18810:
18809:
18804:
18789:
18787:
18786:
18781:
18763:
18761:
18760:
18755:
18738:
18737:
18709:
18707:
18706:
18701:
18686:
18684:
18683:
18678:
18660:
18658:
18657:
18652:
18633:
18631:
18630:
18625:
18583:
18581:
18580:
18575:
18557:
18555:
18554:
18549:
18514:
18511:
18493:
18491:
18490:
18485:
18432:
18429:
18405:
18403:
18402:
18397:
18389:
18388:
18309:
18307:
18306:
18301:
18296:
18285:in the real TVS
18284:
18282:
18281:
18276:
18260:
18258:
18257:
18252:
18247:
18229:
18227:
18226:
18221:
18207:
18199:
18191:
18174:
18173:
18157:
18155:
18154:
18149:
18137:
18135:
18134:
18129:
18118:neighborhood of
18113:
18111:
18110:
18105:
18093:
18091:
18090:
18085:
18071:
18068:
18038:
18036:
18035:
18030:
18019:
18016:
18011:
18010:
17922:
17920:
17919:
17914:
17903:
17902:
17880:
17878:
17877:
17872:
17854:
17852:
17851:
17846:
17832:
17830:
17829:
17824:
17816:
17808:
17796:
17794:
17793:
17788:
17776:
17774:
17773:
17768:
17766:
17765:
17749:
17747:
17746:
17741:
17726:
17719:
17711:
17689:
17688:
17670:
17669:
17653:
17651:
17650:
17645:
17633:
17631:
17630:
17625:
17605:
17603:
17602:
17597:
17586:
17585:
17515:
17513:
17512:
17507:
17492:
17490:
17489:
17484:
17472:
17470:
17469:
17464:
17452:
17450:
17449:
17444:
17436:
17435:
17413:
17411:
17410:
17405:
17394:
17393:
17371:
17369:
17368:
17363:
17351:
17349:
17348:
17343:
17331:
17329:
17328:
17323:
17307:
17301:
17254:
17252:
17251:
17246:
17219:
17217:
17216:
17211:
17197:
17195:
17194:
17189:
17141:
17139:
17138:
17133:
17082:
17080:
17079:
17074:
17069:
17068:
17063:
17044:
17042:
17041:
17036:
17024:
17022:
17021:
17016:
17008:
17007:
16985:
16983:
16982:
16977:
16965:
16963:
16962:
16957:
16949:
16948:
16926:
16924:
16923:
16918:
16910:
16909:
16893:
16891:
16890:
16885:
16877:
16876:
16848:
16846:
16845:
16840:
16828:
16826:
16825:
16820:
16806:
16805:
16785:
16783:
16782:
16777:
16755:
16753:
16752:
16747:
16724:
16723:
16705:
16704:
16687:
16681:
16668:
16667:
16643:
16642:
16626:
16624:
16623:
16618:
16616:
16612:
16605:
16604:
16587:
16586:
16575:
16569:
16562:
16561:
16550:
16549:
16546:
16542:
16541:
16537:
16530:
16529:
16512:
16511:
16500:
16494:
16487:
16486:
16470:
16468:
16467:
16462:
16450:
16448:
16447:
16442:
16410:
16408:
16407:
16402:
16390:
16388:
16387:
16382:
16380:
16379:
16374:
16355:
16353:
16352:
16347:
16323:
16321:
16320:
16315:
16300:
16298:
16297:
16292:
16277:
16275:
16274:
16269:
16261:
16260:
16244:
16242:
16241:
16236:
16234:
16233:
16220:
16218:
16217:
16212:
16200:
16198:
16197:
16192:
16190:
16189:
16188:
16187:
16174:
16170:
16169:
16152:
16151:
16135:
16133:
16132:
16127:
16119:
16118:
16096:
16094:
16093:
16088:
16067:
16065:
16064:
16059:
16038:
16036:
16035:
16030:
16015:
16013:
16012:
16007:
16005:
16004:
15989:
15987:
15986:
15981:
15954:
15952:
15951:
15946:
15938:
15937:
15921:
15919:
15918:
15913:
15889:
15887:
15886:
15881:
15869:
15867:
15866:
15861:
15849:
15847:
15846:
15841:
15839:
15838:
15827:
15823:
15822:
15805:
15804:
15788:
15786:
15785:
15780:
15762:
15760:
15759:
15754:
15739:
15737:
15736:
15731:
15719:
15717:
15716:
15711:
15696:
15694:
15693:
15688:
15676:
15674:
15673:
15668:
15656:
15654:
15653:
15648:
15632:
15630:
15629:
15624:
15616:
15615:
15604:
15598:
15559:
15553:
15547:
15541:
15502:
15496:
15489:
15488:
15472:
15470:
15469:
15464:
15448:
15446:
15445:
15440:
15438:
15437:
15415:
15413:
15412:
15407:
15388:
15386:
15385:
15380:
15366:
15364:
15363:
15358:
15343:
15341:
15340:
15335:
15320:
15318:
15317:
15312:
15294:
15292:
15291:
15286:
15240:
15234:
15184:
15182:
15181:
15176:
15161:
15159:
15158:
15153:
15135:
15133:
15132:
15127:
15109:
15107:
15106:
15101:
15089:
15087:
15086:
15081:
15063:
15061:
15060:
15055:
15043:
15041:
15040:
15035:
15023:
15021:
15020:
15015:
14972:
14970:
14969:
14964:
14952:
14950:
14949:
14944:
14924:
14923:
14905:
14904:
14888:
14886:
14885:
14880:
14872:
14871:
14853:
14852:
14836:
14834:
14833:
14828:
14810:
14808:
14807:
14802:
14800:
14799:
14790:
14789:
14771:
14770:
14761:
14760:
14740:
14738:
14737:
14732:
14711:
14709:
14708:
14703:
14691:
14689:
14688:
14683:
14659:
14657:
14656:
14651:
14634:
14632:
14631:
14626:
14606:
14604:
14603:
14598:
14578:
14576:
14575:
14570:
14559:
14558:
14542:
14540:
14539:
14534:
14526:
14525:
14509:
14507:
14506:
14501:
14485:
14483:
14482:
14477:
14462:
14460:
14459:
14454:
14442:
14440:
14439:
14434:
14401:
14399:
14398:
14393:
14378:
14376:
14375:
14370:
14338:
14336:
14335:
14330:
14325:
14310:
14308:
14307:
14302:
14300:
14286:
14271:
14269:
14268:
14263:
14243:
14241:
14240:
14235:
14233:
14225:
14210:
14208:
14207:
14202:
14197:
14181:
14179:
14178:
14173:
14171:
14129:
14127:
14126:
14121:
14109:
14107:
14106:
14101:
14086:
14084:
14083:
14078:
14076:
14075:
14070:
14057:
14055:
14054:
14049:
14047:
14046:
14041:
14027:Euclidean spaces
14018:
14016:
14015:
14010:
14005:
14004:
13980:
13978:
13977:
13972:
13970:
13969:
13953:
13951:
13950:
13945:
13943:
13942:
13923:: these have an
13917:
13915:
13914:
13909:
13904:
13903:
13887:
13885:
13884:
13879:
13877:
13876:
13861:, whose dual is
13859:
13857:
13856:
13851:
13849:
13848:
13813:
13811:
13810:
13805:
13790:
13788:
13787:
13782:
13755:
13753:
13752:
13747:
13745:
13744:
13699:nuclear operator
13668:
13666:
13665:
13660:
13655:
13641:
13640:
13625:
13619:
13586:
13585:
13557:
13555:
13554:
13549:
13544:
13536:
13535:
13487:Stereotype space
13463:Barrelled spaces
13459:
13457:
13456:
13451:
13427:
13425:
13424:
13419:
13398:
13396:
13395:
13390:
13388:
13387:
13350:
13348:
13347:
13342:
13322:
13320:
13319:
13314:
13312:
13311:
13255:
13253:
13252:
13247:
13235:
13233:
13232:
13227:
13215:
13213:
13212:
13207:
13192:
13190:
13189:
13184:
13165:
13163:
13162:
13157:
13145:
13143:
13142:
13137:
13116:
13114:
13113:
13108:
13089:
13087:
13086:
13081:
13062:
13060:
13059:
13054:
13026:
13024:
13023:
13018:
13002:
13000:
12999:
12994:
12966:
12964:
12963:
12958:
12946:
12944:
12943:
12938:
12926:
12924:
12923:
12918:
12888:
12886:
12885:
12880:
12865:
12863:
12862:
12857:
12829:
12827:
12826:
12821:
12794:
12792:
12791:
12786:
12772:
12770:
12769:
12764:
12749:
12747:
12746:
12741:
12729:
12727:
12726:
12721:
12705:pseudometrizable
12702:
12700:
12699:
12694:
12689:
12681:
12651:
12649:
12648:
12643:
12641:
12624:
12607:
12599:
12587:
12585:
12584:
12579:
12577:
12563:
12555:
12540:
12538:
12537:
12532:
12527:
12526:
12521:
12508:
12506:
12505:
12500:
12485:
12483:
12482:
12477:
12458:
12456:
12455:
12450:
12438:
12436:
12435:
12430:
12396:
12394:
12393:
12388:
12369:
12367:
12366:
12361:
12359:
12358:
12342:
12340:
12339:
12334:
12319:
12317:
12316:
12311:
12291:
12290:
12275:
12273:
12272:
12267:
12262:
12250:
12248:
12247:
12242:
12240:
12222:
12220:
12219:
12214:
12187:
12185:
12184:
12179:
12167:
12165:
12164:
12159:
12151:
12150:
12128:
12126:
12125:
12120:
12108:
12106:
12105:
12100:
12069:
12067:
12066:
12061:
12049:
12047:
12046:
12041:
12018:
12017:
12001:
11999:
11998:
11993:
11981:
11979:
11978:
11973:
11961:
11959:
11958:
11953:
11941:
11939:
11938:
11933:
11931:
11923:
11922:
11906:
11904:
11903:
11898:
11884:
11872:
11870:
11869:
11864:
11849:
11847:
11846:
11841:
11829:
11827:
11826:
11821:
11809:
11807:
11806:
11801:
11796:
11778:
11776:
11775:
11770:
11759:
11758:
11733:
11731:
11730:
11725:
11714:every subset of
11713:
11711:
11710:
11705:
11690:
11688:
11687:
11682:
11677:
11659:
11657:
11656:
11651:
11631:
11630:
11614:
11612:
11611:
11606:
11598:
11590:
11589:
11573:
11571:
11570:
11565:
11538:
11536:
11535:
11530:
11518:
11516:
11515:
11510:
11508:
11496:
11494:
11493:
11488:
11486:
11477:
11472:
11461:
11457:
11456:
11435:
11433:
11432:
11427:
11403:
11401:
11400:
11395:
11392:
11387:
11376:
11372:
11368:
11367:
11345:
11343:
11342:
11337:
11325:
11323:
11322:
11317:
11293:
11291:
11290:
11285:
11273:
11271:
11270:
11265:
11263:
11245:
11243:
11242:
11237:
11219:
11217:
11216:
11211:
11206:
11194:
11192:
11191:
11186:
11174:
11172:
11171:
11166:
11154:
11152:
11151:
11146:
11128:
11126:
11125:
11120:
11109:
11101:
11093:
11076:
11075:
11059:
11057:
11056:
11051:
11049:
11037:
11035:
11034:
11029:
11027:
10997:
10995:
10994:
10989:
10987:
10975:
10973:
10972:
10967:
10955:
10953:
10952:
10947:
10945:
10934:Since the field
10926:
10924:
10923:
10918:
10906:
10904:
10903:
10898:
10872:
10870:
10869:
10864:
10847:trivial topology
10844:
10842:
10841:
10836:
10812:
10810:
10809:
10804:
10772:
10770:
10769:
10764:
10752:
10750:
10749:
10744:
10720:
10718:
10717:
10712:
10688:
10686:
10685:
10680:
10669:
10667:
10666:
10661:
10641:product topology
10638:
10636:
10635:
10630:
10628:
10627:
10622:
10609:
10607:
10606:
10601:
10596:
10595:
10590:
10571:
10569:
10568:
10563:
10551:
10549:
10548:
10543:
10541:
10540:
10535:
10522:
10520:
10519:
10514:
10498:
10496:
10495:
10490:
10488:
10487:
10482:
10469:
10467:
10466:
10461:
10449:
10447:
10446:
10441:
10417:
10415:
10414:
10409:
10407:
10395:
10393:
10392:
10387:
10371:
10369:
10368:
10363:
10361:
10349:
10347:
10346:
10341:
10339:
10327:
10325:
10324:
10319:
10317:
10305:
10303:
10302:
10297:
10295:
10260:
10258:
10257:
10252:
10234:
10232:
10231:
10226:
10221:
10192:
10160:
10158:
10157:
10152:
10147:
10135:
10133:
10132:
10127:
10106:
10104:
10103:
10098:
10087:
10086:
10071:
10069:
10068:
10063:
10048:
10046:
10045:
10040:
10028:
10026:
10025:
10020:
10005:
10003:
10002:
9997:
9995:
9994:
9978:
9976:
9975:
9970:
9952:
9950:
9949:
9944:
9924:
9922:
9921:
9916:
9913:
9908:
9897:
9893:
9892:
9870:
9868:
9867:
9862:
9857:
9842:the topology of
9839:
9837:
9836:
9831:
9829:
9828:
9827:
9821:
9801:product topology
9798:
9796:
9795:
9790:
9782:
9781:
9780:
9774:
9758:
9756:
9755:
9750:
9734:
9732:
9731:
9726:
9724:
9712:
9710:
9709:
9704:
9702:
9694:
9676:
9674:
9673:
9668:
9655:product topology
9626:
9624:
9623:
9618:
9616:
9615:
9595:
9593:
9592:
9587:
9575:
9573:
9572:
9567:
9565:
9561:
9560:
9559:
9532:
9530:
9529:
9524:
9522:
9521:
9505:
9503:
9502:
9497:
9485:
9483:
9482:
9477:
9465:
9463:
9462:
9457:
9441:
9440:
9434:
9432:
9431:
9426:
9411:
9409:
9408:
9403:
9401:
9400:
9377:
9375:
9374:
9369:
9334:pseudometrizable
9324:
9322:
9321:
9316:
9304:
9302:
9301:
9296:
9284:
9282:
9281:
9276:
9247:trivial topology
9240:Trivial topology
9235:
9233:
9232:
9227:
9202:
9200:
9199:
9194:
9179:
9177:
9176:
9171:
9163:
9162:
9146:
9144:
9143:
9138:
9126:
9124:
9123:
9118:
9116:
9115:
9105:
9100:
9076:
9074:
9073:
9068:
9053:
9051:
9050:
9045:
9027:
9025:
9024:
9019:
9003:
9001:
9000:
8995:
8980:
8978:
8977:
8972:
8960:
8958:
8957:
8952:
8940:
8938:
8937:
8932:
8911:Every TVS has a
8876:
8874:
8873:
8868:
8847:
8845:
8844:
8839:
8821:
8819:
8818:
8813:
8805:
8804:
8792:
8791:
8775:
8773:
8772:
8767:
8755:
8753:
8752:
8747:
8732:
8730:
8729:
8724:
8708:
8706:
8705:
8700:
8698:
8697:
8686:
8682:
8681:
8664:
8663:
8598:
8596:
8595:
8590:
8575:
8573:
8572:
8567:
8551:
8549:
8548:
8543:
8500:
8498:
8497:
8492:
8477:
8475:
8474:
8469:
8467:
8466:
8461:
8444:
8442:
8441:
8436:
8434:
8411:
8409:
8408:
8403:
8401:
8369:pseudometrizable
8350:
8348:
8347:
8342:
8317:
8315:
8314:
8309:
8294:
8292:
8291:
8286:
8271:
8269:
8268:
8263:
8252:that induces on
8251:
8249:
8248:
8243:
8220:
8218:
8217:
8212:
8187:
8185:
8184:
8179:
8157:
8155:
8154:
8149:
8137:
8135:
8134:
8129:
8107:
8105:
8104:
8099:
8075:
8059:
8057:
8056:
8051:
8036:
8034:
8033:
8028:
8016:
8014:
8013:
8008:
7996:
7994:
7993:
7988:
7973:
7971:
7970:
7965:
7947:
7945:
7944:
7939:
7924:
7922:
7921:
7916:
7904:
7902:
7901:
7896:
7880:
7878:
7877:
7872:
7860:
7858:
7857:
7852:
7828:
7826:
7825:
7820:
7808:
7806:
7805:
7800:
7788:
7786:
7785:
7780:
7768:
7766:
7765:
7760:
7745:
7743:
7742:
7737:
7716:
7714:
7713:
7708:
7696:
7694:
7693:
7688:
7670:
7668:
7667:
7662:
7650:
7648:
7647:
7642:
7619:if the space is
7618:
7616:
7615:
7610:
7591:
7589:
7588:
7583:
7572:neighborhood of
7560:
7558:
7557:
7552:
7528:
7526:
7525:
7520:
7488:
7486:
7485:
7480:
7454:
7452:
7451:
7446:
7422:
7420:
7419:
7414:
7365:
7363:
7362:
7357:
7349:
7341:
7329:
7327:
7326:
7321:
7289:
7287:
7286:
7281:
7270:
7262:
7250:
7248:
7247:
7242:
7230:
7228:
7227:
7222:
7201:
7199:
7198:
7193:
7175:
7173:
7172:
7167:
7147:): if for every
7146:
7144:
7143:
7138:
7117:
7115:
7114:
7109:
7097:
7095:
7094:
7089:
7069:
7067:
7066:
7061:
7049:
7047:
7046:
7041:
7029:
7027:
7026:
7021:
7005:
7003:
7002:
6997:
6979:
6977:
6976:
6971:
6953:
6951:
6950:
6945:
6937:
6936:
6900:
6899:
6883:
6881:
6880:
6875:
6854:
6852:
6851:
6846:
6825:
6823:
6822:
6817:
6793:
6791:
6790:
6785:
6767:
6765:
6764:
6759:
6738:
6736:
6735:
6730:
6709:
6707:
6706:
6701:
6683:
6681:
6680:
6675:
6653:
6651:
6650:
6645:
6637:
6636:
6616:
6614:
6613:
6608:
6600:
6599:
6577:
6575:
6574:
6569:
6551:
6549:
6548:
6543:
6532:
6531:
6510:
6509:
6496:
6494:
6493:
6488:
6473:
6471:
6470:
6465:
6460:
6446:
6444:
6443:
6438:
6417:
6415:
6414:
6409:
6397:
6395:
6394:
6389:
6384:
6369:
6367:
6366:
6361:
6349:
6347:
6346:
6341:
6325:
6323:
6322:
6317:
6312:
6297:
6295:
6294:
6289:
6265:
6263:
6262:
6257:
6230:
6228:
6227:
6222:
6191:
6189:
6188:
6183:
6172:
6171:
6170:
6153:
6151:
6150:
6145:
6121:
6119:
6118:
6113:
6111:
6095:
6093:
6092:
6087:
6082:
6068:
6066:
6065:
6060:
6058:
6057:
6056:
6039:
6037:
6036:
6031:
6012:
6010:
6009:
6004:
6002:
5990:
5988:
5987:
5982:
5980:
5962:
5960:
5959:
5954:
5942:
5940:
5939:
5934:
5932:
5920:
5918:
5917:
5912:
5893:
5891:
5890:
5885:
5872:
5870:
5869:
5864:
5833:
5831:
5830:
5825:
5823:
5811:
5809:
5808:
5803:
5791:
5789:
5788:
5783:
5781:
5769:
5767:
5766:
5761:
5737:
5734:
5721:
5719:
5718:
5713:
5708:
5696:
5694:
5693:
5688:
5686:
5685:
5669:
5668:
5660:
5659:
5642:
5619:defined above).
5618:
5616:
5615:
5610:
5592:
5590:
5589:
5584:
5582:
5570:
5568:
5567:
5562:
5560:
5559:
5547:
5546:
5530:
5528:
5527:
5522:
5520:
5519:
5507:
5506:
5490:
5488:
5487:
5482:
5480:
5472:
5471:
5455:
5453:
5452:
5447:
5442:
5434:
5433:
5421:
5420:
5404:
5402:
5401:
5396:
5394:
5366:
5364:
5363:
5358:
5356:
5344:
5342:
5341:
5336:
5321:
5319:
5318:
5313:
5311:
5296:
5294:
5293:
5288:
5283:
5282:
5281:
5269:
5265:
5264:
5263:
5251:
5250:
5232:
5231:
5219:
5218:
5207:
5194:
5192:
5191:
5186:
5181:
5180:
5179:
5167:
5163:
5162:
5161:
5149:
5148:
5130:
5129:
5117:
5116:
5105:
5092:
5090:
5089:
5084:
5079:
5078:
5077:
5065:
5061:
5060:
5059:
5038:
5037:
5023:
5010:
5008:
5007:
5002:
4997:
4996:
4986:
4985:
4966:
4965:
4948:
4935:
4933:
4932:
4927:
4922:
4918:
4917:
4903:
4902:
4885:
4884:
4867:
4854:
4852:
4851:
4846:
4832:for every index
4831:
4829:
4828:
4823:
4821:
4820:
4808:
4807:
4791:
4789:
4788:
4783:
4781:
4780:
4768:
4767:
4756:
4748:
4746:
4745:
4740:
4738:
4737:
4719:
4717:
4716:
4711:
4709:
4708:
4689:
4687:
4686:
4681:
4669:
4667:
4666:
4661:
4649:
4647:
4646:
4641:
4639:
4638:
4637:
4625:
4621:
4620:
4603:
4602:
4586:
4584:
4583:
4578:
4576:
4575:
4574:
4562:
4558:
4557:
4540:
4539:
4513:
4511:
4510:
4505:
4490:
4488:
4487:
4482:
4434:
4432:
4431:
4426:
4414:
4412:
4411:
4406:
4404:
4403:
4387:
4385:
4384:
4379:
4367:
4365:
4364:
4359:
4347:
4345:
4344:
4339:
4337:
4336:
4320:
4318:
4317:
4312:
4300:
4298:
4297:
4292:
4271:
4269:
4268:
4263:
4255:
4247:
4235:
4233:
4232:
4227:
4216:for all scalars
4215:
4213:
4212:
4207:
4168:
4166:
4165:
4160:
4158:
4157:
4141:
4139:
4138:
4133:
4090:
4088:
4087:
4082:
4080:
4079:
4063:
4061:
4060:
4055:
4028:
4026:
4025:
4020:
4015:
4014:
4004:
3982:
3980:
3979:
3974:
3956:
3954:
3953:
3948:
3924:
3922:
3921:
3916:
3859:
3857:
3856:
3851:
3837:
3835:
3834:
3829:
3824:
3820:
3810:
3802:
3798:
3797:
3796:
3778:
3777:
3760:
3759:
3748:
3742:
3741:
3740:
3739:
3738:
3715:
3714:
3713:
3712:
3689:
3661:
3659:
3658:
3653:
3651:
3650:
3628:
3626:
3625:
3620:
3593:
3591:
3590:
3585:
3547:
3545:
3544:
3539:
3534:
3530:
3529:
3528:
3527:
3526:
3503:
3502:
3501:
3500:
3480:
3477:
3469:
3466:
3458:
3457:
3434:
3428:
3427:
3423:
3422:
3421:
3403:
3402:
3385:
3384:
3358:
3346:
3344:
3343:
3338:
3333:
3332:
3310:
3308:
3307:
3302:
3284:
3282:
3281:
3276:
3274:
3273:
3261:
3260:
3242:
3241:
3219:
3217:
3216:
3211:
3209:
3208:
3186:
3184:
3183:
3178:
3175:
3170:
3159:
3155:
3154:
3137:
3136:
3120:
3117:
3115:
3113:
3112:
3107:
3105:
3073:
3071:
3070:
3065:
3052:
3050:
3049:
3044:
3030:
3028:
3027:
3022:
3011:
3010:
2994:
2992:
2991:
2986:
2981:
2980:
2962:
2961:
2945:
2943:
2942:
2937:
2918:
2916:
2915:
2910:
2895:
2893:
2892:
2887:
2872:
2870:
2869:
2864:
2862:
2861:
2845:
2843:
2842:
2837:
2824:
2823:
2816:
2815:
2807:
2805:
2804:
2799:
2797:
2796:
2779:
2778:
2771:
2769:
2768:
2763:
2758:
2757:
2698:
2696:
2695:
2690:
2676:for every index
2675:
2673:
2672:
2667:
2665:
2664:
2652:
2651:
2633:
2632:
2609:
2608:
2599:
2597:
2596:
2591:
2589:
2588:
2572:
2570:
2569:
2564:
2559:
2558:
2538:
2536:
2535:
2530:
2528:
2527:
2511:
2509:
2508:
2503:
2498:
2497:
2479:
2477:
2476:
2471:
2458:
2456:
2455:
2450:
2448:
2447:
2432:
2430:
2429:
2424:
2409:
2407:
2406:
2401:
2399:
2398:
2381:
2380:
2374:
2372:
2371:
2366:
2364:
2363:
2347:
2345:
2344:
2339:
2324:
2322:
2321:
2316:
2313:
2308:
2297:
2293:
2292:
2275:
2274:
2258:
2256:
2255:
2250:
2224:
2222:
2221:
2216:
2201:
2199:
2198:
2193:
2173:
2171:
2170:
2165:
2163:
2162:
2146:
2144:
2143:
2138:
2111:
2109:
2108:
2103:
2101:
2100:
2081:
2079:
2078:
2073:
2071:
2070:
2051:
2049:
2048:
2043:
2041:
2040:
2017:
2015:
2014:
2009:
2007:
2006:
1979:
1977:
1976:
1971:
1969:
1968:
1955:
1953:
1952:
1947:
1932:
1930:
1929:
1924:
1908:
1906:
1905:
1900:
1898:
1897:
1884:
1882:
1881:
1876:
1852:
1850:
1849:
1844:
1842:
1841:
1828:
1826:
1825:
1820:
1808:
1805:
1786:
1784:
1783:
1778:
1770:
1769:
1747:
1745:
1744:
1739:
1721:
1719:
1718:
1713:
1702:
1701:
1673:
1671:
1670:
1665:
1637:
1635:
1634:
1629:
1611:
1609:
1608:
1603:
1567:
1565:
1564:
1559:
1534:product topology
1531:
1529:
1528:
1523:
1505:
1503:
1502:
1497:
1482:
1480:
1479:
1474:
1458:
1456:
1455:
1450:
1426:
1422:
1420:
1419:
1414:
1398:
1396:
1395:
1390:
1363:
1361:
1360:
1355:
1353:
1352:
1333:
1331:
1330:
1325:
1320:
1319:
1296:
1294:
1293:
1288:
1286:
1285:
1233:
1232:
1225:
1224:
1217:
1216:
1209:
1208:
1190:
1189:
1179:
1178:
1171:
1170:
1162:
1160:
1159:
1154:
1135:
1133:
1132:
1127:
1112:
1110:
1109:
1104:
1064:
1062:
1061:
1056:
1026:
1024:
1023:
1018:
987:
986:
977:
976:
975:TVS homomorphism
969:
968:
957:
955:
954:
949:
947:
929:
927:
926:
921:
919:
903:
901:
900:
895:
890:
889:
888:
882:
857:
855:
854:
849:
847:
846:
845:
839:
820:
818:
817:
812:
810:
777:
776:
755:
753:
752:
747:
724:under addition.
717:
715:
714:
709:
693:
692:
685:
684:
670:
668:
667:
662:
648:
630:
628:
627:
622:
572:(most often the
571:
569:
568:
563:
561:
542:
540:
539:
534:
468:, and spaces of
435:
433:
432:
427:
412:
410:
409:
404:
402:
390:
388:
387:
382:
343:
341:
340:
335:
321:
296:
294:
293:
288:
252:
250:
249:
244:
176:
174:
173:
168:
163:
148:
146:
145:
140:
138:
111:linear operators
21:
36365:
36364:
36360:
36359:
36358:
36356:
36355:
36354:
36325:
36324:
36323:
36318:
36300:
36062:B-complete/Ptak
36045:
35989:
35953:
35945:
35924:Bounding points
35887:
35829:Densely defined
35775:
35764:Bounded inverse
35710:
35644:
35638:
35608:
35603:
35585:
35549:Advanced topics
35544:
35468:
35447:
35406:
35372:Hilbert–Schmidt
35345:
35336:Gelfand–Naimark
35283:
35233:
35168:
35154:
35110:
35092:
35058:
35024:
34994:
34967:
34937:
34915:
34885:
34851:
34824:
34800:Dunford, Nelson
34784:
34774:Springer-Verlag
34762:Conway, John B.
34746:
34717:
34715:
34699:
34697:Further reading
34694:
34680:
34650:
34623:
34609:Schechter, Eric
34593:
34559:
34523:
34488:
34458:
34431:
34421:Springer-Verlag
34410:
34405:
34404:
34396:
34392:
34384:
34380:
34372:
34363:
34355:
34351:
34343:
34339:
34331:
34327:
34319:
34310:
34302:
34295:
34287:
34283:
34275:
34271:
34263:
34256:
34248:
34241:
34233:
34220:
34212:
34208:
34200:
34185:
34177:
34173:
34165:
34158:
34150:
34146:
34138:
34131:
34123:
34116:
34108:
34104:
34096:
34092:
34084:
34080:
34072:
34068:
34060:
34051:
34043:
34036:
34028:
34024:
34016:
34012:
34004:
33995:
33987:
33980:
33971:
33969:
33960:
33959:
33955:
33947:
33943:
33935:
33926:
33918:
33914:
33906:
33902:
33894:
33890:
33881:
33879:
33864:
33863:
33859:
33851:
33847:
33839:
33835:
33827:
33814:
33806:
33802:
33794:
33785:
33777:
33773:
33765:
33761:
33753:
33738:
33730:
33726:
33722:, pp. 5–9.
33718:
33705:
33697:
33654:
33646:
33637:
33629:
33625:
33617:
33613:
33605:
33596:
33588:
33584:
33579:
33574:
33573:
33555:
33552:
33551:
33401:
33398:
33397:
33366:
33363:
33362:
33334:
33331:
33330:
33173:
33170:
33169:
33168:
33164:
33145:
33142:
33141:
33119:
33116:
33115:
33080:
33076:
33068:
33065:
33064:
33048:
33045:
33044:
33028:
33025:
33024:
32995:
32991:
32989:
32986:
32985:
32969:
32966:
32965:
32939:
32935:
32914:
32910:
32896:
32893:
32892:
32866:
32862:
32841:
32837:
32804:
32800:
32779:
32775:
32767:
32764:
32763:
32741:
32738:
32737:
32735:
32731:
32699:
32695:
32693:
32690:
32689:
32687:
32683:
32661:
32658:
32657:
32641:
32638:
32637:
32611:
32607:
32602:
32599:
32598:
32563:
32559:
32554:
32551:
32550:
32518:
32514:
32509:
32506:
32505:
32474:
32471:
32470:
32454:
32451:
32450:
32427:
32423:
32379:
32376:
32375:
32347:
32344:
32343:
32320:
32316:
32304:
32300:
32285:
32281:
32273:
32270:
32269:
32253:
32250:
32249:
32223:
32219:
32214:
32211:
32210:
32182:
32179:
32178:
32158:
32154:
32133:
32129:
32094:
32091:
32090:
32062:
32059:
32058:
32032:
32028:
32023:
32020:
32019:
31973:
31972:
31968:
31959:
31955:
31953:
31950:
31949:
31927:
31924:
31923:
31880:
31879:
31875:
31851:
31843:
31840:
31839:
31808:
31805:
31804:
31776:
31773:
31772:
31746:
31742:
31730:
31707:
31706:
31702:
31693:
31689:
31687:
31684:
31683:
31659:
31656:
31655:
31633:
31630:
31629:
31589:
31583:
31575:
31572:
31571:
31540:
31534:
31521:
31519:
31514:
31512:
31511:
31503:
31500:
31499:
31483:
31480:
31479:
31430:
31427:
31426:
31409:
31405:
31390:
31386:
31371:
31367:
31359:
31356:
31355:
31327:
31324:
31323:
31297:
31293:
31291:
31288:
31287:
31236:
31234:
31229:
31227:
31226:
31217:
31213:
31211:
31208:
31207:
31179:
31176:
31175:
31173:
31169:
31141:
31137:
31110:
31106:
31104:
31101:
31100:
31099:if and only if
31077:
31073:
31065:
31062:
31061:
31059:
31055:
31022:
31019:
31018:
31002:
31000:
30997:
30996:
30994:
30990:
30985:
30980:
30979:
30958:
30956:
30953:
30952:
30936:
30934:
30931:
30930:
30914:
30907:
30899:
30897:
30894:
30893:
30871:
30869:
30866:
30865:
30845:
30840:
30839:
30837:
30834:
30833:
30817:
30814:
30813:
30789:
30781:
30778:
30777:
30761:
30758:
30757:
30756:the sum of the
30737:
30732:
30731:
30729:
30726:
30725:
30723:
30719:
30698:
30695:
30694:
30665:
30661:
30653:
30650:
30649:
30633:
30630:
30629:
30583:
30580:
30579:
30550:
30546:
30538:
30535:
30534:
30505:
30501:
30493:
30490:
30489:
30483:
30479:
30460:
30457:
30456:
30435:
30431:
30425:
30414:
30408:
30405:
30404:
30402:
30398:
30389:
30385:
30380:
30376:
30369:
30349:
30346:
30345:
30323:
30320:
30319:
30303:
30300:
30299:
30298:In particular,
30297:
30293:
30274:
30271:
30270:
30268:
30264:
30259:
30245:
30236:
30216:
30207:
30187:
30178:
30164:
30143:
30130:
30088:
30085:
30084:
30080:
30058:
30051:
30035:
30024:
30018:
30016:
30006:
30004:
29999:
29994:
29986:
29978:
29973:
29930:
29927:
29926:
29727:
29724:
29723:
26995:
26992:
26991:
26847:
26844:
26843:
26664:
26661:
26660:
26659:
26490:
26487:
26486:
26485:
26460:
26457:
26456:
26267:
26264:
26263:
26049:
26046:
26045:
26044:
25840:
25837:
25836:
25835:
25810:
25807:
25806:
25602:
25599:
25598:
25560:
25557:
25556:
25552:
25530:
25523:
25507:
25496:
25490:
25488:
25478:
25476:
25471:
25466:
25458:
25450:
25445:
25404:
25401:
25400:
25381:
25378:
25377:
25359:
25356:
25355:
25321:
25318:
25317:
25298:
25295:
25294:
25272:(Minkowski) sum
25265:totally bounded
25260:
25233:
25230:
25229:
25213:
25210:
25209:
25193:
25190:
25189:
25167:
25164:
25163:
25124:
25123:
25119:
25103:
25099:
25061:
25058:
25057:
25035:
25032:
25031:
24934:
24931:
24930:
24906:
24903:
24902:
24901:" then for any
24874:
24871:
24870:
24845:
24842:
24841:
24750:
24747:
24746:
24722:
24719:
24718:
24717:) then for any
24696:
24693:
24692:
24663:
24660:
24659:
24619:
24616:
24615:
24596:
24593:
24592:
24535:
24532:
24531:
24479:
24476:
24475:
24444:
24441:
24440:
24412:
24409:
24408:
24355: and
24353:
24296: and
24294:
24250:
24247:
24246:
24230:
24227:
24226:
24198:
24195:
24194:
24174:
24169:
24168:
24166:
24163:
24162:
24073:
24070:
24069:
24032:
24029:
24028:
24009:
24006:
24005:
23979:
23976:
23975:
23941:
23938:
23937:
23891:
23888:
23887:
23861: and
23859:
23848:
23845:
23844:
23795:
23792:
23791:
23775:
23772:
23771:
23731:
23728:
23727:
23688:
23685:
23684:
23668:
23665:
23664:
23644:
23641:
23640:
23609:
23606:
23605:
23583:
23580:
23579:
23568:barrelled space
23536:
23529:
23517:
23514:
23513:
23497:
23494:
23493:
23469:
23466:
23465:
23449:
23446:
23445:
23426:
23423:
23422:
23421:if and only if
23402:
23399:
23398:
23378:
23375:
23374:
23351:
23339:totally bounded
23284:
23280:
23253:
23249:
23248:
23244:
23193:
23189:
23187:
23184:
23183:
23148:
23144:
23142:
23139:
23138:
23103:
23099:
23097:
23094:
23093:
23065:
23062:
23061:
23023:
23019:
22992:
22988:
22943:
22939:
22937:
22934:
22933:
22917:
22914:
22913:
22897:
22894:
22893:
22865:
22862:
22861:
22820:
22816:
22814:
22811:
22810:
22766:
22762:
22760:
22757:
22756:
22716:
22712:
22710:
22707:
22706:
22676:
22673:
22672:
22656:
22653:
22652:
22617:
22613:
22611:
22608:
22607:
22591:
22588:
22587:
22571:
22563:
22560:
22559:
22537:
22489:
22485:
22483:
22480:
22479:
22460:
22457:
22456:
22439:
22438:
22436:
22433:
22432:
22398:
22397:
22390:
22365:
22361:
22359:
22356:
22355:
22331:
22328:
22327:
22326:For any subset
22308:
22305:
22304:
22285:
22282:
22281:
22258:
22254:
22233:
22227:
22224:
22223:
22198:
22195:
22194:
22178:
22175:
22174:
22140:
22136:
22115:
22111:
22109:
22106:
22105:
22083:
22080:
22079:
22044:
22040:
22008:
22004:
21983:
21979:
21978:
21974:
21965:
21961:
21955: and
21953:
21926:
21922:
21895:
21891:
21870:
21866:
21864:
21861:
21860:
21832:
21829:
21828:
21802:
21798:
21796:
21793:
21792:
21769:
21765:
21747: and
21745:
21734:
21731:
21730:
21693:
21689:
21669:
21665:
21664:
21660:
21645:
21641:
21639:
21636:
21635:
21612: nor
21610:
21599:
21596:
21595:
21579:
21576:
21575:
21553:
21550:
21549:
21523:
21519:
21492:
21488:
21486:
21483:
21482:
21457:
21443:
21440:
21439:
21423:
21420:
21419:
21384:
21380:
21365:
21361:
21356:
21353:
21352:
21336:
21333:
21332:
21310:
21307:
21306:
21284:
21281:
21280:
21264:
21261:
21260:
21244:
21241:
21240:
21218:
21215:
21214:
21198:
21195:
21194:
21178:
21175:
21174:
21158:
21155:
21154:
21138:
21135:
21134:
21088:
21085:
21084:
21083:of 0 such that
21068:
21065:
21064:
21045:
21042:
21041:
21025:
21022:
21021:
21005:
21002:
21001:
20985:
20982:
20981:
20965:
20962:
20961:
20942:
20939:
20938:
20922:
20919:
20918:
20902:
20899:
20898:
20882:
20879:
20878:
20862:
20859:
20858:
20819:
20815:
20810:
20799:
20796:
20795:
20772:
20768:
20766:
20763:
20762:
20761:if and only if
20759:totally bounded
20742:
20739:
20738:
20722:
20719:
20718:
20711:totally bounded
20702:
20666:
20662:
20648:
20645:
20644:
20615:
20611:
20603:
20600:
20599:
20583:
20580:
20579:
20574:is a Hausdorff
20559:
20556:
20555:
20554:. Moreover, if
20522:
20518:
20516:
20513:
20512:
20496:
20493:
20492:
20452:
20449:
20448:
20410:
20406:
20398:
20395:
20394:
20378:
20375:
20374:
20345:
20341:
20339:
20336:
20335:
20319:
20316:
20315:
20283:
20279:
20277:
20274:
20273:
20240:
20236:
20222:
20219:
20218:
20189:
20185:
20165:
20162:
20161:
20160:that satisfies
20145:
20142:
20141:
20112:
20108:
20106:
20103:
20102:
20086:
20083:
20082:
20046:
20042:
20037:
20020:
20017:
20016:
19994:
19991:
19990:
19974:
19971:
19970:
19951:
19946:
19943:
19942:
19926:
19923:
19922:
19906:
19903:
19902:
19863:
19859:
19854:
19843:
19840:
19839:
19823:
19820:
19819:
19793:
19789:
19787:
19784:
19783:
19752:
19748:
19740:
19737:
19736:
19731:totally bounded
19714:
19711:
19710:
19688:
19685:
19684:
19662:
19659:
19658:
19629:
19625:
19623:
19620:
19619:
19599:
19596:
19595:
19594:closed subsets
19552:
19548:
19540:
19537:
19536:
19520:
19517:
19516:
19494:
19491:
19490:
19467:
19463:
19442:
19438:
19430:
19427:
19426:
19402:
19399:
19398:
19358:
19354:
19333:
19329:
19327:
19324:
19323:
19301:
19298:
19297:
19263:
19259:
19257:
19254:
19253:
19237:
19234:
19233:
19209:
19206:
19205:
19172:
19168:
19166:
19163:
19162:
19137:
19134:
19133:
19114:
19111:
19110:
19073:
19069:
19067:
19064:
19063:
19030:
19026:
19024:
19021:
19020:
18987:
18983:
18981:
18978:
18977:
18961:
18958:
18957:
18926:
18925:
18918:
18893:
18889:
18887:
18884:
18883:
18864:
18861:
18860:
18824:
18820:
18818:
18815:
18814:
18795:
18792:
18791:
18769:
18766:
18765:
18733:
18729:
18715:
18712:
18711:
18692:
18689:
18688:
18666:
18663:
18662:
18646:
18643:
18642:
18639:
18589:
18586:
18585:
18563:
18560:
18559:
18512: and
18510:
18499:
18496:
18495:
18430: and
18428:
18411:
18408:
18407:
18384:
18380:
18315:
18312:
18311:
18292:
18290:
18287:
18286:
18270:
18267:
18266:
18243:
18235:
18232:
18231:
18203:
18195:
18187:
18169:
18165:
18163:
18160:
18159:
18143:
18140:
18139:
18123:
18120:
18119:
18099:
18096:
18095:
18067:
18044:
18041:
18040:
18015:
18006:
18002:
17928:
17925:
17924:
17898:
17894:
17886:
17883:
17882:
17860:
17857:
17856:
17840:
17837:
17836:
17812:
17804:
17802:
17799:
17798:
17782:
17779:
17778:
17761:
17757:
17755:
17752:
17751:
17715:
17707:
17700:
17684:
17680:
17665:
17661:
17659:
17656:
17655:
17639:
17636:
17635:
17613:
17610:
17609:
17581:
17577:
17521:
17518:
17517:
17498:
17495:
17494:
17478:
17475:
17474:
17458:
17455:
17454:
17431:
17427:
17419:
17416:
17415:
17389:
17385:
17377:
17374:
17373:
17357:
17354:
17353:
17337:
17334:
17333:
17260:
17257:
17256:
17225:
17222:
17221:
17205:
17202:
17201:
17147:
17144:
17143:
17088:
17085:
17084:
17064:
17059:
17058:
17050:
17047:
17046:
17030:
17027:
17026:
17003:
16999:
16991:
16988:
16987:
16971:
16968:
16967:
16944:
16940:
16932:
16929:
16928:
16905:
16901:
16899:
16896:
16895:
16872:
16868:
16854:
16851:
16850:
16834:
16831:
16830:
16801:
16797:
16795:
16792:
16791:
16771:
16768:
16767:
16719:
16715:
16700:
16696:
16663:
16659:
16638:
16634:
16632:
16629:
16628:
16600:
16596:
16595:
16591:
16582:
16578:
16557:
16553:
16547: and
16545:
16525:
16521:
16520:
16516:
16507:
16503:
16482:
16478:
16476:
16473:
16472:
16456:
16453:
16452:
16424:
16421:
16420:
16396:
16393:
16392:
16375:
16370:
16369:
16361:
16358:
16357:
16341:
16338:
16337:
16330:nonmeager space
16326:second category
16309:
16306:
16305:
16283:
16280:
16279:
16256:
16252:
16250:
16247:
16246:
16229:
16228:
16226:
16223:
16222:
16206:
16203:
16202:
16183:
16182:
16175:
16165:
16161:
16157:
16156:
16147:
16143:
16141:
16138:
16137:
16114:
16110:
16102:
16099:
16098:
16073:
16070:
16069:
16044:
16041:
16040:
16021:
16018:
16017:
16000:
15999:
15997:
15994:
15993:
15960:
15957:
15956:
15933:
15929:
15927:
15924:
15923:
15895:
15892:
15891:
15875:
15872:
15871:
15855:
15852:
15851:
15828:
15818:
15814:
15810:
15809:
15800:
15796:
15794:
15791:
15790:
15768:
15765:
15764:
15745:
15742:
15741:
15725:
15722:
15721:
15705:
15702:
15701:
15682:
15679:
15678:
15662:
15659:
15658:
15642:
15639:
15638:
15611:
15607:
15484:
15480:
15478:
15475:
15474:
15458:
15455:
15454:
15433:
15429:
15421:
15418:
15417:
15401:
15398:
15397:
15374:
15371:
15370:
15349:
15346:
15345:
15329:
15326:
15325:
15300:
15297:
15296:
15190:
15187:
15186:
15170:
15167:
15166:
15141:
15138:
15137:
15115:
15112:
15111:
15095:
15092:
15091:
15069:
15066:
15065:
15049:
15046:
15045:
15029:
15026:
15025:
15003:
15000:
14999:
14979:
14958:
14955:
14954:
14919:
14915:
14900:
14896:
14894:
14891:
14890:
14867:
14863:
14848:
14844:
14842:
14839:
14838:
14837:is an integer,
14816:
14813:
14812:
14795:
14791:
14785:
14781:
14766:
14762:
14756:
14752:
14750:
14747:
14746:
14723:
14720:
14719:
14718:of elements in
14697:
14694:
14693:
14671:
14668:
14667:
14639:
14636:
14635:
14611:
14608:
14607:
14583:
14580:
14579:
14554:
14550:
14548:
14545:
14544:
14543:(respectively,
14521:
14517:
14515:
14512:
14511:
14495:
14492:
14491:
14468:
14465:
14464:
14448:
14445:
14444:
14428:
14425:
14424:
14384:
14381:
14380:
14358:
14355:
14354:
14351:
14345:
14318:
14316:
14313:
14312:
14296:
14279:
14277:
14274:
14273:
14257:
14254:
14253:
14246:weak-* topology
14229:
14218:
14216:
14213:
14212:
14193:
14191:
14188:
14187:
14164:
14162:
14159:
14158:
14157:—the set
14151:
14139:Main articles:
14137:
14115:
14112:
14111:
14092:
14089:
14088:
14071:
14066:
14065:
14063:
14060:
14059:
14042:
14037:
14036:
14034:
14031:
14030:
13994:
13990:
13988:
13985:
13984:
13965:
13961:
13959:
13956:
13955:
13938:
13934:
13932:
13929:
13928:
13899:
13895:
13893:
13890:
13889:
13872:
13868:
13866:
13863:
13862:
13844:
13840:
13838:
13835:
13834:
13796:
13793:
13792:
13761:
13758:
13757:
13740:
13736:
13734:
13731:
13730:
13651:
13630:
13626:
13621:
13594:
13575:
13571:
13563:
13560:
13559:
13540:
13531:
13527:
13525:
13522:
13521:
13433:
13430:
13429:
13404:
13401:
13400:
13383:
13379:
13377:
13374:
13373:
13330:
13327:
13326:
13307:
13303:
13301:
13298:
13297:
13270:
13241:
13238:
13237:
13221:
13218:
13217:
13201:
13198:
13197:
13178:
13175:
13174:
13166:of the origin.
13151:
13148:
13147:
13122:
13119:
13118:
13102:
13099:
13098:
13095:
13072:
13069:
13068:
13048:
13045:
13044:
13012:
13009:
13008:
12988:
12985:
12984:
12976:
12952:
12949:
12948:
12932:
12929:
12928:
12894:
12891:
12890:
12871:
12868:
12867:
12839:
12836:
12835:
12800:
12797:
12796:
12780:
12777:
12776:
12755:
12752:
12751:
12735:
12732:
12731:
12715:
12712:
12711:
12685:
12677:
12657:
12654:
12653:
12637:
12620:
12603:
12595:
12593:
12590:
12589:
12573:
12559:
12551:
12549:
12546:
12545:
12522:
12517:
12516:
12514:
12511:
12510:
12491:
12488:
12487:
12471:
12468:
12467:
12461:infinitely many
12444:
12441:
12440:
12406:
12403:
12402:
12398:
12382:
12379:
12378:
12354:
12350:
12348:
12345:
12344:
12325:
12322:
12321:
12286:
12282:
12280:
12277:
12276:
12258:
12256:
12253:
12252:
12251:center at 0 in
12236:
12228:
12225:
12224:
12193:
12190:
12189:
12173:
12170:
12169:
12146:
12142:
12134:
12131:
12130:
12114:
12111:
12110:
12079:
12076:
12075:
12055:
12052:
12051:
12013:
12009:
12007:
12004:
12003:
11987:
11984:
11983:
11967:
11964:
11963:
11947:
11944:
11943:
11927:
11918:
11914:
11912:
11909:
11908:
11880:
11878:
11875:
11874:
11855:
11852:
11851:
11835:
11832:
11831:
11815:
11812:
11811:
11792:
11784:
11781:
11780:
11754:
11750:
11739:
11736:
11735:
11719:
11716:
11715:
11696:
11693:
11692:
11673:
11665:
11662:
11661:
11626:
11622:
11620:
11617:
11616:
11594:
11585:
11581:
11579:
11576:
11575:
11544:
11541:
11540:
11524:
11521:
11520:
11504:
11502:
11499:
11498:
11482:
11473:
11462:
11452:
11448:
11444:
11441:
11438:
11437:
11409:
11406:
11405:
11388:
11377:
11363:
11359:
11358:
11354:
11351:
11348:
11347:
11331:
11328:
11327:
11299:
11296:
11295:
11279:
11276:
11275:
11259:
11251:
11248:
11247:
11225:
11222:
11221:
11202:
11200:
11197:
11196:
11180:
11177:
11176:
11160:
11157:
11156:
11134:
11131:
11130:
11105:
11097:
11089:
11071:
11067:
11065:
11062:
11061:
11045:
11043:
11040:
11039:
11023:
11021:
11018:
11017:
10983:
10981:
10978:
10977:
10961:
10958:
10957:
10941:
10939:
10936:
10935:
10912:
10909:
10908:
10880:
10877:
10876:
10858:
10855:
10854:
10818:
10815:
10814:
10786:
10783:
10782:
10758:
10755:
10754:
10726:
10723:
10722:
10694:
10691:
10690:
10674:
10671:
10670:
10652:
10649:
10648:
10623:
10618:
10617:
10615:
10612:
10611:
10591:
10586:
10585:
10583:
10580:
10579:
10557:
10554:
10553:
10536:
10531:
10530:
10528:
10525:
10524:
10508:
10505:
10504:
10483:
10478:
10477:
10475:
10472:
10471:
10455:
10452:
10451:
10423:
10420:
10419:
10403:
10401:
10398:
10397:
10381:
10378:
10377:
10357:
10355:
10352:
10351:
10335:
10333:
10330:
10329:
10313:
10311:
10308:
10307:
10291:
10289:
10286:
10285:
10281:of the origin.
10275:locally compact
10267:
10240:
10237:
10236:
10217:
10188:
10186:
10183:
10182:
10143:
10141:
10138:
10137:
10112:
10109:
10108:
10082:
10078:
10076:
10073:
10072:
10054:
10051:
10050:
10034:
10031:
10030:
10014:
10011:
10010:
9990:
9986:
9984:
9981:
9980:
9958:
9955:
9954:
9938:
9935:
9934:
9909:
9898:
9888:
9884:
9880:
9877:
9874:
9873:
9853:
9851:
9848:
9847:
9823:
9822:
9817:
9816:
9808:
9805:
9804:
9776:
9775:
9770:
9769:
9767:
9764:
9763:
9744:
9741:
9740:
9720:
9718:
9715:
9714:
9698:
9690:
9682:
9679:
9678:
9662:
9659:
9658:
9647:
9611:
9607:
9605:
9602:
9601:
9581:
9578:
9577:
9555:
9551:
9544:
9540:
9538:
9535:
9534:
9517:
9513:
9511:
9508:
9507:
9491:
9488:
9487:
9471:
9468:
9467:
9448:
9445:
9444:
9438:
9437:
9417:
9414:
9413:
9396:
9392:
9390:
9387:
9386:
9351:
9348:
9347:
9346:if and only if
9327:locally compact
9310:
9307:
9306:
9290:
9287:
9286:
9258:
9255:
9254:
9221:
9218:
9217:
9214:
9209:
9185:
9182:
9181:
9158:
9154:
9152:
9149:
9148:
9132:
9129:
9128:
9111:
9107:
9101:
9090:
9084:
9081:
9080:
9059:
9056:
9055:
9039:
9036:
9035:
9013:
9010:
9009:
8989:
8986:
8985:
8966:
8963:
8962:
8946:
8943:
8942:
8926:
8923:
8922:
8901:linear subspace
8853:
8850:
8849:
8827:
8824:
8823:
8800:
8796:
8787:
8783:
8781:
8778:
8777:
8761:
8758:
8757:
8738:
8735:
8734:
8718:
8715:
8714:
8687:
8677:
8673:
8669:
8668:
8659:
8655:
8653:
8650:
8649:
8627:totally bounded
8581:
8578:
8577:
8561:
8558:
8557:
8525:
8522:
8521:
8512:
8506:
8483:
8480:
8479:
8462:
8457:
8456:
8454:
8451:
8450:
8430:
8428:
8425:
8424:
8417:locally compact
8397:
8395:
8392:
8391:
8365:
8324:
8321:
8320:
8300:
8297:
8296:
8277:
8274:
8273:
8257:
8254:
8253:
8237:
8234:
8233:
8194:
8191:
8190:
8170:
8167:
8166:
8158:and there is a
8143:
8140:
8139:
8117:
8114:
8113:
8081:
8078:
8077:
8073:
8065:
8042:
8039:
8038:
8022:
8019:
8018:
8002:
7999:
7998:
7979:
7976:
7975:
7959:
7956:
7955:
7952:totally bounded
7930:
7927:
7926:
7910:
7907:
7906:
7890:
7887:
7886:
7866:
7863:
7862:
7861:Moreover, when
7834:
7831:
7830:
7814:
7811:
7810:
7794:
7791:
7790:
7774:
7771:
7770:
7754:
7751:
7750:
7722:
7719:
7718:
7702:
7699:
7698:
7682:
7679:
7678:
7656:
7653:
7652:
7636:
7633:
7632:
7627:Bounded subsets
7601:
7598:
7597:
7577:
7574:
7573:
7534:
7531:
7530:
7502:
7499:
7498:
7474:
7471:
7470:
7428:
7425:
7424:
7423:for every real
7378:
7375:
7374:
7345:
7337:
7335:
7332:
7331:
7306:
7303:
7302:
7266:
7258:
7256:
7253:
7252:
7236:
7233:
7232:
7231:for any scalar
7207:
7204:
7203:
7181:
7178:
7177:
7152:
7149:
7148:
7132:
7129:
7128:
7103:
7100:
7099:
7083:
7080:
7079:
7076:
7070:at the origin.
7055:
7052:
7051:
7035:
7032:
7031:
7015:
7012:
7011:
6985:
6982:
6981:
6959:
6956:
6955:
6932:
6928:
6895:
6891:
6889:
6886:
6885:
6860:
6857:
6856:
6855:and any subset
6834:
6831:
6830:
6799:
6796:
6795:
6773:
6770:
6769:
6744:
6741:
6740:
6715:
6712:
6711:
6689:
6686:
6685:
6663:
6660:
6659:
6632:
6628:
6626:
6623:
6622:
6595:
6591:
6583:
6580:
6579:
6557:
6554:
6553:
6527:
6523:
6521:
6518:
6517:
6507:
6506:
6502:
6479:
6476:
6475:
6456:
6451:
6448:
6447:
6423:
6420:
6419:
6403:
6400:
6399:
6380:
6375:
6372:
6371:
6355:
6352:
6351:
6335:
6332:
6331:
6326:with the usual
6308:
6303:
6300:
6299:
6274:
6271:
6270:
6251:
6248:
6247:
6235:. Every TVS is
6213:
6210:
6209:
6202:
6166:
6165:
6161:
6159:
6156:
6155:
6127:
6124:
6123:
6107:
6105:
6102:
6101:
6098:
6078:
6076:
6073:
6072:
6052:
6051:
6047:
6045:
6042:
6041:
6022:
6019:
6018:
5998:
5996:
5993:
5992:
5976:
5968:
5965:
5964:
5948:
5945:
5944:
5928:
5926:
5923:
5922:
5906:
5903:
5902:
5901:Conversely, if
5879:
5876:
5875:
5843:
5840:
5839:
5819:
5817:
5814:
5813:
5797:
5794:
5793:
5777:
5775:
5772:
5771:
5743:
5740:
5739:
5735:
5732:
5704:
5702:
5699:
5698:
5681:
5677:
5664:
5655:
5651:
5650:
5638:
5630:
5627:
5626:
5602:
5599:
5598:
5578:
5576:
5573:
5572:
5555:
5551:
5542:
5538:
5536:
5533:
5532:
5515:
5511:
5502:
5498:
5496:
5493:
5492:
5476:
5467:
5463:
5461:
5458:
5457:
5438:
5429:
5425:
5416:
5412:
5410:
5407:
5406:
5390:
5388:
5385:
5384:
5377:under inclusion
5352:
5350:
5347:
5346:
5327:
5324:
5323:
5307:
5305:
5302:
5301:
5277:
5270:
5259:
5255:
5246:
5242:
5241:
5237:
5236:
5227:
5223:
5214:
5210:
5205:
5202:
5201:
5175:
5168:
5157:
5153:
5144:
5140:
5139:
5135:
5134:
5125:
5121:
5112:
5108:
5103:
5100:
5099:
5073:
5066:
5055:
5051:
5047:
5043:
5042:
5033:
5029:
5021:
5018:
5017:
5014:Scalar multiple
4992:
4988:
4981:
4974:
4961:
4957:
4946:
4943:
4942:
4913:
4898:
4894:
4893:
4889:
4880:
4876:
4865:
4862:
4861:
4837:
4834:
4833:
4816:
4812:
4803:
4799:
4797:
4794:
4793:
4792:if and only if
4776:
4772:
4763:
4759:
4754:
4751:
4750:
4733:
4729:
4727:
4724:
4723:
4704:
4700:
4698:
4695:
4694:
4675:
4672:
4671:
4655:
4652:
4651:
4633:
4626:
4616:
4612:
4608:
4607:
4598:
4594:
4592:
4589:
4588:
4570:
4563:
4553:
4549:
4545:
4544:
4535:
4531:
4529:
4526:
4525:
4515:
4496:
4493:
4492:
4440:
4437:
4436:
4420:
4417:
4416:
4399:
4395:
4393:
4390:
4389:
4373:
4370:
4369:
4353:
4350:
4349:
4332:
4328:
4326:
4323:
4322:
4306:
4303:
4302:
4277:
4274:
4273:
4251:
4243:
4241:
4238:
4237:
4221:
4218:
4217:
4174:
4171:
4170:
4153:
4149:
4147:
4144:
4143:
4100:
4097:
4096:
4075:
4071:
4069:
4066:
4065:
4034:
4031:
4030:
4010:
4006:
3994:
3988:
3985:
3984:
3962:
3959:
3958:
3930:
3927:
3926:
3865:
3862:
3861:
3845:
3842:
3841:
3806:
3792:
3788:
3773:
3769:
3768:
3764:
3755:
3751:
3734:
3730:
3726:
3722:
3708:
3704:
3700:
3696:
3695:
3691:
3688:
3667:
3664:
3663:
3646:
3642:
3634:
3631:
3630:
3599:
3596:
3595:
3555:
3552:
3551:
3522:
3518:
3517:
3513:
3496:
3492:
3491:
3487:
3478: and
3476:
3465:
3453:
3449:
3417:
3413:
3398:
3394:
3393:
3389:
3380:
3376:
3375:
3371:
3354:
3352:
3349:
3348:
3328:
3324:
3316:
3313:
3312:
3290:
3287:
3286:
3269:
3265:
3250:
3246:
3231:
3227:
3225:
3222:
3221:
3204:
3200:
3192:
3189:
3188:
3171:
3160:
3150:
3146:
3142:
3132:
3128:
3126:
3123:
3122:
3118:
3101:
3099:
3096:
3095:
3093:
3059:
3056:
3055:
3038:
3035:
3034:
3006:
3002:
3000:
2997:
2996:
2970:
2966:
2957:
2953:
2951:
2948:
2947:
2931:
2928:
2927:
2904:
2901:
2900:
2878:
2875:
2874:
2857:
2853:
2851:
2848:
2847:
2831:
2828:
2827:
2821:
2820:
2813:
2812:
2792:
2788:
2786:
2783:
2782:
2776:
2775:
2753:
2749:
2747:
2744:
2743:
2681:
2678:
2677:
2660:
2656:
2641:
2637:
2622:
2618:
2616:
2613:
2612:
2606:
2605:
2584:
2580:
2578:
2575:
2574:
2554:
2550:
2548:
2545:
2544:
2523:
2519:
2517:
2514:
2513:
2493:
2489:
2487:
2484:
2483:
2465:
2462:
2461:
2443:
2439:
2437:
2434:
2433:
2415:
2412:
2411:
2394:
2390:
2388:
2385:
2384:
2378:
2377:
2359:
2355:
2353:
2350:
2349:
2330:
2327:
2326:
2309:
2298:
2288:
2284:
2280:
2270:
2266:
2264:
2261:
2260:
2244:
2241:
2240:
2237:
2226:
2207:
2204:
2203:
2187:
2184:
2183:
2158:
2157:
2155:
2152:
2151:
2117:
2114:
2113:
2096:
2095:
2087:
2084:
2083:
2082:there exists a
2066:
2065:
2057:
2054:
2053:
2036:
2035:
2033:
2030:
2029:
2002:
2001:
1993:
1990:
1989:
1964:
1963:
1961:
1958:
1957:
1938:
1935:
1934:
1918:
1915:
1914:
1893:
1892:
1890:
1887:
1886:
1870:
1867:
1866:
1837:
1836:
1834:
1831:
1830:
1814:
1811:
1810:
1806:
1803:
1765:
1761:
1753:
1750:
1749:
1727:
1724:
1723:
1697:
1693:
1691:
1688:
1687:
1684:
1676:
1647:
1644:
1643:
1617:
1614:
1613:
1573:
1570:
1569:
1541:
1538:
1537:
1511:
1508:
1507:
1488:
1485:
1484:
1468:
1465:
1464:
1432:
1429:
1428:
1424:
1408:
1405:
1404:
1369:
1366:
1365:
1348:
1347:
1339:
1336:
1335:
1315:
1314:
1306:
1303:
1302:
1281:
1280:
1278:
1275:
1274:
1251:local convexity
1230:
1229:
1222:
1221:
1215:TVS isomorphism
1214:
1213:
1206:
1205:
1184:
1183:
1176:
1175:
1168:
1167:
1145:
1142:
1141:
1118:
1115:
1114:
1074:
1071:
1070:
1032:
1029:
1028:
1000:
997:
996:
982:
981:
974:
973:
966:
965:
943:
941:
938:
937:
915:
913:
910:
909:
884:
883:
866:
865:
863:
860:
859:
841:
840:
829:
828:
826:
823:
822:
806:
804:
801:
800:
774:
773:
761:
741:
738:
737:
700:
697:
696:
690:
689:
683:vector topology
682:
681:
644:
636:
633:
632:
589:
586:
585:
557:
555:
552:
551:
528:
525:
524:
501:
466:Schwartz spaces
454:
418:
415:
414:
398:
396:
393:
392:
349:
346:
345:
317:
309:
306:
305:
258:
255:
254:
211:
208:
207:
188:
183:
159:
157:
154:
153:
134:
132:
129:
128:
126:complex numbers
84:Hausdorff space
68:vector topology
58:that is also a
38:(also called a
28:
23:
22:
15:
12:
11:
5:
36363:
36353:
36352:
36347:
36342:
36337:
36320:
36319:
36317:
36316:
36305:
36302:
36301:
36299:
36298:
36293:
36288:
36283:
36281:Ultrabarrelled
36273:
36267:
36262:
36256:
36251:
36246:
36241:
36236:
36231:
36222:
36216:
36211:
36209:Quasi-complete
36206:
36204:Quasibarrelled
36201:
36196:
36191:
36186:
36181:
36176:
36171:
36166:
36161:
36156:
36151:
36146:
36145:
36144:
36134:
36129:
36124:
36119:
36114:
36109:
36104:
36099:
36094:
36084:
36079:
36069:
36064:
36059:
36053:
36051:
36047:
36046:
36044:
36043:
36033:
36028:
36023:
36018:
36013:
36003:
35997:
35995:
35994:Set operations
35991:
35990:
35988:
35987:
35982:
35977:
35972:
35967:
35962:
35957:
35949:
35941:
35936:
35931:
35926:
35921:
35916:
35911:
35906:
35901:
35895:
35893:
35889:
35888:
35886:
35885:
35880:
35875:
35870:
35865:
35864:
35863:
35858:
35853:
35843:
35838:
35837:
35836:
35831:
35826:
35821:
35816:
35811:
35806:
35796:
35795:
35794:
35783:
35781:
35777:
35776:
35774:
35773:
35768:
35767:
35766:
35756:
35750:
35741:
35736:
35731:
35729:Banach–Alaoglu
35726:
35724:Anderson–Kadec
35720:
35718:
35712:
35711:
35709:
35708:
35703:
35698:
35693:
35688:
35683:
35678:
35673:
35668:
35663:
35658:
35652:
35650:
35649:Basic concepts
35646:
35645:
35637:
35636:
35629:
35622:
35614:
35605:
35604:
35602:
35601:
35590:
35587:
35586:
35584:
35583:
35578:
35573:
35568:
35566:Choquet theory
35563:
35558:
35552:
35550:
35546:
35545:
35543:
35542:
35532:
35527:
35522:
35517:
35512:
35507:
35502:
35497:
35492:
35487:
35482:
35476:
35474:
35470:
35469:
35467:
35466:
35461:
35455:
35453:
35449:
35448:
35446:
35445:
35440:
35435:
35430:
35425:
35420:
35418:Banach algebra
35414:
35412:
35408:
35407:
35405:
35404:
35399:
35394:
35389:
35384:
35379:
35374:
35369:
35364:
35359:
35353:
35351:
35347:
35346:
35344:
35343:
35341:Banach–Alaoglu
35338:
35333:
35328:
35323:
35318:
35313:
35308:
35303:
35297:
35295:
35289:
35288:
35285:
35284:
35282:
35281:
35276:
35271:
35269:Locally convex
35266:
35252:
35247:
35241:
35239:
35235:
35234:
35232:
35231:
35226:
35221:
35216:
35211:
35206:
35201:
35196:
35191:
35186:
35180:
35174:
35170:
35169:
35153:
35152:
35145:
35138:
35130:
35124:
35123:
35109:
35108:External links
35106:
35105:
35104:
35090:
35070:
35056:
35036:
35022:
35006:
34992:
34971:
34965:
34949:
34935:
34919:
34914:978-0201029857
34913:
34897:
34883:
34863:
34849:
34836:
34822:
34796:
34782:
34758:
34744:
34724:
34698:
34695:
34693:
34692:
34678:
34662:
34648:
34635:
34621:
34605:
34591:
34571:
34557:
34535:
34522:978-1584888666
34521:
34508:
34486:
34470:
34456:
34443:
34429:
34411:
34409:
34406:
34403:
34402:
34390:
34378:
34376:, p. 109.
34361:
34349:
34337:
34335:, pp. 80.
34325:
34308:
34293:
34281:
34269:
34254:
34239:
34218:
34216:, p. 156.
34206:
34183:
34181:, p. 102.
34171:
34156:
34144:
34142:, p. 108.
34129:
34114:
34102:
34090:
34078:
34066:
34049:
34034:
34022:
34010:
33993:
33991:, p. 111.
33978:
33953:
33941:
33924:
33912:
33900:
33888:
33857:
33845:
33833:
33812:
33800:
33783:
33771:
33759:
33736:
33732:Schechter 1996
33724:
33703:
33652:
33635:
33623:
33611:
33594:
33581:
33580:
33578:
33575:
33572:
33571:
33559:
33537:
33534:
33531:
33528:
33525:
33522:
33519:
33516:
33513:
33510:
33507:
33504:
33501:
33498:
33495:
33492:
33489:
33486:
33483:
33480:
33477:
33474:
33471:
33468:
33465:
33462:
33459:
33456:
33453:
33450:
33447:
33444:
33441:
33438:
33435:
33432:
33429:
33426:
33423:
33420:
33417:
33414:
33411:
33408:
33405:
33385:
33382:
33379:
33376:
33373:
33370:
33350:
33347:
33344:
33341:
33338:
33318:
33315:
33312:
33309:
33306:
33303:
33300:
33297:
33294:
33291:
33288:
33285:
33282:
33279:
33276:
33273:
33270:
33267:
33264:
33261:
33258:
33255:
33252:
33249:
33246:
33243:
33240:
33237:
33234:
33231:
33228:
33225:
33222:
33219:
33216:
33213:
33210:
33207:
33204:
33201:
33198:
33195:
33192:
33189:
33186:
33183:
33180:
33177:
33162:
33149:
33129:
33126:
33123:
33103:
33100:
33097:
33094:
33091:
33088:
33083:
33079:
33075:
33072:
33052:
33032:
33012:
33009:
33006:
33003:
32998:
32994:
32973:
32953:
32950:
32947:
32942:
32938:
32934:
32931:
32928:
32925:
32922:
32917:
32913:
32909:
32906:
32903:
32900:
32880:
32877:
32874:
32869:
32865:
32861:
32858:
32855:
32852:
32849:
32844:
32840:
32836:
32833:
32830:
32827:
32824:
32821:
32818:
32815:
32812:
32807:
32803:
32799:
32796:
32793:
32790:
32787:
32782:
32778:
32774:
32771:
32751:
32748:
32745:
32729:
32716:
32713:
32710:
32707:
32702:
32698:
32681:
32668:
32665:
32645:
32624:
32620:
32617:
32614:
32610:
32606:
32586:
32583:
32580:
32576:
32572:
32569:
32566:
32562:
32558:
32538:
32535:
32532:
32529:
32526:
32521:
32517:
32513:
32493:
32490:
32487:
32484:
32481:
32478:
32458:
32438:
32435:
32430:
32426:
32422:
32419:
32416:
32413:
32410:
32407:
32404:
32401:
32398:
32395:
32392:
32389:
32386:
32383:
32363:
32360:
32357:
32354:
32351:
32331:
32328:
32323:
32319:
32315:
32312:
32307:
32303:
32299:
32296:
32293:
32288:
32284:
32280:
32277:
32257:
32236:
32232:
32229:
32226:
32222:
32218:
32198:
32195:
32192:
32189:
32186:
32166:
32161:
32157:
32153:
32150:
32147:
32144:
32141:
32136:
32132:
32128:
32125:
32122:
32119:
32116:
32113:
32110:
32107:
32104:
32101:
32098:
32078:
32075:
32072:
32069:
32066:
32046:
32043:
32040:
32035:
32031:
32027:
32007:
32004:
32001:
31998:
31995:
31991:
31987:
31981:
31978:
31971:
31967:
31962:
31958:
31937:
31934:
31931:
31922:where because
31911:
31908:
31905:
31902:
31898:
31894:
31888:
31885:
31878:
31874:
31871:
31868:
31865:
31859:
31856:
31850:
31847:
31827:
31824:
31821:
31818:
31815:
31812:
31792:
31789:
31786:
31783:
31780:
31760:
31757:
31754:
31749:
31745:
31738:
31735:
31729:
31725:
31721:
31715:
31712:
31705:
31701:
31696:
31692:
31672:
31669:
31666:
31663:
31643:
31640:
31637:
31617:
31614:
31611:
31608:
31605:
31598:
31595:
31592:
31588:
31582:
31579:
31559:
31556:
31549:
31546:
31543:
31539:
31517:
31507:
31487:
31467:
31464:
31461:
31458:
31455:
31452:
31449:
31446:
31443:
31440:
31437:
31434:
31412:
31408:
31404:
31401:
31398:
31393:
31389:
31385:
31382:
31379:
31374:
31370:
31366:
31363:
31343:
31340:
31337:
31334:
31331:
31311:
31308:
31305:
31300:
31296:
31275:
31272:
31269:
31266:
31263:
31260:
31257:
31254:
31251:
31232:
31220:
31216:
31195:
31192:
31189:
31186:
31183:
31167:
31155:
31152:
31149:
31144:
31140:
31136:
31133:
31130:
31127:
31124:
31121:
31118:
31113:
31109:
31088:
31085:
31080:
31076:
31072:
31069:
31053:
31041:
31038:
31035:
31032:
31029:
31026:
31005:
30987:
30986:
30984:
30981:
30978:
30977:
30965:
30961:
30939:
30917:
30911:
30906:
30902:
30878:
30874:
30853:
30848:
30843:
30821:
30801:
30796:
30793:
30788:
30785:
30765:
30745:
30740:
30735:
30717:
30705:
30702:
30682:
30679:
30676:
30673:
30668:
30664:
30660:
30657:
30637:
30617:
30614:
30611:
30608:
30605:
30602:
30599:
30595:
30591:
30587:
30567:
30564:
30561:
30558:
30553:
30549:
30545:
30542:
30522:
30519:
30516:
30513:
30508:
30504:
30500:
30497:
30477:
30464:
30438:
30434:
30428:
30423:
30420:
30417:
30413:
30396:
30390:Also called a
30383:
30374:
30367:
30353:
30333:
30330:
30327:
30307:
30291:
30278:
30261:
30260:
30258:
30255:
30254:
30253:
30248:
30239:
30230:
30225:
30219:
30210:
30201:
30196:
30190:
30181:
30172:
30167:
30158:
30152:
30146:
30140:Complete field
30137:
30129:
30126:
30123:
30122:
30119:
30114:
30109:
30104:
30092:
30076:
30071:
30066:
30061:
30054:
30047:
30042:
30037:
30032:
30027:
30020:
30013:
30008:
30001:
29996:
29991:
29988:
29983:
29980:
29975:
29970:
29965:
29960:
29955:
29950:
29946:
29945:
29934:
29923:
29916:
29914:
29907:
29900:
29893:
29886:
29879:
29877:
29870:
29863:
29861:
29854:
29847:
29840:
29833:
29826:
29819:
29812:
29805:
29798:
29791:
29784:
29777:
29770:
29763:
29756:
29749:
29742:
29731:
29719:
29718:
29715:
29713:
29706:
29704:
29697:
29695:
29693:
29686:
29684:
29682:
29680:
29678:
29671:
29664:
29657:
29655:
29653:
29651:
29649:
29647:
29645:
29643:
29641:
29634:
29627:
29620:
29613:
29606:
29599:
29595:
29594:
29591:
29589:
29582:
29580:
29573:
29571:
29569:
29562:
29560:
29558:
29556:
29554:
29547:
29540:
29533:
29526:
29519:
29512:
29505:
29498:
29491:
29484:
29477:
29470:
29463:
29456:
29449:
29442:
29435:
29431:
29430:
29427:
29420:
29413:
29411:
29409:
29407:
29405:
29398:
29396:
29394:
29392:
29385:
29378:
29371:
29364:
29357:
29350:
29343:
29336:
29329:
29322:
29315:
29308:
29301:
29294:
29287:
29280:
29273:
29266:
29262:
29261:
29256:
29254:
29252:
29245:
29238:
29231:
29224:
29217:
29210:
29208:
29206:
29199:
29192:
29185:
29178:
29171:
29164:
29162:
29160:
29158:
29156:
29149:
29142:
29135:
29128:
29121:
29114:
29107:
29100:
29094:
29093:
29090:
29088:
29086:
29079:
29072:
29065:
29058:
29051:
29049:
29047:
29045:
29043:
29041:
29039:
29037:
29030:
29023:
29016:
29009:
29002:
28995:
28988:
28981:
28974:
28967:
28960:
28953:
28946:
28939:
28935:
28934:
28931:
28929:
28927:
28920:
28913:
28906:
28899:
28892:
28890:
28888:
28886:
28884:
28882:
28880:
28878:
28871:
28864:
28857:
28850:
28843:
28836:
28829:
28822:
28815:
28808:
28801:
28794:
28787:
28780:
28776:
28775:
28772:
28770:
28768:
28761:
28754:
28747:
28740:
28733:
28731:
28729:
28727:
28725:
28723:
28721:
28719:
28712:
28705:
28698:
28691:
28684:
28677:
28670:
28663:
28656:
28649:
28642:
28635:
28628:
28621:
28617:
28616:
28611:
28609:
28607:
28600:
28593:
28586:
28579:
28572:
28565:
28563:
28561:
28559:
28557:
28555:
28553:
28546:
28539:
28532:
28530:
28523:
28516:
28509:
28502:
28495:
28488:
28481:
28474:
28467:
28460:
28454:
28453:
28448:
28446:
28444:
28437:
28430:
28423:
28416:
28409:
28402:
28400:
28398:
28396:
28389:
28387:
28385:
28378:
28371:
28364:
28357:
28355:
28348:
28341:
28334:
28327:
28320:
28313:
28306:
28299:
28292:
28286:
28285:
28280:
28278:
28276:
28269:
28262:
28255:
28248:
28241:
28234:
28232:
28230:
28228:
28221:
28214:
28207:
28200:
28193:
28186:
28184:
28177:
28170:
28163:
28156:
28149:
28142:
28135:
28128:
28121:
28114:
28108:
28107:
28102:
28100:
28098:
28091:
28084:
28082:
28080:
28073:
28071:
28069:
28067:
28065:
28063:
28061:
28054:
28047:
28040:
28033:
28026:
28019:
28012:
28005:
27998:
27991:
27984:
27977:
27970:
27963:
27956:
27950:
27949:
27944:
27942:
27940:
27933:
27926:
27924:
27922:
27915:
27913:
27911:
27909:
27907:
27900:
27893:
27891:
27884:
27877:
27870:
27863:
27856:
27849:
27842:
27835:
27828:
27826:
27819:
27812:
27805:
27798:
27792:
27791:
27786:
27784:
27782:
27780:
27773:
27766:
27759:
27752:
27750:
27748:
27746:
27739:
27732:
27725:
27718:
27711:
27704:
27697:
27690:
27683:
27676:
27669:
27662:
27655:
27648:
27641:
27634:
27627:
27620:
27614:
27613:
27610:
27603:
27596:
27589:
27582:
27575:
27568:
27561:
27559:
27552:
27545:
27543:
27536:
27529:
27522:
27515:
27508:
27501:
27494:
27487:
27480:
27473:
27466:
27459:
27452:
27445:
27438:
27431:
27424:
27420:
27419:
27416:
27409:
27402:
27395:
27388:
27381:
27374:
27367:
27360:
27353:
27346:
27339:
27332:
27325:
27318:
27311:
27304:
27297:
27290:
27283:
27276:
27269:
27262:
27255:
27248:
27241:
27234:
27227:
27220:
27216:
27215:
27212:
27205:
27198:
27191:
27184:
27177:
27170:
27163:
27161:
27154:
27147:
27140:
27133:
27126:
27119:
27112:
27105:
27098:
27091:
27084:
27077:
27070:
27063:
27056:
27049:
27042:
27035:
27028:
27021:
27017:
27016:
27005:
27002:
26999:
26989:
26987:
26985:
26983:
26981:
26979:
26977:
26970:
26968:
26966:
26964:
26962:
26955:
26948:
26946:
26944:
26942:
26940:
26933:
26931:
26924:
26917:
26910:
26903:
26896:
26889:
26882:
26875:
26868:
26857:
26854:
26851:
26840:
26839:
26836:
26829:
26827:
26825:
26818:
26816:
26814:
26807:
26805:
26803:
26801:
26799:
26797:
26795:
26788:
26781:
26774:
26767:
26760:
26753:
26746:
26739:
26732:
26725:
26718:
26711:
26704:
26697:
26690:
26686:
26685:
26668:
26657:
26650:
26648:
26646:
26639:
26637:
26635:
26628:
26626:
26624:
26622:
26620:
26618:
26616:
26609:
26602:
26595:
26588:
26581:
26574:
26567:
26560:
26553:
26546:
26539:
26532:
26525:
26518:
26511:
26494:
26482:
26481:
26470:
26467:
26464:
26454:
26447:
26445:
26438:
26431:
26429:
26427:
26420:
26418:
26411:
26409:
26407:
26400:
26393:
26386:
26379:
26372:
26365:
26358:
26351:
26344:
26337:
26330:
26323:
26316:
26309:
26302:
26295:
26288:
26277:
26274:
26271:
26260:
26259:
26256:
26254:
26252:
26245:
26238:
26231:
26224:
26217:
26215:
26208:
26201:
26194:
26187:
26180:
26173:
26166:
26159:
26152:
26145:
26138:
26131:
26124:
26117:
26110:
26103:
26096:
26089:
26082:
26075:
26071:
26070:
26053:
26042:
26040:
26038:
26031:
26024:
26017:
26010:
26003:
26001:
25994:
25987:
25980:
25973:
25966:
25959:
25952:
25945:
25938:
25931:
25924:
25917:
25910:
25903:
25896:
25889:
25882:
25875:
25868:
25861:
25844:
25832:
25831:
25820:
25817:
25814:
25804:
25797:
25790:
25783:
25776:
25769:
25762:
25755:
25753:
25746:
25744:
25737:
25730:
25723:
25716:
25709:
25707:
25700:
25693:
25686:
25679:
25672:
25665:
25658:
25651:
25644:
25637:
25630:
25623:
25612:
25609:
25606:
25595:
25594:
25591:
25586:
25581:
25576:
25564:
25548:
25543:
25538:
25533:
25526:
25519:
25514:
25509:
25504:
25499:
25492:
25485:
25480:
25473:
25468:
25463:
25460:
25455:
25452:
25447:
25442:
25437:
25432:
25427:
25421:
25420:
25408:
25388:
25385:
25366:
25363:
25352:
25331:
25328:
25325:
25302:
25291:
25290:
25288:
25284:
25279:
25277:
25268:
25259:
25256:
25243:
25240:
25237:
25217:
25197:
25177:
25174:
25171:
25151:
25147:
25142:
25138:
25132:
25129:
25122:
25118:
25115:
25112:
25109:
25106:
25102:
25098:
25095:
25092:
25089:
25086:
25083:
25080:
25077:
25074:
25071:
25068:
25065:
25045:
25042:
25039:
25030:Similarly, if
25019:
25016:
25013:
25010:
25007:
25004:
25001:
24998:
24995:
24992:
24989:
24986:
24983:
24980:
24977:
24974:
24971:
24968:
24965:
24962:
24959:
24956:
24953:
24950:
24947:
24944:
24941:
24938:
24919:
24916:
24913:
24910:
24890:
24887:
24884:
24881:
24878:
24858:
24855:
24852:
24849:
24829:
24826:
24823:
24820:
24817:
24814:
24811:
24808:
24805:
24802:
24799:
24796:
24793:
24790:
24787:
24784:
24781:
24778:
24775:
24772:
24769:
24766:
24763:
24760:
24757:
24754:
24735:
24732:
24729:
24726:
24706:
24703:
24700:
24687:is some unary
24676:
24673:
24670:
24667:
24647:
24626:
24623:
24603:
24600:
24578:
24575:
24572:
24569:
24566:
24563:
24560:
24557:
24554:
24551:
24548:
24545:
24542:
24539:
24519:
24516:
24513:
24510:
24507:
24504:
24501:
24498:
24495:
24492:
24489:
24486:
24483:
24463:
24460:
24457:
24454:
24451:
24448:
24428:
24425:
24422:
24419:
24416:
24396:
24393:
24390:
24387:
24384:
24381:
24378:
24375:
24372:
24369:
24366:
24363:
24349:
24346:
24343:
24340:
24337:
24334:
24331:
24328:
24325:
24322:
24319:
24316:
24313:
24310:
24307:
24304:
24290:
24287:
24284:
24281:
24278:
24275:
24272:
24269:
24266:
24263:
24260:
24257:
24254:
24234:
24214:
24211:
24208:
24205:
24202:
24177:
24172:
24146:
24143:
24140:
24137:
24134:
24131:
24128:
24125:
24119:
24113:
24110:
24107:
24101:
24095:
24092:
24089:
24086:
24083:
24080:
24077:
24057:
24054:
24051:
24048:
24045:
24042:
24039:
24036:
24016:
24013:
23989:
23986:
23983:
23957:
23954:
23951:
23948:
23945:
23925:
23922:
23919:
23916:
23913:
23910:
23907:
23904:
23901:
23898:
23895:
23875:
23872:
23869:
23866:
23858:
23855:
23852:
23832:
23829:
23826:
23823:
23820:
23817:
23814:
23811:
23808:
23805:
23802:
23799:
23779:
23756:
23753:
23750:
23747:
23744:
23741:
23738:
23735:
23716:
23713:
23710:
23707:
23704:
23701:
23698:
23695:
23692:
23672:
23662:
23648:
23628:
23625:
23622:
23619:
23616:
23613:
23593:
23590:
23587:
23551:
23548:
23545:
23539:
23535:
23532:
23528:
23524:
23521:
23501:
23473:
23453:
23430:
23406:
23382:
23350:
23347:
23344:
23336:
23315:
23311:
23307:
23304:
23301:
23298:
23295:
23292:
23287:
23283:
23279:
23276:
23273:
23270:
23267:
23264:
23261:
23256:
23252:
23247:
23243:
23240:
23234:
23228:
23225:
23222:
23219:
23216:
23213:
23210:
23207:
23204:
23201:
23196:
23192:
23171:
23168:
23165:
23162:
23159:
23156:
23151:
23147:
23126:
23123:
23120:
23117:
23114:
23111:
23106:
23102:
23081:
23078:
23075:
23072:
23069:
23049:
23046:
23043:
23040:
23037:
23034:
23031:
23026:
23022:
23018:
23015:
23012:
23009:
23006:
23003:
23000:
22995:
22991:
22984:
22978:
22975:
22972:
22969:
22966:
22963:
22960:
22957:
22954:
22951:
22946:
22942:
22921:
22901:
22881:
22878:
22875:
22872:
22869:
22858:
22857:
22846:
22843:
22840:
22837:
22834:
22831:
22828:
22823:
22819:
22803:
22792:
22789:
22786:
22783:
22780:
22777:
22774:
22769:
22765:
22753:
22742:
22739:
22736:
22733:
22730:
22727:
22724:
22719:
22715:
22683:
22680:
22660:
22640:
22637:
22634:
22631:
22628:
22625:
22620:
22616:
22595:
22574:
22570:
22567:
22547:
22544:
22536:
22533:
22530:
22527:
22524:
22521:
22518:
22515:
22512:
22506:
22500:
22497:
22492:
22488:
22467:
22464:
22442:
22420:
22417:
22414:
22411:
22408:
22401:
22396:
22393:
22389:
22382:
22376:
22373:
22368:
22364:
22344:
22341:
22338:
22335:
22312:
22292:
22289:
22269:
22266:
22261:
22257:
22253:
22250:
22247:
22242:
22239:
22236:
22232:
22211:
22208:
22205:
22202:
22182:
22160:
22157:
22154:
22151:
22148:
22143:
22139:
22135:
22132:
22129:
22126:
22123:
22118:
22114:
22093:
22090:
22087:
22067:
22064:
22061:
22058:
22055:
22052:
22047:
22043:
22036:
22029:
22025:
22022:
22019:
22016:
22011:
22007:
22003:
22000:
21997:
21994:
21991:
21986:
21982:
21977:
21973:
21968:
21964:
21949:
21946:
21943:
21940:
21937:
21934:
21929:
21925:
21918:
21912:
21909:
21906:
21903:
21898:
21894:
21890:
21887:
21884:
21881:
21878:
21873:
21869:
21848:
21845:
21842:
21839:
21836:
21813:
21810:
21805:
21801:
21780:
21777:
21772:
21768:
21764:
21761:
21758:
21755:
21752:
21744:
21741:
21738:
21716:
21713:
21710:
21707:
21704:
21701:
21696:
21692:
21688:
21684:
21680:
21677:
21672:
21668:
21663:
21658:
21654:
21651:
21648:
21644:
21623:
21620:
21617:
21609:
21606:
21603:
21583:
21563:
21560:
21557:
21537:
21534:
21531:
21526:
21522:
21518:
21515:
21512:
21509:
21506:
21503:
21500:
21495:
21491:
21470:
21467:
21464:
21459: or
21456:
21453:
21450:
21447:
21438:is Hausdorff,
21427:
21407:
21404:
21401:
21398:
21395:
21392:
21387:
21383:
21379:
21376:
21373:
21368:
21364:
21360:
21340:
21320:
21317:
21314:
21291:
21288:
21268:
21248:
21228:
21225:
21222:
21202:
21182:
21162:
21142:
21107:
21104:
21101:
21098:
21095:
21092:
21072:
21052:
21049:
21029:
21009:
20989:
20969:
20946:
20926:
20906:
20886:
20866:
20842:
20839:
20836:
20833:
20830:
20827:
20822:
20818:
20813:
20809:
20806:
20803:
20783:
20780:
20775:
20771:
20746:
20726:
20701:
20698:
20686:
20683:
20680:
20677:
20674:
20669:
20665:
20661:
20658:
20655:
20652:
20632:
20629:
20626:
20623:
20618:
20614:
20610:
20607:
20587:
20563:
20539:
20536:
20533:
20530:
20525:
20521:
20500:
20480:
20477:
20474:
20471:
20468:
20465:
20462:
20459:
20456:
20436:
20433:
20430:
20427:
20424:
20421:
20418:
20413:
20409:
20405:
20402:
20382:
20362:
20359:
20356:
20353:
20348:
20344:
20323:
20303:
20300:
20297:
20294:
20291:
20286:
20282:
20257:
20254:
20251:
20248:
20243:
20239:
20235:
20232:
20229:
20226:
20206:
20203:
20200:
20197:
20192:
20188:
20184:
20181:
20178:
20175:
20172:
20169:
20149:
20129:
20126:
20123:
20120:
20115:
20111:
20090:
20063:
20060:
20057:
20054:
20049:
20045:
20040:
20036:
20033:
20030:
20027:
20024:
20012:Moreover, the
20001:
19998:
19978:
19958:
19954:
19950:
19930:
19910:
19899:
19898:
19886:
19883:
19880:
19877:
19874:
19871:
19866:
19862:
19857:
19853:
19850:
19847:
19827:
19816:
19804:
19801:
19796:
19792:
19781:
19769:
19766:
19763:
19760:
19755:
19751:
19747:
19744:
19734:
19718:
19695:
19692:
19672:
19669:
19666:
19646:
19643:
19640:
19637:
19632:
19628:
19603:
19593:
19589:
19586:(so that this
19575:
19572:
19569:
19566:
19563:
19560:
19555:
19551:
19547:
19544:
19524:
19504:
19501:
19498:
19478:
19475:
19470:
19466:
19462:
19459:
19456:
19453:
19450:
19445:
19441:
19437:
19434:
19415:
19412:
19409:
19406:
19375:
19372:
19369:
19366:
19361:
19357:
19353:
19350:
19347:
19344:
19341:
19336:
19332:
19311:
19308:
19305:
19283:
19280:
19277:
19274:
19271:
19266:
19262:
19241:
19231:
19227:
19213:
19189:
19186:
19183:
19180:
19175:
19171:
19150:
19147:
19144:
19141:
19121:
19118:
19107:bounded subset
19090:
19087:
19084:
19081:
19076:
19072:
19058:is always the
19047:
19044:
19041:
19038:
19033:
19029:
19004:
19001:
18998:
18995:
18990:
18986:
18965:
18945:
18940:
18937:
18934:
18929:
18924:
18921:
18917:
18913:
18910:
18907:
18904:
18901:
18896:
18892:
18871:
18868:
18858:
18844:
18841:
18838:
18835:
18832:
18827:
18823:
18802:
18799:
18779:
18776:
18773:
18753:
18750:
18747:
18744:
18741:
18736:
18732:
18728:
18725:
18722:
18719:
18699:
18696:
18676:
18673:
18670:
18650:
18638:
18635:
18623:
18620:
18617:
18614:
18611:
18608:
18605:
18602:
18599:
18596:
18593:
18573:
18570:
18567:
18547:
18544:
18541:
18538:
18535:
18532:
18529:
18526:
18523:
18520:
18517:
18509:
18506:
18503:
18483:
18480:
18477:
18474:
18471:
18468:
18465:
18462:
18459:
18456:
18453:
18450:
18447:
18444:
18441:
18438:
18435:
18427:
18424:
18421:
18418:
18415:
18395:
18392:
18387:
18383:
18379:
18376:
18373:
18370:
18367:
18364:
18361:
18358:
18355:
18352:
18349:
18346:
18343:
18340:
18337:
18334:
18331:
18328:
18325:
18322:
18319:
18299:
18295:
18274:
18250:
18246:
18242:
18239:
18219:
18216:
18213:
18210:
18206:
18202:
18198:
18194:
18190:
18186:
18183:
18180:
18177:
18172:
18168:
18147:
18127:
18103:
18083:
18080:
18077:
18074:
18069: if
18066:
18063:
18060:
18057:
18054:
18051:
18048:
18028:
18025:
18022:
18017: if
18014:
18009:
18005:
18001:
17998:
17995:
17992:
17989:
17986:
17983:
17980:
17977:
17974:
17971:
17968:
17965:
17962:
17959:
17956:
17953:
17950:
17947:
17944:
17941:
17938:
17935:
17932:
17912:
17909:
17906:
17901:
17897:
17893:
17890:
17870:
17867:
17864:
17844:
17822:
17819:
17815:
17811:
17807:
17786:
17764:
17760:
17739:
17736:
17733:
17730:
17725:
17722:
17718:
17714:
17710:
17706:
17703:
17699:
17695:
17692:
17687:
17683:
17679:
17676:
17673:
17668:
17664:
17643:
17623:
17620:
17617:
17595:
17592:
17589:
17584:
17580:
17576:
17573:
17570:
17567:
17564:
17561:
17558:
17555:
17552:
17549:
17546:
17543:
17540:
17537:
17534:
17531:
17528:
17525:
17505:
17502:
17482:
17462:
17442:
17439:
17434:
17430:
17426:
17423:
17403:
17400:
17397:
17392:
17388:
17384:
17381:
17361:
17341:
17321:
17318:
17315:
17312:
17306:
17300:
17297:
17294:
17291:
17288:
17285:
17282:
17279:
17276:
17273:
17270:
17267:
17264:
17244:
17241:
17238:
17235:
17232:
17229:
17220:is convex and
17209:
17187:
17184:
17181:
17178:
17175:
17172:
17169:
17166:
17163:
17160:
17157:
17154:
17151:
17131:
17128:
17125:
17122:
17119:
17116:
17113:
17110:
17107:
17104:
17101:
17098:
17095:
17092:
17072:
17067:
17062:
17057:
17054:
17034:
17014:
17011:
17006:
17002:
16998:
16995:
16975:
16955:
16952:
16947:
16943:
16939:
16936:
16916:
16913:
16908:
16904:
16883:
16880:
16875:
16871:
16867:
16864:
16861:
16858:
16838:
16818:
16815:
16812:
16809:
16804:
16800:
16775:
16745:
16742:
16739:
16736:
16733:
16730:
16727:
16722:
16718:
16714:
16711:
16708:
16703:
16699:
16695:
16692:
16686:
16680:
16677:
16674:
16671:
16666:
16662:
16658:
16655:
16652:
16649:
16646:
16641:
16637:
16615:
16611:
16608:
16603:
16599:
16594:
16590:
16585:
16581:
16574:
16568:
16565:
16560:
16556:
16540:
16536:
16533:
16528:
16524:
16519:
16515:
16510:
16506:
16499:
16493:
16490:
16485:
16481:
16460:
16440:
16437:
16434:
16431:
16428:
16400:
16378:
16373:
16368:
16365:
16345:
16313:
16290:
16287:
16267:
16264:
16259:
16255:
16232:
16210:
16186:
16181:
16178:
16173:
16168:
16164:
16160:
16155:
16150:
16146:
16125:
16122:
16117:
16113:
16109:
16106:
16086:
16083:
16080:
16077:
16057:
16054:
16051:
16048:
16028:
16025:
16003:
15979:
15976:
15973:
15970:
15967:
15964:
15944:
15941:
15936:
15932:
15911:
15908:
15905:
15902:
15899:
15879:
15859:
15837:
15834:
15831:
15826:
15821:
15817:
15813:
15808:
15803:
15799:
15778:
15775:
15772:
15752:
15749:
15729:
15709:
15686:
15666:
15646:
15636:
15622:
15619:
15614:
15610:
15603:
15597:
15594:
15591:
15588:
15585:
15582:
15579:
15576:
15573:
15570:
15567:
15564:
15558:
15552:
15546:
15540:
15537:
15534:
15531:
15528:
15525:
15522:
15519:
15516:
15513:
15510:
15507:
15501:
15495:
15492:
15487:
15483:
15462:
15436:
15432:
15428:
15425:
15405:
15378:
15356:
15353:
15333:
15310:
15307:
15304:
15284:
15281:
15278:
15275:
15272:
15269:
15266:
15263:
15260:
15257:
15254:
15251:
15248:
15245:
15239:
15233:
15230:
15227:
15224:
15221:
15218:
15215:
15212:
15209:
15206:
15203:
15200:
15197:
15194:
15174:
15151:
15148:
15145:
15125:
15122:
15119:
15099:
15079:
15076:
15073:
15053:
15033:
15013:
15010:
15007:
14978:
14975:
14962:
14942:
14939:
14936:
14933:
14930:
14927:
14922:
14918:
14914:
14911:
14908:
14903:
14899:
14878:
14875:
14870:
14866:
14862:
14859:
14856:
14851:
14847:
14826:
14823:
14820:
14798:
14794:
14788:
14784:
14780:
14777:
14774:
14769:
14765:
14759:
14755:
14730:
14727:
14717:
14701:
14681:
14678:
14675:
14649:
14646:
14643:
14624:
14621:
14618:
14615:
14596:
14593:
14590:
14587:
14568:
14565:
14562:
14557:
14553:
14532:
14529:
14524:
14520:
14499:
14475:
14472:
14452:
14432:
14391:
14388:
14368:
14365:
14362:
14344:
14341:
14328:
14324:
14321:
14299:
14295:
14292:
14289:
14285:
14282:
14261:
14232:
14228:
14224:
14221:
14200:
14196:
14170:
14167:
14136:
14133:
14132:
14131:
14119:
14099:
14096:
14074:
14069:
14045:
14040:
14024:
14008:
14003:
14000:
13997:
13993:
13982:Sobolev spaces
13968:
13964:
13941:
13937:
13921:Hilbert spaces
13918:
13907:
13902:
13898:
13875:
13871:
13847:
13843:
13831:
13823:
13820:certain spaces
13803:
13800:
13780:
13777:
13774:
13771:
13768:
13765:
13743:
13739:
13720:
13702:
13695:Nuclear spaces
13692:
13689:inverse limits
13681:Fréchet spaces
13670:
13658:
13654:
13650:
13647:
13644:
13639:
13636:
13633:
13629:
13624:
13618:
13615:
13612:
13609:
13606:
13603:
13600:
13597:
13593:
13589:
13584:
13581:
13578:
13574:
13570:
13567:
13547:
13543:
13539:
13534:
13530:
13518:Fréchet spaces
13515:
13498:
13484:
13470:
13460:
13449:
13446:
13443:
13440:
13437:
13417:
13414:
13411:
13408:
13386:
13382:
13360:consisting of
13351:
13340:
13337:
13334:
13310:
13306:
13269:
13266:
13245:
13225:
13205:
13182:
13155:
13135:
13132:
13129:
13126:
13106:
13094:
13091:
13079:
13076:
13066:
13052:
13034:
13016:
12992:
12975:
12972:
12971:
12970:
12969:
12968:
12956:
12936:
12916:
12913:
12910:
12907:
12904:
12901:
12898:
12878:
12875:
12855:
12852:
12849:
12846:
12843:
12833:
12819:
12816:
12813:
12810:
12807:
12804:
12784:
12773:
12762:
12759:
12739:
12719:
12708:locally convex
12692:
12688:
12684:
12680:
12676:
12673:
12670:
12667:
12664:
12661:
12640:
12636:
12633:
12630:
12627:
12623:
12619:
12616:
12613:
12610:
12606:
12602:
12598:
12576:
12572:
12569:
12566:
12562:
12558:
12554:
12530:
12525:
12520:
12498:
12495:
12475:
12462:
12448:
12428:
12425:
12422:
12419:
12416:
12413:
12410:
12386:
12357:
12353:
12332:
12329:
12309:
12306:
12303:
12300:
12297:
12294:
12289:
12285:
12265:
12261:
12239:
12235:
12232:
12212:
12209:
12206:
12203:
12200:
12197:
12177:
12157:
12154:
12149:
12145:
12141:
12138:
12118:
12098:
12095:
12092:
12089:
12086:
12083:
12073:
12059:
12039:
12036:
12033:
12030:
12027:
12024:
12021:
12016:
12012:
11991:
11971:
11951:
11930:
11926:
11921:
11917:
11896:
11893:
11890:
11887:
11883:
11862:
11859:
11839:
11819:
11799:
11795:
11791:
11788:
11768:
11765:
11762:
11757:
11753:
11749:
11746:
11743:
11723:
11703:
11700:
11680:
11676:
11672:
11669:
11649:
11646:
11643:
11640:
11637:
11634:
11629:
11625:
11604:
11601:
11597:
11593:
11588:
11584:
11563:
11560:
11557:
11554:
11551:
11548:
11528:
11507:
11485:
11481:
11476:
11471:
11468:
11465:
11460:
11455:
11451:
11447:
11425:
11422:
11419:
11416:
11413:
11391:
11386:
11383:
11380:
11375:
11371:
11366:
11362:
11357:
11335:
11315:
11312:
11309:
11306:
11303:
11283:
11262:
11258:
11255:
11235:
11232:
11229:
11209:
11205:
11184:
11164:
11144:
11141:
11138:
11118:
11115:
11112:
11108:
11104:
11100:
11096:
11092:
11088:
11085:
11082:
11079:
11074:
11070:
11048:
11026:
11011:
11010:
11009:
11008:
11007:
10986:
10965:
10944:
10916:
10896:
10893:
10890:
10887:
10884:
10873:
10862:
10852:
10834:
10831:
10828:
10825:
10822:
10802:
10799:
10796:
10793:
10790:
10776:
10762:
10753:then although
10742:
10739:
10736:
10733:
10730:
10710:
10707:
10704:
10701:
10698:
10678:
10659:
10656:
10626:
10621:
10599:
10594:
10589:
10577:
10561:
10539:
10534:
10512:
10486:
10481:
10459:
10439:
10436:
10433:
10430:
10427:
10406:
10385:
10360:
10338:
10316:
10294:
10266:
10263:
10250:
10247:
10244:
10224:
10220:
10216:
10213:
10210:
10207:
10204:
10201:
10198:
10195:
10191:
10171:locally convex
10150:
10146:
10125:
10122:
10119:
10116:
10096:
10093:
10090:
10085:
10081:
10061:
10058:
10038:
10018:
9993:
9989:
9968:
9965:
9962:
9942:
9912:
9907:
9904:
9901:
9896:
9891:
9887:
9883:
9871:
9860:
9856:
9826:
9820:
9815:
9812:
9788:
9785:
9779:
9773:
9748:
9723:
9701:
9697:
9693:
9689:
9686:
9666:
9646:
9643:
9637:
9632:locally convex
9630:
9614:
9610:
9585:
9564:
9558:
9554:
9550:
9547:
9543:
9520:
9516:
9495:
9475:
9455:
9452:
9424:
9421:
9399:
9395:
9367:
9364:
9361:
9358:
9355:
9340:locally convex
9314:
9294:
9274:
9271:
9268:
9265:
9262:
9225:
9213:
9210:
9208:
9205:
9204:
9203:
9192:
9189:
9169:
9166:
9161:
9157:
9136:
9114:
9110:
9104:
9099:
9096:
9093:
9089:
9077:
9066:
9063:
9043:
9029:
9017:
8993:
8982:
8970:
8950:
8930:
8919:
8916:
8866:
8863:
8860:
8857:
8837:
8834:
8831:
8811:
8808:
8803:
8799:
8795:
8790:
8786:
8765:
8745:
8742:
8722:
8696:
8693:
8690:
8685:
8680:
8676:
8672:
8667:
8662:
8658:
8603:uniform spaces
8588:
8585:
8565:
8541:
8538:
8535:
8532:
8529:
8508:Main article:
8505:
8502:
8490:
8487:
8465:
8460:
8433:
8400:
8357:
8356:
8340:
8337:
8334:
8331:
8328:
8318:
8307:
8304:
8284:
8281:
8261:
8241:
8226:
8210:
8207:
8204:
8201:
8198:
8188:
8177:
8174:
8147:
8127:
8124:
8121:
8097:
8094:
8091:
8088:
8085:
8066:
8064:
8061:
8049:
8046:
8026:
8017:is bounded in
8006:
7986:
7983:
7963:
7937:
7934:
7925:is bounded on
7914:
7894:
7870:
7850:
7847:
7844:
7841:
7838:
7818:
7798:
7778:
7758:
7735:
7732:
7729:
7726:
7706:
7686:
7660:
7640:
7621:locally convex
7608:
7605:
7581:
7562:
7561:
7550:
7547:
7544:
7541:
7538:
7518:
7515:
7512:
7509:
7506:
7490:
7478:
7455:
7444:
7441:
7438:
7435:
7432:
7412:
7409:
7406:
7403:
7400:
7397:
7394:
7391:
7388:
7385:
7382:
7366:
7355:
7352:
7348:
7344:
7340:
7319:
7316:
7313:
7310:
7290:
7279:
7276:
7273:
7269:
7265:
7261:
7240:
7220:
7217:
7214:
7211:
7191:
7188:
7185:
7165:
7162:
7159:
7156:
7136:
7118:is said to be
7107:
7087:
7075:
7072:
7059:
7039:
7019:
6995:
6992:
6989:
6969:
6966:
6963:
6943:
6940:
6935:
6931:
6927:
6924:
6921:
6918:
6915:
6912:
6909:
6906:
6903:
6898:
6894:
6873:
6870:
6867:
6864:
6844:
6841:
6838:
6815:
6812:
6809:
6806:
6803:
6783:
6780:
6777:
6757:
6754:
6751:
6748:
6728:
6725:
6722:
6719:
6699:
6696:
6693:
6673:
6670:
6667:
6656:
6655:
6643:
6640:
6635:
6631:
6606:
6603:
6598:
6594:
6590:
6587:
6567:
6564:
6561:
6541:
6538:
6535:
6530:
6526:
6511:
6501:
6498:
6486:
6483:
6463:
6459:
6455:
6436:
6433:
6430:
6427:
6407:
6387:
6383:
6379:
6359:
6339:
6315:
6311:
6307:
6287:
6284:
6281:
6278:
6255:
6220:
6217:
6201:
6198:
6181:
6178:
6175:
6169:
6164:
6143:
6140:
6137:
6134:
6131:
6110:
6085:
6081:
6055:
6050:
6029:
6026:
6001:
5979:
5975:
5972:
5952:
5931:
5910:
5897:
5883:
5862:
5859:
5856:
5853:
5850:
5847:
5822:
5801:
5780:
5759:
5756:
5753:
5750:
5747:
5726:
5711:
5707:
5684:
5680:
5676:
5673:
5667:
5663:
5658:
5654:
5649:
5645:
5641:
5637:
5634:
5607:
5581:
5558:
5554:
5550:
5545:
5541:
5518:
5514:
5510:
5505:
5501:
5479:
5475:
5470:
5466:
5445:
5441:
5437:
5432:
5428:
5424:
5419:
5415:
5393:
5367:is said to be
5355:
5334:
5331:
5310:
5298:
5297:
5286:
5280:
5276:
5273:
5268:
5262:
5258:
5254:
5249:
5245:
5240:
5235:
5230:
5226:
5222:
5217:
5213:
5195:
5184:
5178:
5174:
5171:
5166:
5160:
5156:
5152:
5147:
5143:
5138:
5133:
5128:
5124:
5120:
5115:
5111:
5093:
5082:
5076:
5072:
5069:
5064:
5058:
5054:
5050:
5046:
5041:
5036:
5032:
5028:
5011:
5000:
4995:
4991:
4984:
4980:
4977:
4973:
4969:
4964:
4960:
4956:
4953:
4936:
4925:
4921:
4916:
4912:
4909:
4906:
4901:
4897:
4892:
4888:
4883:
4879:
4875:
4872:
4855:
4844:
4841:
4819:
4815:
4811:
4806:
4802:
4779:
4775:
4771:
4766:
4762:
4736:
4732:
4707:
4703:
4679:
4659:
4636:
4632:
4629:
4624:
4619:
4615:
4611:
4606:
4601:
4597:
4573:
4569:
4566:
4561:
4556:
4552:
4548:
4543:
4538:
4534:
4503:
4500:
4480:
4477:
4474:
4471:
4468:
4465:
4462:
4459:
4456:
4453:
4450:
4447:
4444:
4424:
4402:
4398:
4377:
4357:
4335:
4331:
4310:
4290:
4287:
4284:
4281:
4261:
4258:
4254:
4250:
4246:
4225:
4205:
4202:
4199:
4196:
4193:
4190:
4187:
4184:
4181:
4178:
4156:
4152:
4131:
4128:
4125:
4122:
4119:
4116:
4113:
4110:
4107:
4104:
4093:symmetric sets
4078:
4074:
4053:
4050:
4047:
4044:
4041:
4038:
4018:
4013:
4009:
4003:
4000:
3997:
3993:
3972:
3969:
3966:
3946:
3943:
3940:
3937:
3934:
3914:
3911:
3908:
3905:
3902:
3899:
3896:
3893:
3890:
3887:
3884:
3881:
3878:
3875:
3872:
3869:
3849:
3827:
3823:
3819:
3816:
3813:
3809:
3805:
3801:
3795:
3791:
3787:
3784:
3781:
3776:
3772:
3767:
3763:
3758:
3754:
3747:
3737:
3733:
3729:
3725:
3721:
3718:
3711:
3707:
3703:
3699:
3694:
3687:
3683:
3680:
3677:
3674:
3671:
3649:
3645:
3641:
3638:
3618:
3615:
3612:
3609:
3606:
3603:
3583:
3580:
3577:
3574:
3571:
3568:
3565:
3562:
3559:
3537:
3533:
3525:
3521:
3516:
3512:
3509:
3506:
3499:
3495:
3490:
3486:
3483:
3475:
3472:
3464:
3461:
3456:
3452:
3448:
3445:
3442:
3439:
3433:
3426:
3420:
3416:
3412:
3409:
3406:
3401:
3397:
3392:
3388:
3383:
3379:
3374:
3370:
3367:
3364:
3361:
3357:
3336:
3331:
3327:
3323:
3320:
3300:
3297:
3294:
3272:
3268:
3264:
3259:
3256:
3253:
3249:
3245:
3240:
3237:
3234:
3230:
3207:
3203:
3199:
3196:
3174:
3169:
3166:
3163:
3158:
3153:
3149:
3145:
3140:
3135:
3131:
3104:
3087:
3063:
3042:
3020:
3017:
3014:
3009:
3005:
2984:
2979:
2976:
2973:
2969:
2965:
2960:
2956:
2935:
2908:
2897:
2896:
2885:
2882:
2860:
2856:
2835:
2809:
2795:
2791:
2772:
2761:
2756:
2752:
2699:
2688:
2685:
2663:
2659:
2655:
2650:
2647:
2644:
2640:
2636:
2631:
2628:
2625:
2621:
2587:
2583:
2562:
2557:
2553:
2539:is called the
2526:
2522:
2501:
2496:
2492:
2469:
2459:is called the
2446:
2442:
2422:
2419:
2397:
2393:
2362:
2358:
2337:
2334:
2312:
2307:
2304:
2301:
2296:
2291:
2287:
2283:
2278:
2273:
2269:
2248:
2236:
2233:
2214:
2211:
2191:
2181:
2177:
2161:
2148:
2147:
2136:
2133:
2130:
2127:
2124:
2121:
2099:
2094:
2091:
2069:
2064:
2061:
2039:
2027:
2005:
2000:
1997:
1967:
1945:
1942:
1922:
1896:
1874:
1840:
1818:
1797:
1776:
1773:
1768:
1764:
1760:
1757:
1737:
1734:
1731:
1711:
1708:
1705:
1700:
1696:
1683:
1680:
1663:
1660:
1657:
1654:
1651:
1627:
1624:
1621:
1601:
1598:
1595:
1592:
1589:
1586:
1583:
1580:
1577:
1557:
1554:
1551:
1548:
1545:
1521:
1518:
1515:
1495:
1492:
1472:
1448:
1445:
1442:
1439:
1436:
1412:
1400:
1388:
1385:
1382:
1379:
1376:
1373:
1351:
1346:
1343:
1323:
1318:
1313:
1310:
1300:
1284:
1246:TVS embedding
1234:
1226:
1218:
1210:
1191:
1180:
1172:
1152:
1149:
1125:
1122:
1102:
1099:
1096:
1093:
1090:
1087:
1084:
1081:
1078:
1054:
1051:
1048:
1045:
1042:
1039:
1036:
1016:
1013:
1010:
1007:
1004:
988:
978:
970:
946:
918:
893:
887:
881:
878:
875:
872:
869:
844:
838:
835:
832:
809:
778:
759:
745:
707:
704:
660:
657:
654:
651:
647:
643:
640:
620:
617:
614:
611:
608:
605:
601:
597:
593:
560:
532:
500:
497:
470:test functions
453:
450:
446:Hilbert spaces
438:
437:
425:
422:
401:
380:
377:
374:
371:
368:
365:
362:
359:
356:
353:
333:
330:
327:
324:
320:
316:
313:
302:
286:
283:
280:
277:
274:
271:
268:
265:
262:
242:
239:
236:
233:
230:
227:
223:
219:
215:
194:has a natural
187:
184:
182:
179:
166:
162:
137:
100:Sobolev spaces
96:Hilbert spaces
69:
26:
9:
6:
4:
3:
2:
36362:
36351:
36350:Vector spaces
36348:
36346:
36343:
36341:
36338:
36336:
36333:
36332:
36330:
36315:
36307:
36306:
36303:
36297:
36294:
36292:
36289:
36287:
36284:
36282:
36278:
36274:
36272:) convex
36271:
36268:
36266:
36263:
36261:
36257:
36255:
36252:
36250:
36247:
36245:
36244:Semi-complete
36242:
36240:
36237:
36235:
36232:
36230:
36226:
36223:
36221:
36217:
36215:
36212:
36210:
36207:
36205:
36202:
36200:
36197:
36195:
36192:
36190:
36187:
36185:
36182:
36180:
36177:
36175:
36172:
36170:
36167:
36165:
36162:
36160:
36159:Infrabarreled
36157:
36155:
36152:
36150:
36147:
36143:
36140:
36139:
36138:
36135:
36133:
36130:
36128:
36125:
36123:
36120:
36118:
36117:Distinguished
36115:
36113:
36110:
36108:
36105:
36103:
36100:
36098:
36095:
36093:
36089:
36085:
36083:
36080:
36078:
36074:
36070:
36068:
36065:
36063:
36060:
36058:
36055:
36054:
36052:
36050:Types of TVSs
36048:
36042:
36038:
36034:
36032:
36029:
36027:
36024:
36022:
36019:
36017:
36014:
36012:
36008:
36004:
36002:
35999:
35998:
35996:
35992:
35986:
35983:
35981:
35978:
35976:
35973:
35971:
35970:Prevalent/Shy
35968:
35966:
35963:
35961:
35960:Extreme point
35958:
35956:
35950:
35948:
35942:
35940:
35937:
35935:
35932:
35930:
35927:
35925:
35922:
35920:
35917:
35915:
35912:
35910:
35907:
35905:
35902:
35900:
35897:
35896:
35894:
35892:Types of sets
35890:
35884:
35881:
35879:
35876:
35874:
35871:
35869:
35866:
35862:
35859:
35857:
35854:
35852:
35849:
35848:
35847:
35844:
35842:
35839:
35835:
35834:Discontinuous
35832:
35830:
35827:
35825:
35822:
35820:
35817:
35815:
35812:
35810:
35807:
35805:
35802:
35801:
35800:
35797:
35793:
35790:
35789:
35788:
35785:
35784:
35782:
35778:
35772:
35769:
35765:
35762:
35761:
35760:
35757:
35754:
35751:
35749:
35745:
35742:
35740:
35737:
35735:
35732:
35730:
35727:
35725:
35722:
35721:
35719:
35717:
35713:
35707:
35704:
35702:
35699:
35697:
35694:
35692:
35691:Metrizability
35689:
35687:
35684:
35682:
35679:
35677:
35676:Fréchet space
35674:
35672:
35669:
35667:
35664:
35662:
35659:
35657:
35654:
35653:
35651:
35647:
35642:
35635:
35630:
35628:
35623:
35621:
35616:
35615:
35612:
35600:
35592:
35591:
35588:
35582:
35579:
35577:
35574:
35572:
35571:Weak topology
35569:
35567:
35564:
35562:
35559:
35557:
35554:
35553:
35551:
35547:
35540:
35536:
35533:
35531:
35528:
35526:
35523:
35521:
35518:
35516:
35513:
35511:
35508:
35506:
35503:
35501:
35498:
35496:
35495:Index theorem
35493:
35491:
35488:
35486:
35483:
35481:
35478:
35477:
35475:
35471:
35465:
35462:
35460:
35457:
35456:
35454:
35452:Open problems
35450:
35444:
35441:
35439:
35436:
35434:
35431:
35429:
35426:
35424:
35421:
35419:
35416:
35415:
35413:
35409:
35403:
35400:
35398:
35395:
35393:
35390:
35388:
35385:
35383:
35380:
35378:
35375:
35373:
35370:
35368:
35365:
35363:
35360:
35358:
35355:
35354:
35352:
35348:
35342:
35339:
35337:
35334:
35332:
35329:
35327:
35324:
35322:
35319:
35317:
35314:
35312:
35309:
35307:
35304:
35302:
35299:
35298:
35296:
35294:
35290:
35280:
35277:
35275:
35272:
35270:
35267:
35264:
35260:
35256:
35253:
35251:
35248:
35246:
35243:
35242:
35240:
35236:
35230:
35227:
35225:
35222:
35220:
35217:
35215:
35212:
35210:
35207:
35205:
35202:
35200:
35197:
35195:
35192:
35190:
35187:
35185:
35182:
35181:
35178:
35175:
35171:
35166:
35162:
35158:
35151:
35146:
35144:
35139:
35137:
35132:
35131:
35128:
35121:
35116:
35112:
35111:
35101:
35097:
35093:
35087:
35083:
35079:
35075:
35074:Voigt, JĂĽrgen
35071:
35067:
35063:
35059:
35053:
35049:
35045:
35041:
35037:
35033:
35029:
35025:
35019:
35015:
35011:
35007:
35003:
34999:
34995:
34989:
34985:
34981:
34977:
34972:
34968:
34966:0-201-04166-9
34962:
34958:
34954:
34950:
34946:
34942:
34938:
34932:
34928:
34924:
34920:
34916:
34910:
34906:
34902:
34901:Horváth, John
34898:
34894:
34890:
34886:
34880:
34875:
34874:
34868:
34864:
34860:
34856:
34852:
34846:
34842:
34837:
34833:
34829:
34825:
34819:
34815:
34811:
34810:
34805:
34801:
34797:
34793:
34789:
34785:
34779:
34775:
34771:
34767:
34763:
34759:
34755:
34751:
34747:
34745:3-540-13627-4
34741:
34737:
34733:
34729:
34725:
34713:
34709:
34705:
34701:
34700:
34689:
34685:
34681:
34675:
34671:
34667:
34663:
34659:
34655:
34651:
34645:
34641:
34636:
34632:
34628:
34624:
34618:
34614:
34610:
34606:
34602:
34598:
34594:
34588:
34584:
34580:
34576:
34572:
34568:
34564:
34560:
34554:
34550:
34546:
34545:
34540:
34539:Rudin, Walter
34536:
34532:
34528:
34524:
34518:
34514:
34509:
34505:
34501:
34497:
34493:
34489:
34483:
34479:
34475:
34471:
34467:
34463:
34459:
34453:
34449:
34444:
34440:
34436:
34432:
34426:
34422:
34418:
34413:
34412:
34400:, p. 35.
34399:
34394:
34387:
34382:
34375:
34370:
34368:
34366:
34358:
34353:
34346:
34341:
34334:
34329:
34322:
34321:Wilansky 2013
34317:
34315:
34313:
34305:
34300:
34298:
34291:, p. 63.
34290:
34289:Wilansky 2013
34285:
34278:
34273:
34266:
34261:
34259:
34252:, p. 25.
34251:
34246:
34244:
34236:
34231:
34229:
34227:
34225:
34223:
34215:
34210:
34203:
34198:
34196:
34194:
34192:
34190:
34188:
34180:
34175:
34169:, p. 38.
34168:
34163:
34161:
34153:
34148:
34141:
34136:
34134:
34127:, p. 55.
34126:
34121:
34119:
34112:, p. 42.
34111:
34110:Wilansky 2013
34106:
34100:, p. 43.
34099:
34098:Wilansky 2013
34094:
34087:
34082:
34076:, p. 35.
34075:
34070:
34064:, p. 38.
34063:
34058:
34056:
34054:
34046:
34041:
34039:
34031:
34026:
34019:
34014:
34007:
34002:
34000:
33998:
33990:
33985:
33983:
33967:
33963:
33957:
33950:
33945:
33938:
33933:
33931:
33929:
33922:, p. 16.
33921:
33916:
33909:
33904:
33897:
33892:
33877:
33873:
33872:
33867:
33861:
33854:
33849:
33842:
33837:
33830:
33825:
33823:
33821:
33819:
33817:
33809:
33804:
33797:
33792:
33790:
33788:
33781:, p. 53.
33780:
33779:Wilansky 2013
33775:
33768:
33763:
33756:
33751:
33749:
33747:
33745:
33743:
33741:
33733:
33728:
33721:
33716:
33714:
33712:
33710:
33708:
33700:
33695:
33693:
33691:
33689:
33687:
33685:
33683:
33681:
33679:
33677:
33675:
33673:
33671:
33669:
33667:
33665:
33663:
33661:
33659:
33657:
33649:
33648:Wilansky 2013
33644:
33642:
33640:
33632:
33627:
33620:
33615:
33609:, p. 91.
33608:
33603:
33601:
33599:
33591:
33586:
33582:
33557:
33550:
33535:
33526:
33523:
33520:
33514:
33511:
33508:
33505:
33502:
33496:
33487:
33484:
33481:
33475:
33472:
33469:
33466:
33463:
33460:
33457:
33451:
33442:
33439:
33436:
33430:
33427:
33424:
33421:
33418:
33415:
33412:
33409:
33406:
33383:
33380:
33377:
33374:
33371:
33368:
33348:
33345:
33342:
33339:
33336:
33329:and so using
33310:
33307:
33301:
33298:
33295:
33286:
33283:
33280:
33277:
33274:
33271:
33268:
33265:
33262:
33256:
33247:
33241:
33238:
33235:
33232:
33229:
33226:
33223:
33220:
33217:
33211:
33202:
33196:
33193:
33190:
33187:
33184:
33178:
33175:
33166:
33147:
33127:
33121:
33101:
33098:
33092:
33086:
33081:
33077:
33073:
33070:
33050:
33030:
33007:
33001:
32996:
32992:
32971:
32951:
32948:
32945:
32940:
32936:
32932:
32926:
32920:
32915:
32911:
32907:
32904:
32901:
32898:
32878:
32875:
32872:
32867:
32863:
32859:
32853:
32847:
32842:
32838:
32834:
32825:
32819:
32816:
32810:
32805:
32801:
32797:
32791:
32785:
32780:
32776:
32772:
32769:
32749:
32746:
32743:
32733:
32711:
32705:
32700:
32696:
32685:
32666:
32663:
32643:
32622:
32618:
32615:
32612:
32608:
32604:
32584:
32581:
32578:
32574:
32570:
32567:
32564:
32560:
32556:
32536:
32533:
32530:
32527:
32524:
32519:
32515:
32511:
32491:
32488:
32485:
32482:
32479:
32476:
32456:
32436:
32433:
32428:
32424:
32420:
32414:
32411:
32408:
32402:
32399:
32396:
32393:
32387:
32381:
32361:
32358:
32355:
32352:
32349:
32329:
32326:
32321:
32317:
32313:
32310:
32305:
32301:
32297:
32294:
32286:
32282:
32275:
32255:
32234:
32230:
32227:
32224:
32220:
32216:
32196:
32193:
32190:
32187:
32184:
32164:
32159:
32155:
32151:
32148:
32145:
32142:
32139:
32134:
32130:
32126:
32120:
32117:
32114:
32108:
32105:
32102:
32096:
32076:
32070:
32067:
32064:
32044:
32041:
32038:
32033:
32029:
32025:
32005:
32002:
31999:
31996:
31993:
31989:
31985:
31979:
31976:
31969:
31965:
31960:
31956:
31935:
31932:
31929:
31909:
31906:
31903:
31900:
31896:
31892:
31886:
31883:
31876:
31872:
31869:
31866:
31863:
31857:
31854:
31848:
31845:
31825:
31822:
31819:
31816:
31813:
31810:
31790:
31787:
31784:
31781:
31778:
31758:
31755:
31752:
31747:
31743:
31736:
31733:
31727:
31723:
31719:
31713:
31710:
31703:
31699:
31694:
31690:
31670:
31667:
31661:
31641:
31638:
31635:
31615:
31612:
31609:
31606:
31603:
31596:
31593:
31590:
31586:
31580:
31577:
31557:
31554:
31547:
31544:
31541:
31537:
31515:
31505:
31485:
31465:
31462:
31459:
31456:
31450:
31447:
31444:
31438:
31435:
31432:
31410:
31406:
31402:
31399:
31396:
31391:
31387:
31383:
31380:
31377:
31372:
31368:
31364:
31361:
31341:
31338:
31335:
31332:
31329:
31322:By replacing
31309:
31306:
31303:
31298:
31294:
31273:
31267:
31264:
31261:
31255:
31252:
31249:
31230:
31218:
31214:
31193:
31190:
31187:
31184:
31181:
31171:
31153:
31150:
31147:
31142:
31138:
31134:
31128:
31122:
31119:
31116:
31111:
31107:
31086:
31083:
31078:
31074:
31070:
31067:
31057:
31039:
31033:
31030:
31027:
30992:
30988:
30963:
30909:
30904:
30892:
30891:Minkowski sum
30876:
30851:
30846:
30819:
30799:
30794:
30791:
30786:
30783:
30763:
30743:
30738:
30721:
30703:
30700:
30693:is closed in
30677:
30671:
30666:
30662:
30658:
30655:
30635:
30615:
30612:
30606:
30603:
30600:
30597:
30593:
30589:
30585:
30562:
30556:
30551:
30547:
30543:
30540:
30517:
30511:
30506:
30502:
30498:
30495:
30487:
30481:
30462:
30454:
30436:
30432:
30421:
30418:
30415:
30411:
30400:
30393:
30387:
30378:
30371:
30351:
30328:
30305:
30295:
30276:
30266:
30262:
30252:
30249:
30243:
30240:
30234:
30231:
30229:
30226:
30223:
30220:
30214:
30211:
30205:
30202:
30200:
30197:
30194:
30191:
30185:
30182:
30176:
30173:
30171:
30168:
30162:
30159:
30156:
30153:
30150:
30149:Hilbert space
30147:
30141:
30138:
30135:
30132:
30131:
30120:
30118:
30115:
30113:
30110:
30108:
30105:
30090:
30082:
30077:
30075:
30072:
30070:
30067:
30065:
30062:
30060:
30055:
30053:
30048:
30046:
30043:
30041:
30038:
30033:
30031:
30028:
30026:
30021:
30014:
30012:
30009:
30002:
29997:
29992:
29989:
29984:
29981:
29976:
29971:
29969:
29966:
29964:
29961:
29959:
29956:
29954:
29951:
29948:
29947:
29932:
29924:
29915:
29878:
29862:
29729:
29721:
29720:
29716:
29714:
29705:
29696:
29694:
29685:
29683:
29681:
29679:
29656:
29654:
29652:
29650:
29648:
29646:
29644:
29642:
29597:
29596:
29592:
29590:
29581:
29572:
29570:
29561:
29559:
29557:
29555:
29433:
29432:
29428:
29412:
29410:
29408:
29406:
29397:
29395:
29393:
29264:
29263:
29260:
29257:
29255:
29253:
29209:
29207:
29163:
29161:
29159:
29157:
29099:
29096:
29095:
29091:
29089:
29087:
29050:
29048:
29046:
29044:
29042:
29040:
29038:
28937:
28936:
28932:
28930:
28928:
28891:
28889:
28887:
28885:
28883:
28881:
28879:
28778:
28777:
28773:
28771:
28769:
28732:
28730:
28728:
28726:
28724:
28722:
28720:
28619:
28618:
28615:
28612:
28610:
28608:
28564:
28562:
28560:
28558:
28556:
28554:
28531:
28459:
28456:
28455:
28452:
28449:
28447:
28445:
28401:
28399:
28397:
28388:
28386:
28356:
28291:
28288:
28287:
28284:
28283:Balanced hull
28281:
28279:
28277:
28233:
28231:
28229:
28185:
28113:
28112:Balanced hull
28110:
28109:
28106:
28105:Balanced core
28103:
28101:
28099:
28083:
28081:
28072:
28070:
28068:
28066:
28064:
28062:
27955:
27954:Balanced core
27952:
27951:
27948:
27945:
27943:
27941:
27925:
27923:
27914:
27912:
27910:
27908:
27892:
27827:
27797:
27794:
27793:
27790:
27787:
27785:
27783:
27781:
27751:
27749:
27747:
27619:
27616:
27615:
27611:
27560:
27544:
27422:
27421:
27417:
27218:
27217:
27213:
27162:
27019:
27018:
27003:
27000:
26997:
26990:
26988:
26986:
26984:
26982:
26980:
26978:
26969:
26967:
26965:
26963:
26947:
26945:
26943:
26941:
26932:
26855:
26852:
26849:
26842:
26841:
26837:
26828:
26826:
26817:
26815:
26806:
26804:
26802:
26800:
26798:
26796:
26688:
26687:
26684:
26666:
26658:
26649:
26647:
26638:
26636:
26627:
26625:
26623:
26621:
26619:
26617:
26510:
26492:
26484:
26483:
26468:
26465:
26462:
26455:
26446:
26430:
26428:
26419:
26410:
26408:
26275:
26272:
26269:
26262:
26261:
26257:
26255:
26253:
26216:
26073:
26072:
26069:
26051:
26043:
26041:
26039:
26002:
25860:
25842:
25834:
25833:
25818:
25815:
25812:
25805:
25754:
25745:
25708:
25610:
25607:
25604:
25597:
25596:
25592:
25590:
25587:
25585:
25582:
25580:
25577:
25562:
25554:
25549:
25547:
25544:
25542:
25539:
25537:
25534:
25532:
25527:
25525:
25520:
25518:
25515:
25513:
25510:
25505:
25503:
25500:
25498:
25493:
25486:
25484:
25481:
25474:
25469:
25464:
25461:
25456:
25453:
25448:
25443:
25441:
25438:
25436:
25433:
25431:
25428:
25426:
25423:
25422:
25406:
25386:
25383:
25364:
25361:
25349:
25343:
25329:
25326:
25323:
25314:
25300:
25286:
25282:
25280:
25275:
25273:
25269:
25266:
25262:
25261:
25255:
25241:
25238:
25235:
25215:
25195:
25175:
25172:
25169:
25162:The elements
25149:
25145:
25140:
25136:
25130:
25127:
25120:
25116:
25113:
25110:
25107:
25104:
25100:
25096:
25087:
25081:
25078:
25075:
25072:
25069:
25063:
25043:
25040:
25037:
25017:
25011:
25008:
25002:
24999:
24996:
24990:
24987:
24984:
24981:
24975:
24969:
24966:
24960:
24954:
24951:
24948:
24945:
24939:
24936:
24917:
24914:
24911:
24908:
24888:
24885:
24879:
24853:
24847:
24827:
24818:
24815:
24812:
24806:
24803:
24800:
24797:
24794:
24788:
24779:
24773:
24770:
24767:
24764:
24761:
24755:
24752:
24733:
24730:
24727:
24724:
24704:
24701:
24698:
24690:
24671:
24665:
24656:
24654:
24652:
24645:
24642:
24641:
24637:
24624:
24621:
24601:
24598:
24589:
24576:
24570:
24561:
24555:
24552:
24546:
24543:
24540:
24537:
24514:
24505:
24499:
24496:
24490:
24487:
24484:
24481:
24461:
24458:
24455:
24452:
24449:
24446:
24426:
24423:
24420:
24417:
24414:
24394:
24391:
24388:
24385:
24382:
24379:
24373:
24370:
24364:
24361:
24347:
24344:
24341:
24338:
24335:
24332:
24329:
24326:
24323:
24317:
24314:
24311:
24305:
24302:
24288:
24285:
24282:
24279:
24276:
24273:
24270:
24264:
24261:
24258:
24252:
24232:
24212:
24209:
24206:
24203:
24200:
24191:
24175:
24160:
24159:balanced hull
24144:
24138:
24135:
24132:
24126:
24123:
24117:
24111:
24108:
24105:
24099:
24090:
24087:
24084:
24078:
24075:
24055:
24049:
24046:
24043:
24037:
24034:
24014:
24011:
24003:
24002:balanced hull
23987:
23984:
23981:
23973:
23968:
23955:
23952:
23949:
23946:
23943:
23923:
23920:
23917:
23911:
23908:
23905:
23899:
23896:
23893:
23873:
23870:
23867:
23864:
23856:
23853:
23850:
23830:
23827:
23824:
23821:
23818:
23815:
23812:
23806:
23803:
23800:
23777:
23768:
23751:
23748:
23745:
23742:
23736:
23733:
23714:
23708:
23705:
23702:
23699:
23693:
23690:
23670:
23660:
23646:
23626:
23623:
23620:
23617:
23614:
23611:
23591:
23588:
23585:
23576:
23575:
23571:
23569:
23565:
23549:
23546:
23543:
23533:
23530:
23526:
23522:
23519:
23499:
23491:
23490:nowhere dense
23487:
23471:
23451:
23442:
23428:
23420:
23404:
23396:
23380:
23371:
23369:
23368:nowhere dense
23365:
23364:nowhere dense
23361:
23356:
23355:
23346:
23342:
23340:
23334:
23331:
23330:
23326:
23313:
23309:
23302:
23299:
23296:
23290:
23285:
23281:
23277:
23271:
23268:
23265:
23259:
23254:
23250:
23245:
23241:
23238:
23232:
23220:
23217:
23214:
23208:
23205:
23199:
23194:
23190:
23166:
23163:
23160:
23154:
23149:
23145:
23121:
23118:
23115:
23109:
23104:
23100:
23079:
23076:
23073:
23070:
23067:
23047:
23041:
23038:
23035:
23029:
23024:
23020:
23016:
23010:
23007:
23004:
22998:
22993:
22989:
22982:
22970:
22967:
22964:
22958:
22955:
22949:
22944:
22940:
22919:
22899:
22879:
22876:
22873:
22870:
22867:
22844:
22838:
22835:
22832:
22826:
22821:
22817:
22808:
22804:
22790:
22784:
22781:
22778:
22772:
22767:
22763:
22754:
22740:
22734:
22731:
22728:
22722:
22717:
22713:
22704:
22703:
22702:
22699:
22698:
22694:
22681:
22678:
22658:
22638:
22635:
22632:
22629:
22626:
22623:
22618:
22614:
22593:
22568:
22565:
22542:
22534:
22531:
22528:
22525:
22522:
22519:
22516:
22510:
22504:
22498:
22495:
22490:
22486:
22465:
22462:
22415:
22412:
22409:
22394:
22391:
22387:
22380:
22374:
22371:
22366:
22362:
22342:
22339:
22336:
22333:
22324:
22310:
22303:moreover, if
22290:
22287:
22267:
22264:
22259:
22255:
22251:
22248:
22245:
22240:
22237:
22234:
22230:
22209:
22206:
22203:
22200:
22180:
22171:
22158:
22152:
22146:
22141:
22137:
22133:
22127:
22121:
22116:
22112:
22091:
22088:
22085:
22062:
22059:
22056:
22050:
22045:
22041:
22034:
22027:
22020:
22014:
22009:
22005:
22001:
21995:
21989:
21984:
21980:
21975:
21971:
21966:
21962:
21944:
21941:
21938:
21932:
21927:
21923:
21916:
21907:
21901:
21896:
21892:
21888:
21882:
21876:
21871:
21867:
21846:
21843:
21840:
21837:
21834:
21825:
21811:
21808:
21803:
21799:
21778:
21775:
21770:
21766:
21762:
21759:
21756:
21753:
21750:
21742:
21739:
21736:
21727:
21714:
21708:
21705:
21699:
21694:
21690:
21686:
21682:
21678:
21675:
21670:
21666:
21661:
21656:
21652:
21649:
21646:
21642:
21621:
21618:
21615:
21607:
21604:
21601:
21581:
21561:
21558:
21555:
21535:
21532:
21529:
21524:
21520:
21516:
21513:
21507:
21504:
21498:
21493:
21489:
21465:
21462:
21454:
21451:
21448:
21445:
21425:
21405:
21399:
21396:
21390:
21385:
21381:
21377:
21374:
21371:
21366:
21362:
21358:
21338:
21318:
21315:
21312:
21303:
21289:
21286:
21279:is closed in
21266:
21246:
21239:is closed in
21226:
21223:
21220:
21200:
21180:
21160:
21140:
21130:
21128:
21123:
21122:
21118:
21105:
21102:
21099:
21096:
21093:
21090:
21070:
21050:
21047:
21027:
21007:
20987:
20967:
20958:
20944:
20924:
20904:
20884:
20864:
20854:
20834:
20825:
20820:
20816:
20811:
20807:
20801:
20781:
20778:
20773:
20769:
20760:
20744:
20724:
20716:
20713:. Thus, in a
20712:
20707:
20706:
20697:
20684:
20678:
20672:
20667:
20663:
20659:
20656:
20653:
20650:
20627:
20621:
20616:
20612:
20608:
20605:
20585:
20577:
20561:
20553:
20534:
20528:
20523:
20519:
20498:
20478:
20475:
20472:
20463:
20460:
20457:
20434:
20431:
20422:
20416:
20411:
20407:
20403:
20400:
20380:
20357:
20351:
20346:
20342:
20321:
20301:
20295:
20289:
20284:
20280:
20271:
20252:
20246:
20241:
20237:
20233:
20230:
20227:
20224:
20201:
20195:
20190:
20186:
20182:
20179:
20176:
20170:
20147:
20124:
20118:
20113:
20109:
20088:
20079:
20077:
20058:
20052:
20047:
20043:
20038:
20034:
20028:
20025:
20022:
20015:
19999:
19996:
19989:is closed in
19976:
19956:
19952:
19948:
19928:
19908:
19878:
19869:
19864:
19860:
19855:
19851:
19845:
19825:
19818:The image if
19817:
19802:
19799:
19794:
19790:
19782:
19764:
19758:
19753:
19749:
19745:
19742:
19735:
19732:
19716:
19709:
19708:
19707:
19693:
19690:
19670:
19667:
19664:
19641:
19635:
19630:
19626:
19617:
19601:
19591:
19587:
19573:
19570:
19564:
19558:
19553:
19549:
19545:
19542:
19522:
19502:
19499:
19496:
19476:
19473:
19468:
19464:
19460:
19454:
19448:
19443:
19439:
19435:
19432:
19413:
19410:
19407:
19404:
19395:
19393:
19389:
19370:
19364:
19359:
19355:
19351:
19348:
19345:
19342:
19339:
19334:
19330:
19309:
19306:
19303:
19294:
19281:
19275:
19269:
19264:
19260:
19239:
19229:
19225:
19211:
19203:
19184:
19178:
19173:
19169:
19148:
19142:
19119:
19116:
19108:
19104:
19103:compact space
19085:
19079:
19074:
19070:
19061:
19042:
19036:
19031:
19027:
19018:
18999:
18993:
18988:
18984:
18963:
18943:
18935:
18922:
18919:
18915:
18911:
18905:
18899:
18894:
18890:
18869:
18866:
18856:
18842:
18836:
18830:
18825:
18821:
18800:
18797:
18774:
18751:
18745:
18739:
18734:
18730:
18726:
18720:
18697:
18694:
18671:
18648:
18634:
18621:
18618:
18615:
18612:
18606:
18603:
18600:
18597:
18594:
18591:
18568:
18565:
18545:
18542:
18539:
18536:
18533:
18530:
18524:
18521:
18518:
18507:
18504:
18501:
18478:
18475:
18472:
18466:
18463:
18460:
18454:
18451:
18448:
18445:
18436:
18433:
18425:
18422:
18419:
18416:
18413:
18393:
18390:
18385:
18381:
18377:
18374:
18368:
18365:
18362:
18359:
18353:
18350:
18347:
18344:
18338:
18335:
18332:
18326:
18323:
18320:
18317:
18297:
18272:
18264:
18248:
18240:
18237:
18217:
18211:
18208:
18200:
18192:
18184:
18181:
18175:
18170:
18166:
18145:
18125:
18117:
18101:
18081:
18078:
18075:
18072:
18061:
18055:
18052:
18049:
18026:
18023:
18020:
18012:
18007:
18003:
17999:
17993:
17990:
17987:
17984:
17981:
17978:
17975:
17969:
17966:
17963:
17957:
17954:
17951:
17945:
17939:
17936:
17933:
17910:
17907:
17904:
17899:
17895:
17891:
17888:
17868:
17865:
17862:
17842:
17833:
17820:
17817:
17809:
17784:
17762:
17758:
17737:
17734:
17731:
17728:
17723:
17720:
17712:
17704:
17701:
17697:
17693:
17690:
17685:
17681:
17677:
17674:
17671:
17666:
17662:
17641:
17621:
17618:
17615:
17606:
17593:
17590:
17587:
17582:
17578:
17574:
17568:
17565:
17562:
17559:
17556:
17553:
17550:
17544:
17541:
17538:
17532:
17529:
17526:
17503:
17500:
17480:
17460:
17440:
17437:
17432:
17428:
17424:
17421:
17401:
17398:
17395:
17390:
17386:
17382:
17379:
17359:
17339:
17319:
17316:
17313:
17310:
17304:
17298:
17295:
17292:
17286:
17283:
17280:
17274:
17271:
17268:
17265:
17262:
17242:
17239:
17236:
17233:
17230:
17227:
17207:
17198:
17185:
17179:
17176:
17173:
17170:
17167:
17161:
17158:
17155:
17126:
17123:
17120:
17117:
17114:
17108:
17105:
17102:
17093:
17090:
17070:
17065:
17055:
17052:
17032:
17012:
17009:
17004:
17000:
16996:
16993:
16973:
16953:
16950:
16945:
16941:
16937:
16934:
16914:
16911:
16906:
16902:
16881:
16878:
16873:
16869:
16865:
16859:
16836:
16813:
16810:
16807:
16802:
16798:
16789:
16773:
16765:
16761:
16756:
16743:
16737:
16734:
16731:
16725:
16720:
16716:
16712:
16709:
16706:
16701:
16697:
16693:
16690:
16684:
16675:
16669:
16664:
16660:
16656:
16650:
16644:
16639:
16635:
16613:
16609:
16606:
16601:
16597:
16592:
16588:
16583:
16579:
16572:
16566:
16563:
16558:
16554:
16538:
16534:
16531:
16526:
16522:
16517:
16513:
16508:
16504:
16497:
16491:
16488:
16483:
16479:
16458:
16438:
16435:
16432:
16429:
16426:
16417:
16416:
16412:
16398:
16376:
16366:
16363:
16343:
16335:
16331:
16327:
16311:
16302:
16288:
16285:
16265:
16257:
16253:
16208:
16179:
16176:
16171:
16166:
16162:
16158:
16153:
16148:
16144:
16123:
16120:
16115:
16111:
16107:
16104:
16084:
16081:
16078:
16075:
16055:
16052:
16049:
16046:
16026:
16023:
15990:
15977:
15971:
15968:
15965:
15942:
15934:
15930:
15906:
15903:
15900:
15877:
15857:
15835:
15832:
15829:
15824:
15819:
15815:
15811:
15806:
15801:
15797:
15776:
15773:
15770:
15750:
15747:
15727:
15707:
15698:
15684:
15664:
15644:
15637:assumed that
15634:
15620:
15617:
15612:
15608:
15601:
15592:
15589:
15583:
15577:
15574:
15571:
15568:
15565:
15556:
15550:
15544:
15535:
15532:
15526:
15520:
15517:
15514:
15511:
15508:
15499:
15493:
15490:
15485:
15481:
15460:
15452:
15434:
15430:
15426:
15423:
15403:
15395:
15392:
15376:
15367:
15354:
15351:
15331:
15324:
15308:
15305:
15302:
15279:
15276:
15270:
15267:
15264:
15258:
15255:
15252:
15249:
15246:
15237:
15228:
15225:
15219:
15213:
15210:
15207:
15204:
15201:
15195:
15192:
15172:
15163:
15149:
15146:
15143:
15123:
15120:
15117:
15097:
15077:
15074:
15071:
15051:
15031:
15011:
15008:
15005:
14997:
14993:
14989:
14986:Every TVS is
14984:
14983:
14974:
14960:
14937:
14934:
14931:
14925:
14920:
14916:
14912:
14909:
14906:
14901:
14897:
14876:
14873:
14868:
14864:
14860:
14857:
14854:
14849:
14845:
14824:
14821:
14818:
14796:
14792:
14786:
14782:
14778:
14775:
14772:
14767:
14763:
14757:
14753:
14744:
14728:
14725:
14716:
14713:
14699:
14679:
14676:
14673:
14666:
14661:
14647:
14644:
14641:
14622:
14619:
14616:
14613:
14594:
14591:
14588:
14585:
14566:
14563:
14560:
14555:
14551:
14530:
14527:
14522:
14518:
14497:
14489:
14473:
14470:
14450:
14430:
14422:
14418:
14416:
14412:
14407:
14406:
14389:
14386:
14366:
14363:
14360:
14350:
14340:
14326:
14322:
14319:
14290:
14287:
14283:
14280:
14259:
14251:
14247:
14222:
14219:
14198:
14185:
14168:
14165:
14156:
14150:
14146:
14142:
14117:
14097:
14094:
14072:
14043:
14028:
14025:
14022:
14006:
14001:
13998:
13995:
13991:
13983:
13966:
13962:
13939:
13935:
13926:
13925:inner product
13922:
13919:
13905:
13896:
13869:
13860:
13845:
13841:
13832:reflexive is
13829:
13827:
13824:
13821:
13817:
13801:
13798:
13778:
13772:
13769:
13766:
13763:
13741:
13737:
13728:
13724:
13723:Banach spaces
13721:
13718:
13714:
13710:
13706:
13705:Normed spaces
13703:
13700:
13696:
13693:
13690:
13686:
13682:
13678:
13674:
13671:
13656:
13645:
13634:
13627:
13613:
13610:
13607:
13604:
13598:
13595:
13587:
13582:
13579:
13576:
13568:
13528:
13519:
13516:
13514:
13510:
13506:
13502:
13499:
13496:
13492:
13488:
13485:
13482:
13478:
13474:
13471:
13468:
13464:
13461:
13447:
13444:
13441:
13438:
13435:
13415:
13412:
13409:
13406:
13384:
13380:
13371:
13367:
13363:
13359:
13355:
13352:
13338:
13335:
13332:
13324:
13308:
13304:
13294:
13290:
13287:
13286:
13285:
13282:
13280:
13276:
13265:
13263:
13259:
13243:
13223:
13203:
13196:
13180:
13172:
13167:
13153:
13130:
13124:
13104:
13090:
13077:
13074:
13064:
13050:
13042:
13038:
13032:
13030:
13014:
13006:
12990:
12981:
12980:
12954:
12934:
12914:
12911:
12908:
12905:
12902:
12899:
12896:
12876:
12873:
12853:
12850:
12847:
12844:
12841:
12831:
12817:
12814:
12811:
12808:
12805:
12802:
12782:
12774:
12760:
12757:
12737:
12717:
12709:
12706:
12690:
12682:
12674:
12671:
12668:
12665:
12662:
12659:
12631:
12625:
12617:
12611:
12600:
12567:
12564:
12556:
12544:
12528:
12523:
12496:
12493:
12473:
12465:
12464:
12460:
12446:
12426:
12423:
12420:
12417:
12414:
12411:
12408:
12400:
12399:
12397:
12384:
12377:
12373:
12372:homeomorphism
12355:
12351:
12330:
12327:
12307:
12304:
12301:
12295:
12287:
12283:
12263:
12233:
12230:
12210:
12207:
12204:
12201:
12198:
12195:
12175:
12155:
12152:
12147:
12143:
12139:
12136:
12116:
12096:
12093:
12090:
12087:
12084:
12081:
12071:
12057:
12037:
12034:
12031:
12028:
12025:
12022:
12019:
12014:
12010:
11989:
11969:
11949:
11924:
11919:
11915:
11894:
11888:
11885:
11860:
11857:
11837:
11817:
11797:
11789:
11786:
11763:
11755:
11751:
11747:
11744:
11741:
11721:
11701:
11698:
11691:for any such
11678:
11670:
11667:
11647:
11644:
11641:
11635:
11627:
11623:
11602:
11591:
11586:
11582:
11561:
11558:
11555:
11552:
11549:
11546:
11526:
11479:
11469:
11466:
11463:
11458:
11453:
11449:
11445:
11423:
11420:
11417:
11414:
11411:
11384:
11381:
11378:
11373:
11369:
11364:
11360:
11355:
11333:
11313:
11310:
11307:
11304:
11301:
11281:
11256:
11253:
11233:
11230:
11227:
11207:
11182:
11162:
11142:
11139:
11136:
11113:
11110:
11102:
11094:
11086:
11083:
11077:
11072:
11068:
11013:Proof outline
11005:
11001:
11000:absorbing set
10963:
10933:
10932:
10930:
10914:
10894:
10891:
10888:
10885:
10882:
10874:
10860:
10850:
10848:
10829:
10823:
10820:
10800:
10797:
10794:
10791:
10788:
10780:
10779:
10778:
10774:
10760:
10740:
10737:
10734:
10731:
10728:
10708:
10705:
10702:
10699:
10696:
10676:
10657:
10654:
10646:
10642:
10624:
10597:
10592:
10576:
10573:
10559:
10537:
10510:
10502:
10484:
10457:
10437:
10434:
10431:
10428:
10425:
10383:
10375:
10282:
10280:
10276:
10272:
10262:
10248:
10245:
10242:
10214:
10211:
10208:
10205:
10202:
10196:
10193:
10180:
10176:
10172:
10168:
10164:
10148:
10120:
10114:
10107:converges to
10091:
10083:
10079:
10059:
10056:
10036:
10016:
10008:
9991:
9987:
9966:
9963:
9960:
9940:
9932:
9928:
9905:
9902:
9899:
9894:
9889:
9885:
9881:
9858:
9845:
9841:
9813:
9810:
9802:
9786:
9783:
9762:
9746:
9738:
9687:
9684:
9664:
9656:
9652:
9642:
9640:
9635:
9633:
9628:
9612:
9608:
9599:
9583:
9562:
9556:
9552:
9548:
9545:
9541:
9518:
9514:
9493:
9473:
9453:
9450:
9442:
9422:
9419:
9397:
9393:
9383:
9382:
9378:
9365:
9362:
9359:
9356:
9353:
9345:
9341:
9338:
9335:
9332:
9328:
9312:
9292:
9266:
9263:
9253:
9249:
9248:
9242:
9241:
9237:
9223:
9190:
9187:
9167:
9159:
9155:
9134:
9112:
9108:
9097:
9094:
9091:
9087:
9078:
9064:
9061:
9041:
9034:
9030:
9015:
9007:
8991:
8983:
8968:
8948:
8928:
8920:
8917:
8914:
8910:
8909:
8908:
8906:
8902:
8899:
8895:
8891:
8886:
8884:
8883:
8877:
8864:
8861:
8858:
8855:
8835:
8832:
8829:
8809:
8806:
8801:
8797:
8793:
8788:
8784:
8763:
8743:
8740:
8720:
8712:
8694:
8691:
8688:
8683:
8678:
8674:
8670:
8665:
8660:
8656:
8647:
8643:
8638:
8636:
8632:
8628:
8624:
8620:
8616:
8612:
8608:
8604:
8599:
8586:
8583:
8563:
8555:
8536:
8533:
8530:
8519:
8518:
8511:
8501:
8488:
8485:
8463:
8448:
8422:
8418:
8415:
8388:
8386:
8385:
8379:
8377:
8375:
8370:
8364:
8362:
8354:
8335:
8332:
8329:
8319:
8305:
8302:
8282:
8279:
8272:the topology
8259:
8239:
8231:
8227:
8224:
8205:
8202:
8199:
8189:
8175:
8172:
8164:
8161:
8145:
8138:is closed in
8122:
8111:
8110:
8109:
8092:
8089:
8086:
8071:
8063:Metrizability
8060:
8047:
8044:
8024:
8004:
7984:
7981:
7961:
7953:
7948:
7935:
7932:
7912:
7892:
7885:: the subset
7884:
7868:
7848:
7845:
7842:
7839:
7836:
7816:
7796:
7776:
7756:
7747:
7733:
7730:
7727:
7724:
7704:
7684:
7676:
7675:
7658:
7638:
7629:
7628:
7624:
7622:
7606:
7603:
7595:
7594:balanced sets
7579:
7571:
7567:
7566:absorbing set
7548:
7545:
7542:
7539:
7536:
7516:
7513:
7510:
7507:
7504:
7496:
7495:
7491:
7476:
7468:
7467:
7462:
7461:
7456:
7442:
7439:
7436:
7433:
7430:
7410:
7407:
7404:
7398:
7395:
7392:
7386:
7383:
7380:
7372:
7371:
7367:
7353:
7350:
7342:
7317:
7314:
7311:
7308:
7300:
7296:
7295:
7291:
7277:
7274:
7271:
7263:
7238:
7218:
7215:
7212:
7209:
7189:
7186:
7183:
7163:
7160:
7157:
7154:
7134:
7126:
7125:
7121:
7120:
7119:
7105:
7085:
7074:Local notions
7071:
7057:
7037:
7017:
7009:
6993:
6990:
6987:
6967:
6964:
6961:
6941:
6938:
6933:
6929:
6925:
6922:
6919:
6913:
6910:
6907:
6901:
6896:
6892:
6871:
6868:
6865:
6862:
6842:
6839:
6836:
6827:
6813:
6810:
6807:
6801:
6781:
6775:
6755:
6752:
6749:
6746:
6726:
6723:
6717:
6697:
6691:
6671:
6668:
6665:
6641:
6638:
6633:
6629:
6620:
6619:homeomorphism
6604:
6601:
6596:
6592:
6585:
6565:
6559:
6539:
6536:
6533:
6528:
6524:
6515:
6514:
6513:
6505:
6497:
6484:
6481:
6461:
6457:
6453:
6434:
6428:
6405:
6385:
6381:
6377:
6357:
6337:
6329:
6313:
6309:
6305:
6285:
6282:
6279:
6276:
6269:
6253:
6244:
6242:
6238:
6234:
6218:
6215:
6207:
6206:abelian group
6197:
6195:
6179:
6176:
6173:
6162:
6138:
6135:
6132:
6097:
6096:
6083:
6048:
6027:
6024:
6016:
5973:
5970:
5950:
5908:
5899:
5896:
5881:
5874:
5860:
5854:
5851:
5848:
5837:
5799:
5754:
5751:
5748:
5725:
5722:
5709:
5682:
5678:
5674:
5671:
5661:
5656:
5652:
5647:
5643:
5635:
5632:
5624:
5620:
5605:
5596:
5556:
5552:
5548:
5543:
5539:
5516:
5512:
5508:
5503:
5499:
5473:
5468:
5464:
5443:
5435:
5430:
5426:
5422:
5417:
5413:
5382:
5378:
5374:
5370:
5332:
5329:
5284:
5274:
5271:
5266:
5260:
5256:
5252:
5247:
5243:
5238:
5233:
5228:
5224:
5220:
5215:
5211:
5199:
5196:
5182:
5172:
5169:
5164:
5158:
5154:
5150:
5145:
5141:
5136:
5131:
5126:
5122:
5118:
5113:
5109:
5097:
5094:
5080:
5070:
5067:
5062:
5056:
5052:
5048:
5044:
5039:
5034:
5030:
5026:
5015:
5012:
4998:
4993:
4989:
4978:
4975:
4971:
4967:
4962:
4958:
4954:
4951:
4940:
4937:
4923:
4919:
4910:
4907:
4904:
4899:
4895:
4890:
4886:
4881:
4877:
4873:
4870:
4859:
4856:
4842:
4839:
4817:
4813:
4809:
4804:
4800:
4777:
4773:
4769:
4764:
4760:
4734:
4730:
4722:
4705:
4701:
4693:
4692:
4691:
4677:
4657:
4630:
4627:
4622:
4617:
4613:
4609:
4604:
4599:
4595:
4567:
4564:
4559:
4554:
4550:
4546:
4541:
4536:
4532:
4522:
4520:
4514:
4501:
4498:
4475:
4472:
4469:
4463:
4460:
4454:
4451:
4448:
4442:
4422:
4400:
4396:
4375:
4355:
4333:
4329:
4308:
4288:
4285:
4282:
4279:
4259:
4256:
4248:
4223:
4200:
4194:
4191:
4185:
4182:
4176:
4154:
4150:
4126:
4120:
4117:
4111:
4108:
4102:
4094:
4076:
4072:
4051:
4048:
4042:
4036:
4016:
4011:
4007:
4001:
3998:
3995:
3991:
3970:
3967:
3964:
3944:
3941:
3938:
3935:
3932:
3909:
3903:
3900:
3894:
3888:
3885:
3879:
3876:
3873:
3867:
3847:
3838:
3825:
3821:
3814:
3803:
3799:
3793:
3789:
3785:
3782:
3779:
3774:
3770:
3765:
3761:
3756:
3752:
3745:
3735:
3731:
3727:
3723:
3719:
3716:
3709:
3705:
3701:
3697:
3692:
3681:
3675:
3669:
3647:
3643:
3639:
3636:
3616:
3613:
3607:
3601:
3578:
3575:
3572:
3563:
3560:
3557:
3548:
3535:
3531:
3523:
3519:
3514:
3510:
3507:
3504:
3497:
3493:
3488:
3484:
3481:
3473:
3470:
3462:
3459:
3454:
3450:
3446:
3443:
3440:
3437:
3431:
3424:
3418:
3414:
3410:
3407:
3404:
3399:
3395:
3390:
3386:
3381:
3377:
3372:
3368:
3362:
3334:
3329:
3325:
3321:
3318:
3298:
3295:
3292:
3270:
3266:
3262:
3257:
3254:
3251:
3247:
3243:
3238:
3235:
3232:
3228:
3205:
3201:
3197:
3194:
3167:
3164:
3161:
3156:
3151:
3147:
3143:
3138:
3133:
3129:
3086:
3084:
3079:
3077:
3061:
3053:
3040:
3018:
3015:
3012:
3007:
3003:
2982:
2977:
2974:
2971:
2967:
2963:
2958:
2954:
2933:
2925:
2922:
2906:
2883:
2880:
2858:
2854:
2833:
2825:
2817:
2810:
2793:
2789:
2780:
2773:
2759:
2754:
2750:
2741:
2740:
2735:
2734:
2729:
2728:
2723:
2719:
2715:
2711:
2710:
2705:
2704:
2700:
2686:
2683:
2661:
2657:
2653:
2648:
2645:
2642:
2638:
2634:
2629:
2626:
2623:
2619:
2610:
2603:
2602:
2601:
2585:
2581:
2573:The sequence
2560:
2555:
2551:
2542:
2524:
2520:
2499:
2494:
2490:
2481:
2467:
2444:
2440:
2420:
2417:
2395:
2391:
2382:
2360:
2356:
2335:
2332:
2305:
2302:
2299:
2294:
2289:
2285:
2281:
2276:
2271:
2267:
2246:
2232:
2230:
2225:
2212:
2209:
2189:
2179:
2175:
2134:
2131:
2128:
2125:
2122:
2119:
2092:
2089:
2062:
2059:
2028:
2025:
2021:
1998:
1995:
1987:
1986:
1985:
1983:
1943:
1940:
1920:
1912:
1872:
1864:
1860:
1856:
1816:
1809:Suppose that
1796:
1794:
1790:
1789:homeomorphism
1774:
1771:
1766:
1762:
1755:
1735:
1729:
1709:
1706:
1703:
1698:
1694:
1679:
1675:
1658:
1655:
1652:
1641:
1640:neighborhoods
1625:
1622:
1619:
1599:
1596:
1593:
1584:
1581:
1578:
1555:
1549:
1546:
1543:
1535:
1519:
1516:
1513:
1493:
1490:
1470:
1462:
1443:
1440:
1437:
1410:
1399:
1386:
1383:
1380:
1377:
1374:
1371:
1344:
1341:
1321:
1311:
1308:
1301:if for every
1298:
1273:A collection
1271:
1270:
1266:
1264:
1260:
1256:
1255:metrizability
1252:
1247:
1245:
1241:
1240:homeomorphism
1238:
1228:
1220:
1212:
1211:(abbreviated
1204:
1201:
1199:
1195:
1188:
1182:
1177:TVS embedding
1174:
1173:(abbreviated
1166:
1163:
1150:
1147:
1139:
1136:is given the
1123:
1120:
1100:
1094:
1088:
1085:
1082:
1079:
1076:
1068:
1052:
1049:
1046:
1040:
1037:
1034:
1014:
1008:
1005:
1002:
995:
992:
985:
980:
972:
971:(abbreviated
964:
961:
959:
933:
907:
891:
798:
793:
792:
788:
786:
782:
772:
770:
766:
762:
743:
735:
730:
729:
725:
723:
718:
705:
702:
694:
686:
678:
674:
658:
652:
649:
641:
638:
618:
612:
609:
606:
603:
599:
595:
591:
583:
579:
575:
550:
546:
530:
522:
518:
510:
505:
496:
494:
490:
485:
483:
479:
478:Montel spaces
475:
474:distributions
471:
467:
463:
459:
449:
447:
443:
442:Banach spaces
423:
420:
378:
375:
372:
369:
360:
357:
354:
331:
325:
322:
314:
311:
303:
300:
284:
281:
278:
269:
266:
263:
240:
234:
231:
228:
225:
221:
217:
213:
205:
204:
203:
201:
197:
193:
186:Normed spaces
178:
164:
152:
127:
123:
118:
116:
112:
108:
103:
101:
97:
93:
92:Banach spaces
89:
85:
81:
77:
73:
67:
65:
61:
57:
53:
49:
45:
41:
37:
33:
19:
36220:Polynomially
36149:Grothendieck
36142:tame Fréchet
36092:Bornological
35952:Linear cone
35944:Convex cone
35919:Banach disks
35861:Sesquilinear
35716:Main results
35706:Vector space
35700:
35661:Completeness
35656:Banach space
35640:
35561:Balanced set
35535:Distribution
35473:Applications
35326:Krein–Milman
35311:Closed graph
35228:
35077:
35043:
35013:
34975:
34956:
34926:
34904:
34872:
34840:
34807:
34765:
34731:
34718:20 September
34716:. Retrieved
34711:
34669:
34639:
34612:
34578:
34543:
34512:
34477:
34447:
34416:
34408:Bibliography
34393:
34388:, p. 6.
34381:
34357:Jarchow 1981
34352:
34340:
34328:
34284:
34272:
34265:Jarchow 1981
34209:
34174:
34152:Jarchow 1981
34147:
34105:
34093:
34081:
34069:
34025:
34013:
33970:. Retrieved
33968:. 2016-04-22
33965:
33956:
33944:
33915:
33903:
33891:
33880:, retrieved
33869:
33860:
33848:
33836:
33810:, p. 8.
33803:
33774:
33762:
33727:
33626:
33614:
33585:
33165:
32732:
32684:
31170:
31056:
30991:
30720:
30480:
30452:
30399:
30391:
30386:
30377:
30294:
30265:
30161:Normed space
30134:Banach space
30050:Sequentially
29985:Neighborhood
25522:Sequentially
25457:Neighborhood
25354:Property of
25315:
25292:
24657:
24650:
24643:
24639:
24638:
24590:
24192:
23969:
23769:
23767:also works.
23577:
23573:
23572:
23443:
23372:
23357:
23353:
23352:
23332:
23328:
23327:
22859:
22700:
22697:Closed hulls
22696:
22695:
22325:
22172:
21826:
21728:
21304:
21131:
21124:
21120:
21119:
20959:
20855:
20708:
20704:
20703:
20080:
20074:is always a
20014:quotient map
19900:
19683:of this TVS
19396:
19295:
18640:
17834:
17607:
17199:
16757:
16418:
16414:
16413:
16303:
16245:) such that
16221:(indexed by
15991:
15699:
15368:
15164:
14985:
14981:
14980:
14745:of the form
14692:of a subset
14662:
14420:
14409:
14404:
14352:
14152:
14021:Hardy spaces
13954:spaces, the
13822:of measures.
13756:spaces with
13501:Montel space
13428:but not for
13283:
13271:
13168:
13096:
12982:
12978:
12977:
12370:is a linear
11942:centered at
11015:
10956:is itself a
10450:and so that
10283:
10279:neighborhood
10268:
10161:This TVS is
9648:
9436:
9384:
9380:
9379:
9337:seminormable
9251:
9245:
9243:
9239:
9238:
9215:
9079:If a series
8981:is complete.
8887:
8880:
8878:
8710:
8639:
8607:completeness
8600:
8515:
8513:
8389:
8382:
8380:
8373:
8366:
8358:
8067:
7949:
7748:
7672:
7630:
7626:
7625:
7563:
7492:
7464:
7458:
7368:
7298:
7292:
7122:
7077:
7008:neighborhood
6828:
6657:
6503:
6245:
6203:
6099:
6070:
5900:
5894:
5727:
5723:
5622:
5621:
5380:
5376:
5372:
5368:
5299:
5198:Intersection
5197:
5095:
5013:
4938:
4858:Set of knots
4857:
4720:
4523:
4516:
3839:
3549:
3088:
3080:
3032:
2898:
2819:
2811:
2774:
2737:
2731:
2725:
2721:
2717:
2713:
2707:
2701:
2604:
2540:
2460:
2376:
2375:is called a
2238:
2231:
2227:
2149:
1798:
1685:
1677:
1568:(defined by
1401:
1272:
1268:
1267:
1259:completeness
1248:
1202:
1187:monomorphism
1185:topological
1164:
1067:open mapping
962:
958:-linear maps
794:
790:
789:
734:Walter Rudin
731:
727:
726:
719:
691:TVS topology
688:
680:
545:vector space
520:
516:
514:
486:
455:
439:
189:
151:real numbers
119:
104:
80:completeness
56:vector space
47:
43:
39:
35:
29:
36214:Quasinormed
36127:FK-AK space
36021:Linear span
36016:Convex hull
36001:Affine hull
35804:Almost open
35744:Hahn–Banach
35490:Heat kernel
35480:Hardy space
35387:Trace class
35301:Hahn–Banach
35263:Topological
34953:Lang, Serge
34398:Swartz 1992
34179:Conway 1990
33949:Swartz 1992
33882:26 February
33043:of any set
32469:is convex,
31286:belongs to
30451:is said to
30069:Bornivorous
29259:Linear span
29098:Linear span
28451:Convex hull
28290:Convex hull
25541:Bornivorous
25289:be bounded.
23419:Baire space
22805:The closed
21824:is convex.
21040:containing
20447:defined by
14665:convex hull
14488:convex hull
13725:: Complete
13509:bounded set
13362:convex sets
13093:Linear maps
12830:there are,
12588:defined by
9739:. This set
9598:Hamel basis
9435:called the
8228:There is a
8112:The origin
7251:satisfying
6794:defined by
6710:defined by
6578:defined by
5895:fundamental
4142:and if all
3083:subadditive
1982:filter base
1865:subsets of
1748:defined by
1263:normability
1140:induced by
936:continuous
767:, and even
509:translation
344:defined by
253:defined by
115:convergence
32:mathematics
36329:Categories
36254:Stereotype
36112:(DF)-space
36107:Convenient
35846:Functional
35814:Continuous
35799:Linear map
35739:F. Riesz's
35681:Linear map
35423:C*-algebra
35238:Properties
35100:1145563701
34386:Rudin 1991
34125:Rudin 1991
34062:Rudin 1991
34030:Rudin 1991
34018:Rudin 1991
34006:Rudin 1991
33972:2020-10-07
33966:What's new
33896:Rudin 1991
33853:Köthe 1983
33841:Rudin 1991
33808:Rudin 1991
33796:Rudin 1991
33607:Köthe 1983
33590:Rudin 1991
30121:Operation
29949:Operation
25593:Operation
25351:Operation
25278:be closed.
24653:-seminorms
23663:hold, let
23512:such that
21213:such that
20576:completion
20076:closed map
19230:not closed
17797:such that
16336:subset of
15870:converges
14347:See also:
14343:Properties
14135:Dual space
13791:the space
13685:ILH spaces
13358:local base
13171:hyperplane
13029:metrizable
12541:induces a
12129:such that
12002:such that
10350:and endow
10175:metrizable
9639:metrizable
8913:completion
8776:such that
8631:precompact
8554:uniformity
8223:metrizable
7829:such that
7717:such that
7202:such that
6194:metrizable
5491:such that
5379:or simply
4236:such that
2112:such that
1364:such that
1244:surjective
994:linear map
991:continuous
499:Definition
181:Motivation
36270:Uniformly
36229:Reflexive
36077:Barrelled
36073:Countably
35985:Symmetric
35883:Transpose
35397:Unbounded
35392:Transpose
35350:Operators
35279:Separable
35274:Reflexive
35259:Algebraic
35245:Barrelled
35066:316568534
35032:853623322
35012:(2006) .
34945:180577972
34730:(1987) .
34688:849801114
34631:175294365
34601:840278135
34531:144216834
34504:840293704
34476:(1983) .
34439:297140003
33876:EMS Press
33577:Citations
33558:◼
33524:−
33506:∈
33485:−
33467:∈
33440:−
33422:∈
33416:−
33308:−
33278:∈
33233:∈
33188:∈
33125:∖
33087:
33002:
32946:
32933:⊆
32921:
32902:⊆
32873:
32860:⊆
32848:
32811:
32786:
32747:∈
32706:
32616:
32579:⊆
32568:
32531:∈
32434:∈
32412:−
32359:
32353:∈
32311:−
32228:
32149:−
32118:−
32100:↦
32074:→
32039:∈
32000:
31994:∩
31966:∈
31933:
31907:
31901:∩
31873:
31867:∈
31820:
31814:∈
31788:
31782:∈
31753:
31700:
31665:→
31639:≠
31594:−
31545:−
31448:−
31403:−
31384:−
31365:−
31304:
31265:−
31148:
31135:∪
31117:
31084:
31071:∈
31034:τ
30672:
30610:→
30604:×
30594:⋅
30586:⋅
30557:
30544:×
30512:
30455:in a TVS
30427:∞
30412:∑
30403:A series
30289:be a TVS.
30112:Separable
30019:subspace
30007:Balanced
29995:Balanced
29979:subspace
29974:Balanced
29968:Symmetric
29953:Absorbing
26667:∩
26493:∩
26466:∩
26273:∩
26052:∪
25843:∪
25816:∪
25608:∪
25584:Separable
25491:subspace
25479:Balanced
25467:Balanced
25451:subspace
25446:Balanced
25440:Symmetric
25425:Absorbing
25327:∪
25239:∈
25173:∈
25108:∈
25073:∈
25041:≠
25006:‖
25000:−
24994:‖
24985:∈
24964:‖
24958:‖
24949:∈
24912:∈
24883:‖
24877:‖
24869:denotes "
24816:−
24798:∈
24765:∈
24728:∈
24702:∈
24689:predicate
24574:∅
24556:∪
24547:
24541:∩
24518:∅
24500:∪
24491:
24485:∩
24456:∪
24424:⊆
24389:
24365:
24342:
24330:
24306:
24210:⊆
24136:
24127:
24109:
24100:⊆
24088:
24079:
24047:
24038:
23985:⊆
23974:of a set
23953:≤
23947:≤
23921:⊆
23909:−
23770:A subset
23700:−
23618:⊆
23589:⊆
23564:nonmeager
23534:∈
23527:⋃
23486:nonmeager
23300:
23291:
23278:∪
23269:
23260:
23242:
23218:∪
23209:
23200:
23164:
23155:
23119:
23110:
23077:⊆
23039:
23030:
23008:
22999:
22959:
22950:
22877:⊆
22836:
22827:
22782:
22773:
22732:
22723:
22630:⊆
22624:
22526:⊆
22511:⋂
22505:⊇
22496:
22478:However,
22395:∈
22388:⋂
22372:
22337:⊆
22265:
22252:⊆
22231:⋂
22204:⊆
22147:
22122:
22051:
22015:
21990:
21972:
21933:
21917:⊆
21902:
21877:
21844:⊆
21809:
21776:
21760:⊆
21740:⊆
21700:
21676:
21650:
21619:
21605:
21559:⊆
21530:
21499:
21469:∅
21449:≠
21418:where if
21391:
21378:⊆
21372:
21316:⊆
21224:∩
21100:⊆
20826:
20805:→
20779:
20737:of a TVS
20673:
20660:×
20654:≅
20622:
20609:×
20529:
20470:↦
20429:→
20417:
20404:×
20352:
20290:
20247:
20196:
20183:∩
20119:
20053:
20032:→
19870:
19849:→
19800:
19759:
19668:⊆
19636:
19588:arbitrary
19559:
19500:⊆
19474:
19461:⊆
19449:
19408:⊆
19365:
19340:
19307:⊆
19270:
19179:
19080:
19037:
18994:
18923:∈
18916:⋂
18900:
18831:
18740:
18616:
18610:∖
18604:
18598:∈
18572:∞
18569:≠
18540:
18534:⊆
18476:⊆
18423:
18417:∈
18360:−
18348:
18321:
18263:symmetric
18241:∩
18185:∈
18065:∅
18024:≠
18013:
18000:⊆
17991:≤
17967:−
17905:
17892:∈
17866:⊆
17735:⊆
17698:⋃
17678:⊆
17672:
17619:⊆
17588:
17575:⊆
17542:−
17516:that is,
17438:
17425:∈
17396:
17383:∈
17314:
17305:⊆
17296:
17284:−
17269:
17237:≤
17124:≥
17010:
16997:∈
16951:
16938:∈
16912:
16879:
16866:∪
16829:in a TVS
16817:∅
16814:≠
16808:
16726:
16713:⊆
16707:
16685:⊆
16670:
16645:
16607:
16589:
16564:
16532:
16514:
16489:
16436:⊆
16334:absorbing
16263:→
16258:∙
16180:∈
16149:∙
16121:
16108:∈
16079:∈
16050:⊆
15972:τ
15940:→
15935:∙
15907:ν
15833:∈
15802:∙
15777:ν
15774:⊆
15771:τ
15728:ν
15708:τ
15618:
15602:⊆
15590:≤
15569:∈
15557:⊆
15545:⊆
15512:∈
15500:⊆
15491:
15396:in a TVS
15391:absorbing
15306:∈
15295:for some
15268:−
15250:∈
15205:∈
15147:−
15121:⊆
15009:⊆
14988:connected
14926:∈
14910:…
14874:∈
14858:…
14822:≥
14776:⋯
14677:
14645:
14617:
14589:
14561:
14528:
14379:of a TVS
14364:⊆
14294:→
14288:×
14227:→
13901:∞
13874:∞
13776:∞
13773:≤
13767:≤
13673:LF-spaces
13635:ℓ
13605:−
13599:∈
13583:ℓ
13573:‖
13566:‖
13533:∞
13410:≥
12906:
12851:
12812:≥
12806:
12675:
12663:
12571:→
12424:≥
12412:
12385:◼
12234:⊆
12205:∈
12199:≠
12140:⊆
12091:∈
12085:≠
12032:≠
12026:⊆
12020:⋅
11925:⊆
11892:→
11886:×
11790:⊆
11615:given by
11600:→
11556:∈
11550:≠
11480:⊆
11475:∞
11421:∈
11415:≠
11390:∞
11326:whenever
11305:⋅
11257:⊆
11231:⊆
11087:∈
10886:
10792:
10775:Hausdorff
10738:≠
10732:
10700:
10575:Hausdorff
10435:
10246:≠
10215:∈
10167:Hausdorff
10007:converges
9964:∈
9911:∞
9696:→
9609:τ
9553:τ
9515:τ
9394:τ
9357:
9344:Hausdorff
9270:∅
9165:→
9160:∙
9103:∞
9088:∑
8859:≥
8833:≥
8822:whenever
8807:∈
8794:−
8692:∈
8661:∙
8619:Tychonoff
8564:τ
8537:τ
8520:on a TVS
8421:Hausdorff
8412:be a non-
8376:-seminorm
8367:A TVS is
8336:τ
8280:τ
8206:τ
8160:countable
8093:τ
7883:seminorms
7840:⊆
7728:⊆
7631:A subset
7537:−
7511:⊆
7505:−
7494:symmetric
7440:≤
7434:≤
7408:⊆
7396:−
7351:≤
7315:⊆
7272:≤
7216:∈
7158:∈
7124:absorbing
7078:A subset
6965:∈
6939:
6902:
6866:⊆
6840:∈
6808:−
6805:↦
6779:→
6753:−
6721:↦
6695:→
6669:≠
6639:≠
6621:, but if
6589:↦
6563:→
6534:∈
6280:⊆
6216:−
6177:τ
6163:τ
6139:τ
6049:τ
5974:
5882:τ
5855:τ
5755:τ
5683:∙
5675:
5662:∈
5657:∙
5648:⋃
5636:
5606:⊆
5595:prefilter
5557:∙
5549:⊆
5544:∙
5517:∙
5509:⊆
5504:∙
5474:∈
5469:∙
5436:∈
5431:∙
5418:∙
5373:downwards
5275:∈
5253:∩
5229:∙
5221:∩
5216:∙
5173:∈
5127:∙
5114:∙
5071:∈
5035:∙
4979:∈
4972:⋂
4963:∙
4955:
4911:∈
4882:∙
4874:
4810:⊆
4778:∙
4770:⊆
4765:∙
4735:∙
4706:∙
4631:∈
4600:∙
4568:∈
4537:∙
4473:−
4401:∙
4283:∈
4257:≤
4192:≤
4109:−
3999:≥
3992:⋂
3942:∈
3886:≤
3804:∈
3783:…
3757:∙
3728:−
3720:⋯
3702:−
3567:→
3508:⋯
3485:∈
3460:≥
3441:≥
3408:…
3382:∙
3322:∈
3296:≥
3263:⊆
3198:∈
3173:∞
3134:∙
2975:−
2921:absorbing
2859:∙
2826:in a TVS
2794:∙
2733:barrelled
2727:symmetric
2709:absorbing
2654:⊆
2607:Summative
2600:is/is a:
2586:∙
2556:∙
2541:beginning
2495:∙
2396:∙
2361:∙
2311:∞
2272:∙
2182:basis at
2129:⊆
2093:∈
2063:∈
2024:absorbing
1999:∈
1863:absorbing
1855:non-empty
1759:↦
1733:→
1704:∈
1659:τ
1623:×
1591:↦
1553:→
1547:×
1517:×
1471:τ
1381:⊆
1345:∈
1312:∈
1194:injective
1080:
1050:
1044:→
1012:→
932:morphisms
781:separable
775:separated
769:Tychonoff
765:Hausdorff
656:→
650:×
639:⋅
616:→
610:×
600:⋅
592:⋅
493:subfields
440:Thus all
373:⋅
367:↦
329:→
323:×
312:⋅
276:↦
238:→
232:×
222:⋅
214:⋅
107:functions
36314:Category
36265:Strictly
36239:Schwartz
36179:LF-space
36174:LB-space
36132:FK-space
36102:Complete
36082:BK-space
36007:Relative
35954:(subset)
35946:(subset)
35873:Seminorm
35856:Bilinear
35599:Category
35411:Algebras
35293:Theorems
35250:Complete
35219:Schwartz
35165:glossary
35076:(2020).
35048:Elsevier
34955:(1972).
34925:(1979).
34903:(1966).
34869:(1973).
34859:30593138
34832:18412261
34806:(1988).
34792:21195908
34764:(1990).
34754:17499190
34706:(1988).
34668:(2013).
34658:24909067
34611:(1996).
34567:21163277
34541:(1991).
32891:Because
32209:The set
31570:so that
30453:converge
30128:See also
30052:Complete
30045:Complete
29958:Balanced
27947:Interior
27796:Interior
26679: of
26505: of
26064: of
25855: of
25524:Complete
25517:Complete
25430:Balanced
24450:∉
23936:for all
23373:Suppose
20550:has the
19202:subspace
18764:Because
18116:balanced
16788:balanced
16415:Interior
16391:) or if
16068:and let
14411:balanced
14353:For any
14323:′
14284:′
14223:′
14169:′
13717:seminorm
13325:for all
13293:complete
13289:F-spaces
12543:seminorm
11574:the map
11129:for all
10179:normable
10173:but not
10163:complete
9927:sequence
9331:complete
9207:Examples
8890:open map
8646:sequence
8414:discrete
8384:normable
7570:balanced
7294:balanced
6552:the map
6516:for all
6268:subspace
6013:forms a
5623:Notation
5369:directed
4721:contains
4272:and all
3925:for all
3640:∉
3311:For all
3285:for all
3078:string.
2703:Balanced
2512:The set
2480:-th knot
2020:balanced
1859:balanced
1722:the map
1299:additive
1192:, is an
934:are the
930:and the
797:category
582:topology
36279:)
36227:)
36169:K-space
36154:Hilbert
36137:Fréchet
36122:F-space
36097:Brauner
36090:)
36075:)
36057:Asplund
36039:)
36009:)
35929:Bounded
35824:Compact
35809:Bounded
35746: (
35402:Unitary
35382:Nuclear
35367:Compact
35362:Bounded
35357:Adjoint
35331:Min–max
35224:Sobolev
35209:Nuclear
35199:Hilbert
35194:Fréchet
35159: (
34496:0248498
34466:8210342
33878:, 2001
32057:Define
30079:Nowhere
30064:Bounded
30036:Convex
30034:Compact
30030:Compact
30025:bounded
30023:Totally
30000:Convex
29990:Closed
27789:Closure
27618:Closure
25551:Nowhere
25536:Bounded
25508:Convex
25506:Compact
25502:Compact
25497:bounded
25495:Totally
25472:Convex
25462:Closed
23397:. Then
21574:and if
20268:) is a
18558:and if
15449:is the
15416:and if
15110:and if
14953:sum to
14408:(resp.
13513:compact
12834:, only
10376:. Let
10306:denote
9953:and if
8623:compact
7674:bounded
7299:circled
5729:Theorem
4670:and if
4064:If all
3550:Define
3090:Theorem
2706:(resp.
1800:Theorem
989:, is a
906:objects
578:complex
547:over a
149:or the
36291:Webbed
36277:Quasi-
36199:Montel
36189:Mackey
36088:Ultra-
36067:Banach
35975:Radial
35939:Convex
35909:Affine
35851:Linear
35819:Closed
35643:(TVSs)
35377:Normal
35214:Orlicz
35204:Hölder
35184:Banach
35173:Spaces
35161:topics
35098:
35088:
35064:
35054:
35030:
35020:
35002:589250
35000:
34990:
34963:
34943:
34933:
34911:
34893:886098
34891:
34881:
34857:
34847:
34830:
34820:
34790:
34780:
34752:
34742:
34686:
34676:
34656:
34646:
34629:
34619:
34599:
34589:
34565:
34555:
34529:
34519:
34502:
34494:
34484:
34464:
34454:
34437:
34427:
33549:Q.E.D.
32688:Since
32679:Q.E.D.
32449:since
31771:Since
31532:
31509:
31247:
31224:
30983:Proofs
30107:Meager
30057:Banach
30017:Vector
30015:Closed
30011:Barrel
30005:Convex
30003:Closed
29998:Closed
29993:Closed
29977:Vector
29972:Convex
29963:Convex
25579:Meager
25529:Banach
25489:Vector
25487:Closed
25483:Barrel
25477:Convex
25475:Closed
25470:Closed
25465:Closed
25449:Vector
25444:Convex
25435:Convex
24646:family
24359:
24351:
24300:
24292:
24121:
24115:
24103:
24097:
23562:Every
23444:A TVS
23343:convex
23236:
23230:
22986:
22980:
22807:disked
22508:
22502:
22431:where
22384:
22378:
22038:
22032:
21959:
21951:
21920:
21914:
21127:barrel
19616:"tube"
17750:where
17308:
17302:
16688:
16682:
16576:
16570:
16551:
16543:
16501:
16495:
15605:
15599:
15560:
15554:
15548:
15542:
15503:
15497:
15389:is an
15241:
15235:
14811:where
14415:disked
14405:convex
14147:, and
13818:, and
13677:limits
13505:closed
13469:holds.
13372:. The
13323:spaces
13277:, the
13262:closed
13258:kernel
12652:where
12376:Q.E.D.
11404:where
10645:finest
10610:where
10169:, and
9713:where
8711:Cauchy
7950:Every
7370:convex
6398:where
6241:normal
5731:
5625:: Let
5208:
5106:
5024:
4949:
4939:Kernel
4868:
4757:
3957:) and
3749:
3743:
3435:
3429:
3092:
2919:is an
2777:String
2718:convex
2714:closed
1988:Every
1802:
1261:, and
1237:linear
1227:or an
1065:is an
756:to be
464:, the
391:where
200:metric
190:Every
122:scalar
48:t.v.s.
36249:Smith
36234:Riesz
36225:Semi-
36037:Quasi
36031:Polar
35189:Besov
32762:then
32597:Thus
32374:then
31354:with
30370:space
30364:is a
30257:Notes
30081:dense
29987:of 0
29982:Open
26683:chain
26509:chain
26068:chain
25859:chain
25553:dense
25459:of 0
25454:Open
24474:then
24106:cobal
23639:; if
23604:then
23417:is a
22833:cobal
22222:then
21859:then
21791:then
21259:then
20937:then
20598:then
19941:then
19590:open
19535:then
18584:then
18494:then
18114:is a
17654:then
17255:then
16849:then
16786:is a
16762:of a
16097:Then
15922:then
15763:Then
15473:then
15064:then
14998:. If
14642:cobal
13268:Types
13031:) is
12795:when
12439:then
12070:does
11294:then
11175:be a
10907:then
10813:then
10235:with
9979:then
9925:is a
9600:then
9147:then
8903:of a
8898:dense
8351:is a
7497:: if
7469:: if
7373:: if
7301:: if
7006:is a
6980:then
6884:then
6617:is a
6154:then
5971:Knots
5834:is a
5672:Knots
5633:Knots
5593:is a
5345:then
4871:Knots
4435:then
4095:then
3840:Then
2818:or a
1980:is a
1909:is a
1885:then
1853:is a
1787:is a
1461:group
1459:is a
1069:when
785:below
687:or a
543:is a
109:, or
35868:Norm
35792:form
35780:Maps
35537:(or
35255:Dual
35096:OCLC
35086:ISBN
35062:OCLC
35052:ISBN
35028:OCLC
35018:ISBN
34998:OCLC
34988:ISBN
34961:ISBN
34941:OCLC
34931:ISBN
34909:ISBN
34889:OCLC
34879:ISBN
34855:OCLC
34845:ISBN
34828:OCLC
34818:ISBN
34788:OCLC
34778:ISBN
34750:OCLC
34740:ISBN
34720:2020
34684:OCLC
34674:ISBN
34654:OCLC
34644:ISBN
34627:OCLC
34617:ISBN
34597:OCLC
34587:ISBN
34563:OCLC
34553:ISBN
34527:OCLC
34517:ISBN
34500:OCLC
34482:ISBN
34462:OCLC
34452:ISBN
34435:OCLC
34425:ISBN
33884:2021
32504:and
32486:<
32480:<
32194:<
32188:<
31803:and
31555:<
31191:<
31185:<
31174:Fix
30889:the
30083:(in
30059:disk
25555:(in
25531:disk
25270:The
25009:<
24967:<
24886:<
24225:and
23970:The
23868:>
23854:>
23492:set
23360:disk
23335:fail
23137:and
22586:and
22238:>
21331:and
21173:and
21000:and
20217:and
19228:but
18505:>
18449:>
18209:<
18158:and
18039:and
17985:<
17881:and
17818:<
17721:<
17705:<
17566:<
17560:<
17473:and
17414:and
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17177:>
16764:disk
16758:The
16627:and
16451:and
16039:let
15992:Let
15720:and
15700:Let
15533:<
15394:disk
15277:<
15226:<
15024:and
14990:and
14889:and
14663:The
14421:hull
14402:the
14019:and
13713:norm
13707:and
13687:are
13675:are
13507:and
13445:<
13439:<
13336:>
13291:are
12459:has
11436:and
11246:and
11155:Let
11140:>
11111:<
10851:only
10523:and
10284:Let
9634:and
9244:The
9216:Let
8848:and
8644:(or
8514:The
8390:Let
7460:disk
7187:>
7127:(in
6246:Let
5531:and
4587:and
4091:are
3347:let
3220:and
3121:Let
2924:disk
2722:open
2379:knot
2239:Let
2022:and
1861:and
1506:and
904:The
795:The
671:are
574:real
444:and
98:and
78:and
34:, a
34583:GTM
32964:if
32736:If
32613:int
32565:int
32356:int
32342:If
32225:int
32089:by
31997:int
31930:int
31904:int
31785:int
31523:def
31295:int
31238:def
31139:Int
31108:Int
31075:Int
30864:In
30724:In
25287:not
25283:not
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25254:).
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24655:.
24648:of
24530:or
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24193:If
24190:).
24133:bal
24076:bal
24044:bal
24004:of
23661:not
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23060:If
22912:or
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21305:If
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20757:is
20578:of
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19232:in
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18613:Int
18537:Int
18440:sup
18420:Int
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18318:Int
18138:in
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18004:Int
17835:If
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17579:int
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17311:Int
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16942:Int
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16870:Int
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16717:Int
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16636:Int
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16480:Int
16419:If
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14660:).
14614:bal
14552:Int
14423:of
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13830:not
13814:of
13715:or
13679:of
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13511:is
13260:is
13065:not
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13033:not
13007:on
12983:If
12903:dim
12848:dim
12803:dim
12672:ker
12660:ker
12486:on
12409:dim
12401:If
12072:not
11220:If
10883:dim
10875:If
10789:dim
10781:If
10729:dim
10721:If
10697:dim
10432:dim
10328:or
10269:By
10261:).
10136:in
10029:in
10009:to
9931:net
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9636:not
9629:not
9627:is
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9412:on
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9250:or
9180:in
8921:If
8733:of
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8642:net
8637:).
8576:on
8232:on
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8076:If
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7030:in
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5300:If
5096:Sum
4952:ker
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34625:.
34595:.
34581:.
34561:.
34551:.
34525:.
34498:.
34492:MR
34490:.
34460:.
34433:.
34423:.
34364:^
34311:^
34296:^
34257:^
34242:^
34221:^
34186:^
34159:^
34132:^
34117:^
34052:^
34037:^
33996:^
33981:^
33964:.
33927:^
33874:,
33868:,
33815:^
33786:^
33739:^
33706:^
33655:^
33638:^
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33160:).
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