12114:
11399:
3642:(TVS) is a linear operator so all of the properties described above for continuous linear operators apply to them. However, because of their specialized nature, we can say even more about continuous linear functionals than we can about more general continuous linear operators.
1067:
7991:
8236:
2137:
is bounded on some (equivalently, on every) non-degenerate open or closed ball (not necessarily centered at the origin, and of any radius) if and only if it is bounded on the closed unit ball centered at the origin
2836:
equivalent concepts. For this reason, this article will avoid the term "locally bounded" and instead say "locally bounded at every point" (there is no disagreement about the definition of "locally bounded
5246:
2550:
8446:
4984:
5525:
4668:
5378:
1752:
537:
3063:
that is not bounded on any neighborhood of any point. Thus although every linear map that is bounded on a neighborhood is necessarily continuous, the converse is not guaranteed in general.
1335:
971:
10806:
7617:
8348:
5728:
4264:
3143:
is a bounded neighborhood of the origin in a (locally bounded) TVS then its image under any continuous linear map will be a bounded set (so this map is thus bounded on this neighborhood
7403:
5943:
5662:
4579:
3385:
In general, without additional information about either the linear map or its domain or codomain, the map being "bounded" is not equivalent to it being "bounded on a neighborhood". If
2943:
10589:
4385:
8048:
10679:
8667:
5452:
2828:" is sometimes used to refer to a map that is locally bounded at every point of its domain, but some functional analysis authors define "locally bounded" to instead be a synonym of "
2189:
8514:
7072:
6997:
6918:
4457:
7098:
6450:
6350:
6324:
4129:
7828:
4625:
4431:
4200:
3756:
2975:
2429:
1848:
1989:
1482:
7859:
5979:
1939:
10716:
8615:
8092:
405:
8563:
4485:
2103:
1692:
5798:
4918:
4041:
7854:
3694:
2283:
1539:
1383:
1361:
966:
598:
10755:
10523:
5855:
2065:
568:
10882:
6213:
1183:
910:
7763:
4735:
7792:
6840:
5577:
4757:
3470:
3415:
2600:
2391:
2234:
2135:
1571:
1416:
1223:
124:
7472:
5757:
5407:
5292:
4518:
2015:
209:
8374:
7151:
6271:
4860:
3996:
3254:
2634:
1874:
940:
4324:
4293:
4158:
3970:
2359:
5551:
5050:
4889:
3539:
3259:
Any linear map from a TVS into a locally bounded TVS (such as any linear functional) is continuous if and only if it is bounded on a neighborhood. Conversely, if
2733:
2312:
1627:
6085:
8097:
7727:
7573:
7499:
7044:
6944:
6865:
6377:
6132:
6057:
5882:
5315:
4709:
3899:
3851:
3783:
3582:
2998:
2760:
2452:
1654:
833:
330:
253:
8466:
8256:
7704:
7684:
7664:
7640:
7550:
7526:
7443:
7423:
7289:
7265:
7245:
7225:
7198:
7171:
7021:
6811:
6791:
6767:
6743:
6719:
6693:
6667:
6640:
6612:
6584:
6549:
6521:
6494:
6470:
6420:
6400:
6296:
6235:
6174:
6154:
6105:
6034:
6007:
5818:
5601:
5266:
5128:
5104:
5024:
5004:
4834:
4807:
4783:
4084:
4064:
4016:
3943:
3920:
3872:
3828:
3806:
3714:
3668:
3622:
3602:
3559:
3510:
3490:
3438:
3321:
3297:
3277:
3225:
3205:
3185:
3161:
3141:
3045:
3021:
2812:
2792:
2704:
2684:
2661:
2472:
2332:
1792:
1772:
1595:
1502:
1436:
1277:
1154:
1130:
1110:
1090:
876:
856:
810:
779:
755:
735:
711:
680:
652:
621:
475:
448:
428:
370:
350:
307:
276:
230:
177:
155:
3346:. But importantly, in the most general setting of a linear operator between arbitrary topological vector spaces, it is possible for a linear operator to be
9633:
11435:
11288:
10913:
9967:
10612:
3323:
must be a locally bounded TVS. In particular, a linear functional on a arbitrary TVS is continuous if and only if it is bounded on a neighborhood.
2334:
is a normed (or seminormed) space then it suffices to check this condition for the open or closed unit ball centered at the origin. Explicitly, if
11124:
10089:
9234:. Grundlehren der mathematischen Wissenschaften. Vol. 159. Translated by Garling, D.J.H. New York: Springer Science & Business Media.
11945:
9621:
10951:
10908:
17:
11562:
11537:
11114:
10022:
2896:(and thus also bounded) but not bounded on any neighborhood. In particular, it demonstrates that being "bounded on a neighborhood" is
11519:
11241:
11096:
9994:
8261:
5139:
2480:
7320:
The proof uses the facts that the translation of an open set in a linear topological space is again an open set, and the equality
2314:
is a bounded subset of it codomain; or said more briefly, if it is bounded on every bounded subset of its domain. When the domain
12153:
11987:
11489:
11428:
11072:
9628:
9460:
8728:
8379:
7300:
6643:
3721:
3374:
3088:
3056:
2946:
279:
7323:
4923:
11732:
11556:
10341:
7314:
7268:
5457:
4892:
3112:
2736:
2257:
1630:
1248:
1242:
10283:
9421:
9391:
9361:
9334:
9308:
9239:
9209:
9182:
9148:
9094:
9022:
8995:
3001:
10099:
7153:
between TVS that is identically equal to zero is a linear map that is continuous, bounded, and bounded on the neighborhood
4630:
10182:
11997:
11494:
11464:
9649:
8743:
7303:
6619:
6552:
5082:; that is, for a linear functional on a normed space, being "bounded" is equivalent to being "bounded on a neighborhood".
3358:
3092:
683:
628:
1697:
487:
12117:
11768:
11421:
10964:
9274:
5320:
11905:
11053:
10944:
9823:
9606:
9121:
9056:
2853:" (of some point) if and only if it is locally bounded at every point of its domain, in which case it is necessarily
1289:
10771:
7578:
11810:
11323:
10418:
9741:
9584:
6696:
6524:
655:
9048:
10968:
10271:
10207:
9758:
7109:
5669:
4205:
3123:
is a locally bounded TVS since the unit ball centered at the origin is a bounded neighborhood of the origin. If
10052:
5887:
5606:
4523:
3163:). Consequently, a linear map from a locally bounded TVS into any other TVS is continuous if and only if it is
2916:
11840:
10545:
10463:
10266:
9945:
9724:
8710:
4329:
3071:
To summarize the discussion below, for a linear map on a normed (or seminormed) space, being continuous, being
2893:
2854:
74:
44:
7996:
11972:
11574:
11551:
11119:
10642:
10304:
10082:
10037:
10027:
9326:
9082:
8699:
5412:
2141:
8621:
2880:. The converse statements are not true in general but they are both true when the linear map's domain is a
12138:
12023:
11402:
11175:
11109:
10937:
10139:
10129:
10057:
9984:
9860:
9529:
7048:
6949:
6870:
4436:
10134:
8474:
7077:
6429:
6329:
6303:
4089:
11844:
11139:
10477:
10467:
10067:
9453:
8693:
7799:
4588:
4394:
4163:
4044:
3727:
3083:
codomain is normable (or seminormable) is continuous if and only if it bounded on a neighborhood. And a
2960:
2396:
1797:
1062:{\displaystyle {\text{ for all }}x,y\in X,{\text{ if }}\|x-y\|<\delta {\text{ then }}\|Fx-Fy\|<r.}
12080:
11532:
11527:
11384:
11338:
11262:
11144:
10836:
10638:
10300:
10114:
10007:
10002:
9897:
9870:
9687:
9580:
8734:
2664:
1944:
1441:
5948:
1879:
12143:
11876:
11686:
11379:
11195:
10694:
10594:
10288:
10261:
10244:
10062:
9907:
9576:
8053:
5071:
5059:
375:
8569:
4462:
2070:
1659:
12148:
11649:
11644:
11637:
11632:
11504:
11444:
11231:
11129:
11032:
10811:
10104:
10094:
10012:
9950:
9877:
9831:
9746:
9571:
8705:
8520:
7121:
5762:
4898:
4021:
3671:
3639:
2819:
2237:
1230:
131:
78:
51:
7833:
3677:
2262:
1518:
1366:
1344:
945:
573:
11910:
11891:
11567:
11547:
11328:
11104:
10733:
10496:
10077:
10017:
8678:
7177:(although it is possible for the constant zero map to be its only continuous linear functional).
7105:
6670:
6587:
5830:
5063:
3366:
3347:
3343:
3084:
3072:
2977:
is bounded on a neighborhood if and only if there exists a bounded neighborhood of the origin in
2954:
2901:
2877:
2866:
2862:
2829:
2553:
2250:
2242:
2199:
2020:
714:
546:
62:
10845:
6179:
1159:
889:
12099:
12089:
12073:
11773:
11722:
11622:
11607:
11359:
11303:
11267:
10119:
10032:
9813:
9729:
9565:
9559:
9446:
9166:
7732:
4718:
10359:
7768:
6816:
5556:
4740:
3443:
3388:
2573:
2364:
2207:
2108:
1544:
1391:
1341:
if it is norm-bounded (or equivalently, von
Neumann bounded). For example, the scalar field (
1196:
97:
12068:
11755:
11737:
11702:
11542:
11066:
10903:
10898:
10373:
10321:
10278:
10202:
10155:
9892:
9554:
9521:
9494:
7448:
7174:
7101:
5821:
5733:
5383:
5271:
4711:). This is one of several reasons why many definitions involving linear functionals, such as
4490:
3875:
1994:
185:
180:
47:
11062:
10192:
8353:
7130:
6240:
4839:
3975:
3233:
2613:
1853:
919:
12084:
12028:
12007:
11342:
10841:
10047:
10042:
9753:
9637:
9543:
9379:
9249:
4302:
4271:
4136:
3948:
2337:
10929:
5530:
5029:
4865:
3515:
2709:
2288:
1603:
8:
11967:
11962:
11920:
11499:
11308:
11246:
10960:
10684:
10485:
10442:
10256:
9979:
9709:
9516:
9086:
7986:{\displaystyle f(U):=\{f(u):u\in U\}\quad {\text{ and }}\quad |f(U)|:=\{|f(u)|:u\in U\},}
6064:
3331:
3076:
28:
10630:
7709:
7555:
7481:
7026:
6926:
6847:
6359:
6114:
6039:
5864:
5297:
4673:
3881:
3833:
3765:
3564:
2980:
2742:
2434:
1636:
815:
312:
235:
11952:
11895:
11829:
11814:
11681:
11671:
11333:
11200:
10918:
10829:
10452:
10422:
10239:
10197:
9804:
9714:
9659:
9506:
9318:
9171:
8748:
8719:
8451:
8241:
7689:
7669:
7649:
7625:
7619:(where in particular, one side is infinite if and only if the other side is infinite).
7535:
7511:
7428:
7408:
7274:
7250:
7230:
7210:
7183:
7156:
7124:(TVS) is continuous. This is not true if the finite-dimensional TVS is not Hausdorff.
7006:
6796:
6776:
6752:
6728:
6704:
6678:
6652:
6625:
6615:
6597:
6569:
6534:
6506:
6479:
6455:
6405:
6385:
6281:
6220:
6159:
6139:
6090:
6019:
5992:
5803:
5586:
5251:
5113:
5089:
5009:
4989:
4819:
4792:
4768:
4069:
4049:
4001:
3928:
3905:
3857:
3813:
3791:
3699:
3653:
3633:
3607:
3587:
3544:
3495:
3475:
3423:
3370:
3306:
3282:
3262:
3210:
3190:
3170:
3146:
3126:
3096:
3030:
3024:
3006:
2797:
2777:
2689:
2669:
2646:
2457:
2317:
1941:
and any translation of a bounded set is again bounded) if and only if it is bounded on
1777:
1757:
1580:
1487:
1421:
1281:
1262:
1139:
1115:
1095:
1075:
861:
841:
795:
782:
764:
740:
720:
696:
665:
637:
606:
460:
433:
413:
355:
335:
292:
261:
215:
162:
140:
10760:
10730:
10691:
9227:
3167:. Moreover, any TVS with this property must be a locally bounded TVS. Explicitly, if
11664:
11590:
11313:
10599:
10072:
9853:
9796:
9776:
9427:
9417:
9397:
9387:
9367:
9357:
9340:
9330:
9304:
9297:
9280:
9270:
9253:
9235:
9215:
9205:
9188:
9178:
9154:
9144:
9127:
9117:
9100:
9090:
9062:
9052:
9028:
9018:
9001:
8991:
7204:
3759:
3228:
2565:
478:
6590:(that is, it maps bounded subsets of its domain to bounded subsets of its codomain).
12057:
11627:
11612:
11413:
11318:
11236:
11205:
11185:
11170:
11165:
11160:
10229:
10224:
10212:
10124:
10109:
9972:
9912:
9887:
9818:
9808:
9671:
9329:. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer.
9040:
8687:
6497:
5079:
3120:
1256:
883:
624:
89:
11940:
11479:
10997:
9616:
9177:. Translated by Chaljub, Orlando. New York: Gordon and Breach Science Publishers.
6556:
3330:
the codomain of a linear map is normable or seminormable, then continuity will be
12032:
11880:
11180:
11134:
11082:
11077:
11048:
10249:
10234:
10160:
9962:
9955:
9922:
9882:
9848:
9840:
9768:
9736:
9601:
9533:
9409:
9245:
8987:
5054:
Importantly, a linear functional being "bounded on a neighborhood" is in general
3717:
2950:
1507:
Any translation, scalar multiple, and subset of a bounded set is again bounded.
451:
127:
11007:
1247:
The notion of a "bounded set" for a topological vector space is that of being a
12063:
12012:
11727:
11369:
11221:
11022:
10819:
10767:
10427:
10293:
9940:
9930:
9549:
9501:
9074:
5982:
3048:
2844:
1386:
540:
3187:
is a TVS such that every continuous linear map (into any TVS) whose domain is
2067:
and any scalar multiple of a bounded set is again bounded). Consequently, if
12132:
12047:
11957:
11900:
11860:
11788:
11763:
11707:
11659:
11595:
11374:
11298:
11027:
11012:
11002:
10824:
10437:
10391:
10326:
10177:
10172:
10165:
9786:
9719:
9692:
9511:
9484:
9431:
9401:
9344:
9284:
9269:. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press.
9257:
9051:. Translated by Eggleston, H.G.; Madan, S. Berlin New York: Springer-Verlag.
9032:
9005:
2911:: A continuous and bounded linear map that is not bounded on any neighborhood
2475:
9371:
9158:
9131:
9104:
9066:
5981:
Polar sets, and so also this particular inequality, play important roles in
4715:
for example, involve closed (rather than open) neighborhoods and non-strict
3279:
is a TVS such that every continuous linear map (from any TVS) with codomain
2794:
in its domain at which it is locally bounded, in which case this linear map
12094:
12042:
12002:
11992:
11870:
11717:
11712:
11509:
11459:
11364:
11017:
10987:
10336:
10331:
9791:
9781:
9654:
9644:
9489:
9469:
9292:
9219:
8231:{\displaystyle \sup |f(U)|~=~\sup\{|f(u)|:u\in U\}~=~\sup _{u\in U}|f(u)|.}
7529:
7475:
6746:
6560:
5075:
3418:
3362:
3116:
3060:
2887:
2881:
2858:
1504:
is contained in some open (or closed) ball centered at the origin (zero).
1252:
879:
786:
687:
58:
9192:
3369:
if and only if it is continuous. The same is true of a linear map from a
12052:
12037:
11930:
11824:
11819:
11804:
11783:
11747:
11654:
11474:
11293:
11283:
11190:
10992:
10625:
10541:
10447:
10432:
10412:
10386:
10351:
9902:
9865:
9538:
3051:). This shows that it is possible for a linear map to be continuous but
32:
11865:
11778:
11742:
11602:
11484:
11226:
11058:
10381:
10362: ((cs, lcs)-closed, (cs, bcs)-complete, (lower) ideally convex, (H
10346:
10187:
9935:
9697:
8806:
3945:
said to be continuous at the origin if for every open (or closed) ball
1226:
9438:
8863:
8861:
8859:
8857:
8855:
8804:
8802:
8800:
8798:
8796:
8794:
8792:
8790:
8788:
8786:
8773:
8771:
8769:
8767:
8765:
12017:
11834:
10457:
9702:
9666:
6422:
are complex vector spaces then this list may be extended to include:
6353:
5858:
5062:" because (as described above) it is possible for a linear map to be
4712:
482:
8928:
8916:
7310:
if and only if every bounded linear functional on it is continuous.
3055:
bounded on any neighborhood. Indeed, this example shows that every
11982:
11977:
11935:
11915:
11885:
11676:
10723:
10607:
10533:
10493:
10396:
10219:
9017:. Graduate Texts in Mathematics. Vol. 15. New York: Springer.
8852:
8783:
8762:
7643:
7307:
4296:
287:
8984:
Topological Vector Spaces: The Theory
Without Convexity Conditions
8731: – A vector space with a topology defined by convex open sets
8713: – A vector space with a topology defined by convex open sets
7227:
is necessarily continuous if and only if every vector subspace of
3103:
Guaranteeing that "continuous" implies "bounded on a neighborhood"
2892:
The next example shows that it is possible for a linear map to be
2474:
is also a (semi)normed space then this happens if and only if the
11925:
10528:
9611:
3624:
of the origin, which (as mentioned above) guarantees continuity.
454:
locally convex spaces then this list may be extended to include:
8986:. Lecture Notes in Mathematics. Vol. 639. Berlin New York:
8964:
7267:
is necessarily a bounded linear functional if and only if every
7120:
Every linear map whose domain is a finite-dimensional
Hausdorff
8828:
8681: – Linear transformation between topological vector spaces
3645:
3381:
Guaranteeing that "bounded" implies "bounded on a neighborhood"
8722: – Linear map from a vector space to its field of scalars
7503:
1132:
finite-dimensional then this list may be extended to include:
5241:{\displaystyle \sup _{x\in sU}|f(x)|=|s|\sup _{u\in U}|f(u)|}
3561:
is a bounded linear map) and a neighborhood of the origin in
2545:{\displaystyle \|F\|:=\sup _{\|x\|\leq 1}\|F(x)\|<\infty }
2845:
Bounded on a neighborhood implies continuous implies bounded
10959:
8894:
8892:
8890:
8888:
8441:{\displaystyle B_{\leq r}:=\{c\in \mathbb {F} :|c|\leq r\}}
6769:
is real-valued) then this list may be extended to include:
4979:{\textstyle \displaystyle \sup _{u\in U}|f(u)|<\infty .}
4816:
Explicitly, this means that there exists some neighborhood
8751: – Linear operator defined on a dense linear subspace
8468:
centered at the origin then the following are equivalent:
2949:
then this linear map is always continuous (indeed, even a
2876:
then it is continuous, and if it is continuous then it is
5520:{\displaystyle \displaystyle \sup _{n\in N_{r}}|f(n)|=r.}
5317:
will be neighborhood of the origin. So in particular, if
3472:
is necessarily continuous; this is because any open ball
1694:
is a normed (or seminormed) space happens if and only if
8885:
8873:
8840:
7622:
Every non-trivial continuous linear functional on a TVS
7173:
of the origin. In particular, every TVS has a non-empty
6527:
at some (or equivalently, at every) point of its domain.
2888:
Continuous and bounded but not bounded on a neighborhood
2884:. Examples and additional details are now given below.
2560:
Function bounded on a neighborhood and local boundedness
658:
at some (or equivalently, at every) point of its domain.
8724:
Pages displaying short descriptions of redirect targets
8715:
Pages displaying short descriptions of redirect targets
8683:
Pages displaying short descriptions of redirect targets
5380:
is a positive real number then for every positive real
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is contained in a finite-dimensional vector subspace.
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4663:{\displaystyle X=\mathbb {R} ,f=\operatorname {Id} ,}
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A subset of a normed (or seminormed) space is called
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9384:
Topological Vector Spaces, Distributions and
Kernels
8982:
Adasch, Norbert; Ernst, Bruno; Keim, Dieter (1978).
8739:
Pages displaying wikidata descriptions as a fallback
65:if and only if it is a continuous linear operator.
9015:
Lectures in
Functional Analysis and Operator Theory
2105:is a normed or seminormed space, then a linear map
812:maps some neighborhood of 0 to a bounded subset of
11289:Spectral theory of ordinary differential equations
10876:
10800:
10749:
10710:
10673:
10583:
10517:
9296:
9170:
9116:(in Romanian). New York: Interscience Publishers.
8696: – Mathematical method in functional analysis
8661:
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8557:
8508:
8460:
8440:
8368:
8342:
8250:
8230:
8086:
8042:
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7856:is a non-empty subset, then by defining the sets
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5595:
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5401:
5373:{\textstyle R:=\displaystyle \sup _{u\in U}|f(u)|}
5372:
5309:
5286:
5260:
5240:
5122:
5098:
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5018:
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3248:
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3199:
3179:
3155:
3135:
3039:
3015:
2992:
2969:
2937:
2865:(because a continuous linear operator is always a
2806:
2786:
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2727:
2698:
2678:
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2628:
2594:
2544:
2466:
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2183:
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2009:
1983:
1933:
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1747:{\displaystyle \sup _{s\in S}\|F(s)\|<\infty .}
1746:
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1565:
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532:{\displaystyle {}^{t}F:Y^{\prime }\to X^{\prime }}
531:
469:
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8737: – ordered vector space with a partial order
5107:
12130:
8573:
8524:
8315:
8265:
8188:
8135:
8101:
8058:
8002:
6699:(or equivalently, at every) point of its domain.
5892:
5674:
5611:
5463:
5332:
5201:
5144:
4929:
4528:
4334:
4210:
3627:
3338:Guaranteeing that "bounded" implies "continuous"
2497:
1702:
1446:
1294:
8981:
5759:which shows that the positive scalar multiples
3207:is necessarily bounded on a neighborhood, then
1330:{\displaystyle \sup _{s\in S}\|s\|<\infty .}
10801:{\displaystyle S\left(\mathbb {R} ^{n}\right)}
9416:. Mineola, New York: Dover Publications, Inc.
9265:Narici, Lawrence; Beckenstein, Edward (2011).
7666:is a linear functional on a real vector space
7612:{\displaystyle \|f\|=\|\operatorname {Re} f\|}
4786:
3300:
3164:
3111:if there exists a neighborhood that is also a
3075:, and being bounded on a neighborhood are all
2873:
2850:
68:
11429:
10945:
10909:Mathematical formulation of quantum mechanics
9454:
9317:
8343:{\displaystyle \sup |f(sU)|~=~|s|\sup |f(U)|}
6563:) then this list may be extended to include:
3512:is both a bounded subset (which implies that
1188:
912:) then this list may be extended to include:
789:) then this list may be extended to include:
631:) then this list may be extended to include:
9165:
9141:Functional Analysis: Theory and Applications
8702: – Bounded operators with sub-unit norm
8435:
8399:
8175:
8138:
7977:
7940:
7905:
7878:
7606:
7594:
7588:
7582:
6986:
6953:
6907:
6874:
5787:
5766:
3646:Characterizing continuous linear functionals
2533:
2518:
2507:
2501:
2490:
2484:
2393:is a bounded linear operator if and only if
2175:
2166:
2160:
2145:
2089:
2083:
1978:
1957:
1837:
1813:
1732:
1717:
1678:
1672:
1315:
1309:
1279:is von Neumann bounded if and only if it is
1047:
1029:
1015:
1003:
899:
893:
9414:Modern Methods in Topological Vector Spaces
7504:Properties of continuous linear functionals
6646:then this list may be extended to include:
6500:then this list may be extended to include:
5723:{\displaystyle \sup _{x\in rU}|f(x)|\leq r}
4259:{\displaystyle \sup _{u\in U}|f(u)|\leq r.}
3227:must be a locally bounded TVS (because the
282:then this list may be extended to include:
11436:
11422:
10952:
10938:
9461:
9447:
8690: – Type of continuous linear operator
7398:{\displaystyle F^{-1}(D)+x=F^{-1}(D+F(x))}
6237:is continuous if and only if the seminorm
5938:{\displaystyle \sup _{u\in U}|f(u)|\leq 1}
5657:{\displaystyle \sup _{u\in U}|f(u)|\leq 1}
4574:{\displaystyle \sup _{u\in U}|f(u)|\leq r}
3059:that is not seminormable has a linear TVS-
2938:{\displaystyle \operatorname {Id} :X\to X}
10784:
10584:{\displaystyle B_{p,q}^{s}(\mathbb {R} )}
10574:
9085:. Vol. 96 (2nd ed.). New York:
9012:
8409:
8083:
8057:
8039:
8000:
7816:
6721:is sequentially continuous at the origin.
4903:
4748:
4744:
4726:
4722:
4641:
4473:
4380:{\displaystyle \sup _{u\in U}|f(u)|<r}
4026:
3744:
3682:
3342:A continuous linear operator is always a
3066:
1371:
1349:
1112:are Hausdorff locally convex spaces with
11242:Group algebra of a locally compact group
9408:
9303:. McGraw-Hill Science/Engineering/Math.
9039:
8958:
8946:
8910:
8898:
8879:
8846:
8822:
8043:{\displaystyle \,\sup _{u\in U}|f(u)|\,}
1484:is finite, which happens if and only if
134:(TVSs). The following are equivalent:
10674:{\displaystyle L^{\lambda ,p}(\Omega )}
9468:
9199:
9138:
9111:
9045:Topological Vector Spaces: Chapters 1–5
8729:Locally convex topological vector space
8662:{\textstyle f(rU)\subseteq B_{\leq r}.}
5447:{\displaystyle N_{r}:={\tfrac {r}{R}}U}
5131:
5108:bounded on a neighborhood of the origin
4810:
2947:locally convex topological vector space
2184:{\displaystyle \{x\in X:\|x\|\leq 1\}.}
14:
12131:
11575:Uniform boundedness (Banach–Steinhaus)
10914:Ordinary Differential Equations (ODEs)
10028:Banach–Steinhaus (Uniform boundedness)
9378:
9354:An introduction to Functional Analysis
9351:
9073:
7247:is closed. Every linear functional on
4433:to be true (consider for example when
1243:Bounded set (topological vector space)
11417:
10933:
9442:
9386:. Mineola, N.Y.: Dover Publications.
9291:
9226:
8509:{\textstyle f(U)\subseteq B_{\leq 1}}
7067:{\displaystyle \operatorname {Re} f,}
6992:{\displaystyle \{x\in X:f(x)\leq r\}}
6913:{\displaystyle \{x\in X:f(x)\leq r\}}
5666:This inequality holds if and only if
4520:), whereas the non-strict inequality
4452:{\displaystyle f=\operatorname {Id} }
3604:is thus bounded on this neighborhood
3047:is Hausdorff, is the same as being a
1251:. If the space happens to also be a
717:(that is, it maps bounded subsets of
7093:{\displaystyle \operatorname {Im} f}
7023:is complex then either all three of
6445:{\displaystyle \operatorname {Im} f}
6345:{\displaystyle \operatorname {Re} f}
6319:{\displaystyle \operatorname {Re} f}
6010:
5454:is a neighborhood of the origin and
5070:continuous. However, continuity and
4986:This supremum over the neighborhood
4160:is a closed ball then the condition
4124:{\displaystyle f(U)\subseteq B_{r}.}
3417:is a bounded linear operator from a
3334:to being bounded on a neighborhood.
3256:is always a continuous linear map).
3091:will be continuous if its domain is
8744:Topologies on spaces of linear maps
7823:{\displaystyle f:X\to \mathbb {F} }
7304:metrizable topological vector space
6773:There exists a continuous seminorm
6749:(which in particular, implies that
6136:There exists a continuous seminorm
4789:(of some point). Said differently,
4620:{\displaystyle f(U)\subseteq B_{r}}
4426:{\displaystyle f(U)\subseteq B_{r}}
4195:{\displaystyle f(U)\subseteq B_{r}}
3751:{\displaystyle f:X\to \mathbb {F} }
2970:{\displaystyle \operatorname {Id} }
2424:{\displaystyle F\left(B_{1}\right)}
1843:{\displaystyle x+S:=\{x+s:s\in S\}}
686:or metrizable (such as a normed or
332:there exists a continuous seminorm
24:
10742:
10703:
10665:
10509:
8050:can be written more succinctly as
7313:A continuous linear operator maps
5074:are equivalent if the domain is a
4969:
2814:is necessarily locally bounded at
2770:" (of some point) if there exists
2539:
1738:
1321:
582:
555:
524:
511:
25:
12165:
10406:Subsets / set operations
10183:Differentiation in Fréchet spaces
4627:to be true (consider for example
1984:{\displaystyle cS:=\{cs:s\in S\}}
1477:{\displaystyle \sup _{s\in S}|s|}
690:) then we may add to this list:
12113:
12112:
11398:
11397:
11324:Topological quantum field theory
9143:. New York: Dover Publications.
5974:{\displaystyle f\in U^{\circ }.}
4299:characterization. Assuming that
2818:point of its domain. The term "
1934:{\displaystyle F(x+S)=F(x)+F(S)}
1794:if and only if it is bounded on
886:(with both seminorms denoted by
12100:With the approximation property
10711:{\displaystyle \ell ^{\infty }}
9079:A course in functional analysis
9013:Berberian, Sterling K. (1974).
8610:{\textstyle \sup |f(rU)|\leq r}
8087:{\displaystyle \,\sup |f(U)|\,}
7914:
7908:
6697:sequentially continuous at some
5820:will satisfy the definition of
5583:There exists some neighborhood
5553:proves the next statement when
1418:is a normed space, so a subset
400:{\displaystyle q\circ F\leq p.}
232:is continuous at the origin in
84:Characterizations of continuity
12154:Theory of continuous functions
11563:Open mapping (Banach–Schauder)
10871:
10852:
10668:
10662:
10578:
10570:
10512:
10506:
10100:Lomonosov's invariant subspace
10023:Banach–Schauder (open mapping)
8711:Finest locally convex topology
8637:
8628:
8597:
8593:
8584:
8577:
8558:{\textstyle \sup |f(U)|\leq 1}
8545:
8541:
8535:
8528:
8487:
8481:
8425:
8417:
8336:
8332:
8326:
8319:
8311:
8303:
8289:
8285:
8276:
8269:
8221:
8217:
8211:
8204:
8159:
8155:
8149:
8142:
8122:
8118:
8112:
8105:
8079:
8075:
8069:
8062:
8035:
8031:
8025:
8018:
7961:
7957:
7951:
7944:
7933:
7929:
7923:
7916:
7890:
7884:
7872:
7866:
7812:
7745:
7737:
7392:
7389:
7383:
7371:
7346:
7340:
7137:
6977:
6971:
6898:
6892:
6553:metrizable or pseudometrizable
6259:
6251:
6192:
6184:
6011:locally bounded at every point
5925:
5921:
5915:
5908:
5710:
5706:
5700:
5693:
5644:
5640:
5634:
5627:
5503:
5499:
5493:
5486:
5365:
5361:
5355:
5348:
5234:
5230:
5224:
5217:
5196:
5188:
5180:
5176:
5170:
5163:
5132:locally bounded at the origin.
4962:
4958:
4952:
4945:
4878:
4872:
4698:
4683:
4601:
4595:
4561:
4557:
4551:
4544:
4480:{\displaystyle X=\mathbb {F} }
4407:
4401:
4367:
4363:
4357:
4350:
4326:is instead an open ball, then
4243:
4239:
4233:
4226:
4176:
4170:
4102:
4096:
3785:The following are equivalent:
3740:
3528:
3522:
3456:
3401:
3240:
2929:
2900:always synonymous with being "
2722:
2716:
2586:
2530:
2524:
2377:
2301:
2295:
2220:
2121:
2098:{\displaystyle (X,\|\cdot \|)}
2092:
2074:
2054:
2048:
2036:
2027:
1928:
1922:
1913:
1907:
1898:
1886:
1729:
1723:
1687:{\displaystyle (Y,\|\cdot \|)}
1681:
1663:
1616:
1610:
1557:
1470:
1462:
1404:
1396:
1209:
516:
110:
75:Continuous function (topology)
13:
1:
11120:Uniform boundedness principle
9083:Graduate Texts in Mathematics
8971:Narici & Beckenstein 2011
8935:Narici & Beckenstein 2011
8923:Narici & Beckenstein 2011
8868:Narici & Beckenstein 2011
8835:Narici & Beckenstein 2011
8811:Narici & Beckenstein 2011
8778:Narici & Beckenstein 2011
8755:
8700:Contraction (operator theory)
8448:is the closed ball of radius
7294:
5793:{\displaystyle \{rU:r>0\}}
4913:{\displaystyle \mathbb {F} ;}
4811:locally bounded at some point
4036:{\displaystyle \mathbb {F} ,}
3922:is continuous at the origin.
3638:Every linear functional on a
3628:Continuous linear functionals
3357:A linear map whose domain is
3079:. A linear map whose domain
2872:For any linear map, if it is
2857:(even if its domain is not a
2556:linear operator is bounded.
570:to equicontinuous subsets of
9985:Singular value decomposition
9321:; Wolff, Manfred P. (1999).
7849:{\displaystyle U\subseteq X}
5800:of this single neighborhood
4670:and the closed neighborhood
3689:{\displaystyle \mathbb {F} }
2945:is the identity map on some
2606:bounded on a neighborhood of
2278:{\displaystyle B\subseteq X}
2258:(von Neumann) bounded subset
2204:By definition, a linear map
1534:{\displaystyle S\subseteq X}
1378:{\displaystyle \mathbb {C} }
1356:{\displaystyle \mathbb {R} }
961:{\displaystyle \delta >0}
593:{\displaystyle X^{\prime }.}
18:Continuous linear functional
7:
11784:Radially convex/Star-shaped
11769:Pre-compact/Totally bounded
10750:{\displaystyle L^{\infty }}
10518:{\displaystyle ba(\Sigma )}
10387:Radially convex/Star-shaped
9232:Topological Vector Spaces I
9204:. Stuttgart: B.G. Teubner.
9139:Edwards, Robert E. (1995).
8694:Continuous linear extension
8672:
7830:is a linear functional and
7200:is any Hausdorff TVS. Then
7115:
7112:(respectively, unbounded).
6745:is a vector space over the
5850:{\displaystyle U^{\circ },}
4581:is instead a necessary but
3830:is uniformly continuous on
2060:{\displaystyle F(cS)=cF(S)}
563:{\displaystyle Y^{\prime }}
69:Continuous linear operators
10:
12170:
11470:Continuous linear operator
11263:Invariant subspace problem
10877:{\displaystyle W(X,L^{p})}
8735:Positive linear functional
7552:is a linear functional on
6208:{\displaystyle |f|\leq p.}
3631:
3492:centered at the origin in
2563:
2197:
1991:for every non-zero scalar
1438:is bounded if and only if
1240:
1189:Continuity and boundedness
1178:{\displaystyle X\times Y.}
905:{\displaystyle \|\cdot \|}
87:
72:
37:continuous linear operator
12108:
11853:
11815:Algebraic interior (core)
11797:
11695:
11583:
11557:Vector-valued Hahn–Banach
11518:
11452:
11445:Topological vector spaces
11393:
11352:
11276:
11255:
11214:
11153:
11095:
11041:
10983:
10976:
10891:
10476:
10423:Algebraic interior (core)
10405:
10314:
10148:
10038:Cauchy–Schwarz inequality
9993:
9921:
9767:
9681:Function space Topologies
9680:
9594:
9477:
9323:Topological Vector Spaces
9267:Topological Vector Spaces
9173:Topological Vector Spaces
7758:{\displaystyle |f|\leq p}
7474:which is true due to the
7108:), or else all three are
5827:By definition of the set
5060:bounded linear functional
4787:bounded on a neighborhood
4730:{\displaystyle \,\leq \,}
4295:be a closed ball in this
3301:bounded on a neighborhood
3165:bounded on a neighborhood
2874:bounded on a neighborhood
2851:bounded on a neighborhood
2832:", which are related but
2766:bounded on a neighborhood
1511:Function bounded on a set
1231:topological vector spaces
132:topological vector spaces
52:topological vector spaces
41:continuous linear mapping
11645:Topological homomorphism
11505:Topological vector space
11232:Spectrum of a C*-algebra
9352:Swartz, Charles (1992).
9112:Dunford, Nelson (1988).
9049:Éléments de mathématique
8706:Discontinuous linear map
7787:{\displaystyle f\leq p.}
7122:topological vector space
6835:{\displaystyle f\leq p.}
6771:
6648:
6565:
6502:
6424:
5822:continuity at the origin
5603:of the origin such that
5572:{\displaystyle R\neq 0.}
4752:{\displaystyle \,<\,}
3876:continuous at some point
3672:topological vector space
3640:topological vector space
3465:{\displaystyle F:X\to Y}
3410:{\displaystyle F:X\to Y}
2595:{\displaystyle F:X\to Y}
2386:{\displaystyle F:X\to Y}
2229:{\displaystyle F:X\to Y}
2130:{\displaystyle F:X\to Y}
1566:{\displaystyle F:X\to Y}
1411:{\displaystyle |\cdot |}
1218:{\displaystyle F:X\to Y}
1134:
181:continuous at some point
119:{\displaystyle F:X\to Y}
79:Discontinuous linear map
57:An operator between two
11329:Noncommutative geometry
9356:. New York: M. Dekker.
9167:Grothendieck, Alexander
8679:Bounded linear operator
7467:{\displaystyle x\in X,}
6671:bounded linear operator
6588:bounded linear operator
6525:sequentially continuous
5752:{\displaystyle r>0,}
5402:{\displaystyle r>0,}
5287:{\displaystyle s\neq 0}
5058:equivalent to being a "
4513:{\displaystyle U=B_{r}}
4459:is the identity map on
3344:bounded linear operator
3085:bounded linear operator
2867:bounded linear operator
2830:bounded linear operator
2554:sequentially continuous
2431:is a bounded subset of
2361:denotes this ball then
2251:bounded linear operator
2200:Bounded linear operator
2010:{\displaystyle c\neq 0}
1249:von Neumann bounded set
914:
791:
715:bounded linear operator
692:
656:sequentially continuous
633:
456:
284:
204:{\displaystyle x\in X.}
63:bounded linear operator
11703:Absolutely convex/disk
11385:Tomita–Takesaki theory
11360:Approximation property
11304:Calculus of variations
10878:
10802:
10751:
10712:
10675:
10585:
10519:
9688:Banach–Mazur compactum
9478:Types of Banach spaces
9200:Jarchow, Hans (1981).
8663:
8611:
8559:
8510:
8462:
8442:
8370:
8369:{\displaystyle r>0}
8344:
8252:
8232:
8088:
8044:
7987:
7850:
7824:
7788:
7759:
7723:
7700:
7680:
7660:
7636:
7613:
7569:
7546:
7522:
7495:
7468:
7439:
7419:
7399:
7285:
7261:
7241:
7221:
7194:
7167:
7147:
7146:{\displaystyle X\to Y}
7094:
7068:
7040:
7017:
6993:
6940:
6914:
6861:
6836:
6807:
6787:
6763:
6739:
6715:
6689:
6663:
6636:
6608:
6580:
6545:
6517:
6490:
6466:
6446:
6416:
6396:
6373:
6346:
6320:
6292:
6267:
6266:{\displaystyle p:=|f|}
6231:
6209:
6170:
6150:
6128:
6101:
6087:or else the kernel of
6081:
6053:
6030:
6003:
5975:
5939:
5878:
5851:
5814:
5794:
5753:
5724:
5658:
5597:
5573:
5547:
5521:
5448:
5403:
5374:
5311:
5288:
5262:
5248:holds for all scalars
5242:
5124:
5100:
5046:
5020:
5000:
4980:
4914:
4885:
4856:
4855:{\displaystyle x\in X}
4830:
4803:
4779:
4753:
4731:
4705:
4664:
4621:
4575:
4514:
4481:
4453:
4427:
4381:
4320:
4289:
4260:
4196:
4154:
4125:
4080:
4060:
4037:
4012:
3992:
3991:{\displaystyle r>0}
3966:
3939:
3916:
3895:
3868:
3847:
3824:
3802:
3779:
3752:
3710:
3690:
3664:
3618:
3598:
3578:
3555:
3535:
3506:
3486:
3466:
3434:
3411:
3317:
3293:
3273:
3250:
3249:{\displaystyle X\to X}
3221:
3201:
3181:
3157:
3137:
3067:Guaranteeing converses
3041:
3017:
2994:
2971:
2939:
2808:
2788:
2756:
2729:
2700:
2680:
2657:
2630:
2629:{\displaystyle x\in X}
2596:
2546:
2468:
2448:
2425:
2387:
2355:
2328:
2308:
2279:
2230:
2185:
2131:
2099:
2061:
2011:
1985:
1935:
1870:
1869:{\displaystyle x\in X}
1844:
1788:
1768:
1748:
1688:
1650:
1623:
1591:
1567:
1535:
1498:
1478:
1432:
1412:
1379:
1357:
1331:
1273:
1219:
1179:
1150:
1126:
1106:
1086:
1063:
962:
936:
935:{\displaystyle r>0}
906:
872:
852:
829:
806:
775:
751:
737:to bounded subsets of
731:
707:
676:
648:
629:pseudometrizable space
617:
594:
564:
533:
471:
444:
424:
401:
366:
346:
326:
303:
272:
249:
226:
205:
173:
151:
120:
11738:Complemented subspace
11552:hyperplane separation
11380:Banach–Mazur distance
11343:Generalized functions
10904:Finite element method
10899:Differential operator
10879:
10803:
10752:
10713:
10676:
10586:
10520:
10360:Convex series related
10156:Abstract Wiener space
10083:hyperplane separation
9638:Minkowski functionals
9522:Polarization identity
9202:Locally convex spaces
8664:
8612:
8560:
8511:
8463:
8443:
8376:is a real number and
8371:
8345:
8253:
8233:
8089:
8045:
7988:
7851:
7825:
7789:
7760:
7724:
7701:
7681:
7661:
7637:
7614:
7570:
7547:
7523:
7496:
7469:
7440:
7420:
7400:
7286:
7262:
7242:
7222:
7195:
7175:continuous dual space
7168:
7148:
7127:Every (constant) map
7095:
7069:
7041:
7018:
6994:
6941:
6915:
6862:
6837:
6808:
6788:
6764:
6740:
6716:
6690:
6664:
6637:
6609:
6581:
6546:
6518:
6491:
6467:
6447:
6417:
6397:
6374:
6347:
6326:is continuous, where
6321:
6293:
6268:
6232:
6210:
6171:
6151:
6129:
6102:
6082:
6054:
6031:
6004:
5976:
5945:holds if and only if
5940:
5879:
5852:
5815:
5795:
5754:
5725:
5659:
5598:
5574:
5548:
5522:
5449:
5404:
5375:
5312:
5289:
5263:
5243:
5125:
5101:
5047:
5021:
5001:
4981:
4915:
4886:
4857:
4831:
4804:
4780:
4754:
4732:
4706:
4665:
4622:
4576:
4515:
4482:
4454:
4428:
4382:
4321:
4319:{\displaystyle B_{r}}
4290:
4288:{\displaystyle B_{r}}
4268:It is important that
4261:
4202:holds if and only if
4197:
4155:
4153:{\displaystyle B_{r}}
4126:
4081:
4061:
4038:
4013:
3993:
3967:
3965:{\displaystyle B_{r}}
3940:
3917:
3896:
3869:
3848:
3825:
3803:
3780:
3753:
3711:
3691:
3674:(TVS) over the field
3665:
3619:
3599:
3579:
3556:
3536:
3507:
3487:
3467:
3435:
3412:
3326:Thus when the domain
3318:
3294:
3274:
3251:
3222:
3202:
3182:
3158:
3138:
3115:. For example, every
3042:
3018:
2995:
2972:
2940:
2809:
2789:
2757:
2730:
2701:
2681:
2658:
2631:
2597:
2547:
2469:
2449:
2426:
2388:
2356:
2354:{\displaystyle B_{1}}
2329:
2309:
2280:
2231:
2186:
2132:
2100:
2062:
2012:
1986:
1936:
1871:
1845:
1789:
1769:
1749:
1689:
1651:
1624:
1592:
1568:
1536:
1499:
1479:
1433:
1413:
1380:
1358:
1332:
1274:
1220:
1180:
1151:
1127:
1107:
1087:
1064:
963:
937:
907:
873:
853:
830:
807:
776:
752:
732:
708:
677:
649:
618:
595:
565:
534:
472:
445:
425:
402:
367:
347:
327:
304:
286:for every continuous
273:
250:
227:
206:
174:
152:
121:
48:linear transformation
31:and related areas of
11988:Locally convex space
11538:Closed graph theorem
11490:Locally convex space
11125:Kakutani fixed-point
11110:Riesz representation
10846:
10772:
10734:
10695:
10643:
10546:
10497:
10486:Absolute continuity
10140:Schauder fixed-point
10130:Riesz representation
10090:Kakutani fixed-point
10058:Freudenthal spectral
9544:L-semi-inner product
8622:
8570:
8521:
8475:
8452:
8380:
8354:
8262:
8242:
8098:
8054:
7997:
7860:
7834:
7800:
7769:
7733:
7710:
7690:
7670:
7650:
7626:
7579:
7556:
7536:
7512:
7482:
7449:
7429:
7409:
7324:
7275:
7251:
7231:
7211:
7184:
7157:
7131:
7078:
7049:
7027:
7007:
6950:
6927:
6871:
6848:
6817:
6797:
6777:
6753:
6729:
6705:
6679:
6653:
6626:
6620:pseudometrizable TVS
6598:
6570:
6535:
6507:
6480:
6456:
6430:
6406:
6386:
6360:
6330:
6304:
6282:
6241:
6221:
6180:
6160:
6140:
6115:
6091:
6065:
6040:
6020:
5993:
5949:
5888:
5865:
5857:which is called the
5831:
5804:
5763:
5734:
5670:
5607:
5587:
5557:
5546:{\displaystyle r:=1}
5531:
5458:
5413:
5384:
5321:
5298:
5272:
5252:
5140:
5114:
5110:. Said differently,
5090:
5045:{\displaystyle f=0.}
5030:
5010:
4990:
4924:
4899:
4884:{\displaystyle f(U)}
4866:
4840:
4820:
4793:
4769:
4741:
4719:
4674:
4631:
4589:
4524:
4491:
4463:
4437:
4395:
4387:is a sufficient but
4330:
4303:
4272:
4206:
4164:
4137:
4090:
4070:
4050:
4022:
4002:
3976:
3949:
3929:
3906:
3882:
3858:
3834:
3814:
3792:
3766:
3728:
3700:
3678:
3654:
3608:
3588:
3565:
3545:
3534:{\displaystyle F(B)}
3516:
3496:
3476:
3444:
3424:
3389:
3375:locally convex space
3307:
3283:
3263:
3234:
3211:
3191:
3171:
3147:
3127:
3107:A TVS is said to be
3089:locally convex space
3057:locally convex space
3031:
3007:
2981:
2961:
2917:
2798:
2778:
2743:
2728:{\displaystyle F(U)}
2710:
2690:
2670:
2647:
2614:
2574:
2481:
2458:
2435:
2397:
2365:
2338:
2318:
2307:{\displaystyle F(B)}
2289:
2263:
2208:
2142:
2109:
2071:
2021:
1995:
1945:
1880:
1854:
1798:
1778:
1774:is bounded on a set
1758:
1698:
1660:
1637:
1622:{\displaystyle F(S)}
1604:
1581:
1545:
1519:
1488:
1442:
1422:
1392:
1367:
1345:
1290:
1263:
1197:
1160:
1140:
1116:
1096:
1076:
972:
946:
920:
890:
862:
842:
816:
796:
765:
741:
721:
697:
666:
638:
607:
574:
547:
488:
461:
434:
414:
376:
356:
336:
313:
293:
262:
236:
216:
186:
163:
141:
98:
12139:Functional analysis
11968:Interpolation space
11500:Operator topologies
11309:Functional calculus
11268:Mahler's conjecture
11247:Von Neumann algebra
10961:Functional analysis
10569:
10307:measurable function
10257:Functional calculus
10120:Parseval's identity
10033:Bessel's inequality
9980:Polar decomposition
9759:Uniform convergence
9517:Inner product space
9319:Schaefer, Helmut H.
9299:Functional analysis
8937:, pp. 225–273.
8925:, pp. 451–457.
8870:, pp. 441–457.
8813:, pp. 156–175.
8780:, pp. 126–128.
7317:into bounded sets.
6725:and if in addition
6426:The imaginary part
6080:{\displaystyle f=0}
5824:given in (4) above.
4920:that is, such that
4737:(rather than strict
3440:into some TVS then
2570:In contrast, a map
2194:Bounded linear maps
977: for all
29:functional analysis
11998:(Pseudo)Metrizable
11830:Minkowski addition
11682:Sublinear function
11334:Riemann hypothesis
11033:Topological vector
10919:Validated numerics
10874:
10830:Sobolev inequality
10798:
10747:
10708:
10671:
10600:Bounded variation
10581:
10549:
10534:Banach coordinate
10515:
10453:Minkowski addition
10115:M. Riesz extension
9595:Banach spaces are:
8749:Unbounded operator
8720:Linear functionals
8659:
8607:
8555:
8506:
8458:
8438:
8366:
8340:
8248:
8228:
8202:
8084:
8040:
8016:
7983:
7846:
7820:
7784:
7755:
7722:{\displaystyle X,}
7719:
7696:
7676:
7656:
7632:
7609:
7568:{\displaystyle X,}
7565:
7542:
7518:
7494:{\displaystyle F.}
7491:
7464:
7435:
7415:
7395:
7281:
7257:
7237:
7217:
7190:
7163:
7143:
7090:
7064:
7039:{\displaystyle f,}
7036:
7013:
6989:
6939:{\displaystyle r,}
6936:
6910:
6860:{\displaystyle r,}
6857:
6832:
6803:
6783:
6759:
6735:
6711:
6685:
6659:
6632:
6616:bornological space
6604:
6576:
6541:
6513:
6486:
6462:
6442:
6412:
6392:
6372:{\displaystyle f.}
6369:
6342:
6316:
6288:
6263:
6227:
6205:
6166:
6146:
6127:{\displaystyle X.}
6124:
6097:
6077:
6052:{\displaystyle X.}
6049:
6026:
5999:
5971:
5935:
5906:
5877:{\displaystyle U,}
5874:
5847:
5810:
5790:
5749:
5720:
5691:
5654:
5625:
5593:
5569:
5543:
5517:
5516:
5484:
5444:
5439:
5399:
5370:
5369:
5346:
5310:{\displaystyle sU}
5307:
5284:
5258:
5238:
5215:
5161:
5120:
5096:
5042:
5016:
4996:
4976:
4975:
4943:
4910:
4881:
4852:
4826:
4799:
4775:
4749:
4727:
4704:{\displaystyle U=}
4701:
4660:
4617:
4571:
4542:
4510:
4477:
4449:
4423:
4377:
4348:
4316:
4285:
4256:
4224:
4192:
4150:
4121:
4076:
4056:
4043:there exists some
4033:
4008:
3988:
3962:
3935:
3912:
3894:{\displaystyle X.}
3891:
3864:
3846:{\displaystyle X.}
3843:
3820:
3798:
3778:{\displaystyle X.}
3775:
3748:
3706:
3686:
3660:
3634:Sublinear function
3614:
3594:
3577:{\displaystyle X,}
3574:
3551:
3531:
3502:
3482:
3462:
3430:
3407:
3371:bornological space
3313:
3289:
3269:
3246:
3217:
3197:
3177:
3153:
3133:
3093:(pseudo)metrizable
3037:
3025:seminormable space
3013:
2993:{\displaystyle X,}
2990:
2967:
2935:
2804:
2784:
2755:{\displaystyle Y.}
2752:
2725:
2696:
2676:
2663:if there exists a
2653:
2640:locally bounded at
2626:
2592:
2542:
2517:
2464:
2447:{\displaystyle Y;}
2444:
2421:
2383:
2351:
2324:
2304:
2275:
2226:
2181:
2127:
2095:
2057:
2007:
1981:
1931:
1866:
1840:
1784:
1764:
1744:
1716:
1684:
1649:{\displaystyle Y,}
1646:
1619:
1587:
1563:
1531:
1494:
1474:
1460:
1428:
1408:
1375:
1353:
1327:
1308:
1269:
1215:
1175:
1146:
1122:
1102:
1082:
1059:
958:
942:there exists some
932:
902:
868:
848:
828:{\displaystyle Y.}
825:
802:
783:seminormable space
771:
747:
727:
703:
672:
644:
613:
590:
560:
529:
467:
440:
420:
397:
362:
342:
325:{\displaystyle Y,}
322:
299:
268:
248:{\displaystyle X.}
245:
222:
201:
169:
147:
116:
12126:
12125:
11845:Relative interior
11591:Bilinear operator
11475:Linear functional
11411:
11410:
11314:Integral operator
11091:
11090:
10927:
10926:
10639:Morrey–Campanato
10621:compact Hausdorff
10468:Relative interior
10322:Absolutely convex
10289:Projection-valued
9898:Strictly singular
9824:on Hilbert spaces
9585:of Hilbert spaces
9423:978-0-486-49353-4
9393:978-0-486-45352-1
9363:978-0-8247-8643-4
9336:978-1-4612-7155-0
9310:978-0-07-054236-5
9241:978-3-642-64988-2
9211:978-3-519-02224-4
9184:978-0-677-30020-7
9150:978-0-486-68143-6
9096:978-0-387-97245-9
9041:Bourbaki, Nicolas
9024:978-0-387-90081-0
8997:978-3-540-08662-8
8901:, pp. 53–55.
8882:, pp. 54–55.
8849:, pp. 47–50.
8461:{\displaystyle r}
8301:
8295:
8258:is a scalar then
8251:{\displaystyle s}
8187:
8186:
8180:
8134:
8128:
8001:
7912:
7706:is a seminorm on
7699:{\displaystyle p}
7679:{\displaystyle X}
7659:{\displaystyle f}
7635:{\displaystyle X}
7545:{\displaystyle f}
7521:{\displaystyle X}
7438:{\displaystyle Y}
7418:{\displaystyle D}
7284:{\displaystyle X}
7260:{\displaystyle X}
7240:{\displaystyle X}
7220:{\displaystyle X}
7205:linear functional
7193:{\displaystyle X}
7166:{\displaystyle X}
7016:{\displaystyle X}
6806:{\displaystyle X}
6786:{\displaystyle p}
6762:{\displaystyle f}
6738:{\displaystyle X}
6714:{\displaystyle f}
6688:{\displaystyle f}
6662:{\displaystyle f}
6635:{\displaystyle Y}
6607:{\displaystyle X}
6579:{\displaystyle f}
6544:{\displaystyle X}
6516:{\displaystyle f}
6489:{\displaystyle X}
6465:{\displaystyle f}
6415:{\displaystyle Y}
6395:{\displaystyle X}
6291:{\displaystyle f}
6230:{\displaystyle f}
6169:{\displaystyle X}
6149:{\displaystyle p}
6100:{\displaystyle f}
6029:{\displaystyle f}
6002:{\displaystyle f}
5891:
5813:{\displaystyle U}
5673:
5610:
5596:{\displaystyle U}
5462:
5438:
5331:
5261:{\displaystyle s}
5200:
5143:
5123:{\displaystyle f}
5099:{\displaystyle f}
5019:{\displaystyle 0}
4999:{\displaystyle U}
4928:
4829:{\displaystyle U}
4802:{\displaystyle f}
4778:{\displaystyle f}
4527:
4333:
4209:
4079:{\displaystyle X}
4066:of the origin in
4059:{\displaystyle U}
4011:{\displaystyle 0}
3938:{\displaystyle f}
3915:{\displaystyle f}
3867:{\displaystyle f}
3823:{\displaystyle f}
3801:{\displaystyle f}
3760:linear functional
3709:{\displaystyle X}
3663:{\displaystyle X}
3617:{\displaystyle B}
3597:{\displaystyle F}
3554:{\displaystyle F}
3541:is bounded since
3505:{\displaystyle X}
3485:{\displaystyle B}
3433:{\displaystyle X}
3316:{\displaystyle Y}
3292:{\displaystyle Y}
3272:{\displaystyle Y}
3229:identity function
3220:{\displaystyle X}
3200:{\displaystyle X}
3180:{\displaystyle X}
3156:{\displaystyle B}
3136:{\displaystyle B}
3040:{\displaystyle X}
3016:{\displaystyle X}
2849:A linear map is "
2807:{\displaystyle F}
2787:{\displaystyle x}
2699:{\displaystyle X}
2686:of this point in
2679:{\displaystyle U}
2656:{\displaystyle x}
2566:Local boundedness
2552:is finite. Every
2496:
2467:{\displaystyle Y}
2327:{\displaystyle X}
1787:{\displaystyle S}
1767:{\displaystyle F}
1701:
1590:{\displaystyle S}
1497:{\displaystyle S}
1445:
1431:{\displaystyle S}
1293:
1272:{\displaystyle S}
1149:{\displaystyle F}
1125:{\displaystyle Y}
1105:{\displaystyle Y}
1085:{\displaystyle X}
1027:
1001:
978:
884:seminormed spaces
871:{\displaystyle Y}
851:{\displaystyle X}
805:{\displaystyle F}
774:{\displaystyle Y}
750:{\displaystyle Y}
730:{\displaystyle X}
706:{\displaystyle F}
675:{\displaystyle X}
647:{\displaystyle F}
616:{\displaystyle X}
479:weakly continuous
470:{\displaystyle F}
443:{\displaystyle Y}
423:{\displaystyle X}
365:{\displaystyle X}
345:{\displaystyle p}
302:{\displaystyle q}
271:{\displaystyle Y}
225:{\displaystyle F}
172:{\displaystyle F}
150:{\displaystyle F}
16:(Redirected from
12161:
12144:Linear operators
12116:
12115:
12090:Uniformly smooth
11759:
11751:
11718:Balanced/Circled
11708:Absorbing/Radial
11438:
11431:
11424:
11415:
11414:
11401:
11400:
11319:Jones polynomial
11237:Operator algebra
10981:
10980:
10954:
10947:
10940:
10931:
10930:
10883:
10881:
10880:
10875:
10870:
10869:
10837:Triebel–Lizorkin
10807:
10805:
10804:
10799:
10797:
10793:
10792:
10787:
10756:
10754:
10753:
10748:
10746:
10745:
10717:
10715:
10714:
10709:
10707:
10706:
10680:
10678:
10677:
10672:
10661:
10660:
10590:
10588:
10587:
10582:
10577:
10568:
10563:
10524:
10522:
10521:
10516:
10377:
10355:
10337:Balanced/Circled
10135:Robinson-Ursescu
10053:Eberlein–Šmulian
9973:Spectral theorem
9769:Linear operators
9566:Uniformly smooth
9463:
9456:
9449:
9440:
9439:
9435:
9410:Wilansky, Albert
9405:
9380:Trèves, François
9375:
9348:
9314:
9302:
9295:(January 1991).
9288:
9261:
9228:Köthe, Gottfried
9223:
9196:
9176:
9162:
9135:
9114:Linear operators
9108:
9070:
9036:
9009:
8974:
8968:
8962:
8956:
8950:
8944:
8938:
8932:
8926:
8920:
8914:
8908:
8902:
8896:
8883:
8877:
8871:
8865:
8850:
8844:
8838:
8832:
8826:
8820:
8814:
8808:
8781:
8775:
8740:
8725:
8716:
8688:Compact operator
8684:
8668:
8666:
8665:
8660:
8655:
8654:
8616:
8614:
8613:
8608:
8600:
8580:
8564:
8562:
8561:
8556:
8548:
8531:
8515:
8513:
8512:
8507:
8505:
8504:
8467:
8465:
8464:
8459:
8447:
8445:
8444:
8439:
8428:
8420:
8412:
8395:
8394:
8375:
8373:
8372:
8367:
8349:
8347:
8346:
8341:
8339:
8322:
8314:
8306:
8299:
8293:
8292:
8272:
8257:
8255:
8254:
8249:
8237:
8235:
8234:
8229:
8224:
8207:
8201:
8184:
8178:
8162:
8145:
8132:
8126:
8125:
8108:
8093:
8091:
8090:
8085:
8082:
8065:
8049:
8047:
8046:
8041:
8038:
8021:
8015:
7992:
7990:
7989:
7984:
7964:
7947:
7936:
7919:
7913:
7910:
7855:
7853:
7852:
7847:
7829:
7827:
7826:
7821:
7819:
7793:
7791:
7790:
7785:
7764:
7762:
7761:
7756:
7748:
7740:
7728:
7726:
7725:
7720:
7705:
7703:
7702:
7697:
7685:
7683:
7682:
7677:
7665:
7663:
7662:
7657:
7641:
7639:
7638:
7633:
7618:
7616:
7615:
7610:
7574:
7572:
7571:
7566:
7551:
7549:
7548:
7543:
7527:
7525:
7524:
7519:
7500:
7498:
7497:
7492:
7473:
7471:
7470:
7465:
7444:
7442:
7441:
7436:
7424:
7422:
7421:
7416:
7404:
7402:
7401:
7396:
7370:
7369:
7339:
7338:
7290:
7288:
7287:
7282:
7266:
7264:
7263:
7258:
7246:
7244:
7243:
7238:
7226:
7224:
7223:
7218:
7199:
7197:
7196:
7191:
7172:
7170:
7169:
7164:
7152:
7150:
7149:
7144:
7099:
7097:
7096:
7091:
7073:
7071:
7070:
7065:
7045:
7043:
7042:
7037:
7022:
7020:
7019:
7014:
6998:
6996:
6995:
6990:
6945:
6943:
6942:
6937:
6919:
6917:
6916:
6911:
6866:
6864:
6863:
6858:
6841:
6839:
6838:
6833:
6812:
6810:
6809:
6804:
6792:
6790:
6789:
6784:
6768:
6766:
6765:
6760:
6744:
6742:
6741:
6736:
6720:
6718:
6717:
6712:
6694:
6692:
6691:
6686:
6668:
6666:
6665:
6660:
6641:
6639:
6638:
6633:
6618:(for example, a
6613:
6611:
6610:
6605:
6585:
6583:
6582:
6577:
6555:(for example, a
6550:
6548:
6547:
6542:
6522:
6520:
6519:
6514:
6498:sequential space
6495:
6493:
6492:
6487:
6471:
6469:
6468:
6463:
6451:
6449:
6448:
6443:
6421:
6419:
6418:
6413:
6401:
6399:
6398:
6393:
6378:
6376:
6375:
6370:
6351:
6349:
6348:
6343:
6325:
6323:
6322:
6317:
6297:
6295:
6294:
6289:
6273:is a continuous.
6272:
6270:
6269:
6264:
6262:
6254:
6236:
6234:
6233:
6228:
6214:
6212:
6211:
6206:
6195:
6187:
6175:
6173:
6172:
6167:
6155:
6153:
6152:
6147:
6133:
6131:
6130:
6125:
6106:
6104:
6103:
6098:
6086:
6084:
6083:
6078:
6058:
6056:
6055:
6050:
6035:
6033:
6032:
6027:
6008:
6006:
6005:
6000:
5980:
5978:
5977:
5972:
5967:
5966:
5944:
5942:
5941:
5936:
5928:
5911:
5905:
5883:
5881:
5880:
5875:
5859:(absolute) polar
5856:
5854:
5853:
5848:
5843:
5842:
5819:
5817:
5816:
5811:
5799:
5797:
5796:
5791:
5758:
5756:
5755:
5750:
5729:
5727:
5726:
5721:
5713:
5696:
5690:
5663:
5661:
5660:
5655:
5647:
5630:
5624:
5602:
5600:
5599:
5594:
5578:
5576:
5575:
5570:
5552:
5550:
5549:
5544:
5526:
5524:
5523:
5518:
5506:
5489:
5483:
5482:
5481:
5453:
5451:
5450:
5445:
5440:
5431:
5425:
5424:
5408:
5406:
5405:
5400:
5379:
5377:
5376:
5371:
5368:
5351:
5345:
5316:
5314:
5313:
5308:
5293:
5291:
5290:
5285:
5267:
5265:
5264:
5259:
5247:
5245:
5244:
5239:
5237:
5220:
5214:
5199:
5191:
5183:
5166:
5160:
5129:
5127:
5126:
5121:
5105:
5103:
5102:
5097:
5080:seminormed space
5051:
5049:
5048:
5043:
5025:
5023:
5022:
5017:
5005:
5003:
5002:
4997:
4985:
4983:
4982:
4977:
4965:
4948:
4942:
4919:
4917:
4916:
4911:
4906:
4890:
4888:
4887:
4882:
4861:
4859:
4858:
4853:
4835:
4833:
4832:
4827:
4808:
4806:
4805:
4800:
4784:
4782:
4781:
4776:
4758:
4756:
4755:
4750:
4736:
4734:
4733:
4728:
4710:
4708:
4707:
4702:
4669:
4667:
4666:
4661:
4644:
4626:
4624:
4623:
4618:
4616:
4615:
4580:
4578:
4577:
4572:
4564:
4547:
4541:
4519:
4517:
4516:
4511:
4509:
4508:
4486:
4484:
4483:
4478:
4476:
4458:
4456:
4455:
4450:
4432:
4430:
4429:
4424:
4422:
4421:
4386:
4384:
4383:
4378:
4370:
4353:
4347:
4325:
4323:
4322:
4317:
4315:
4314:
4294:
4292:
4291:
4286:
4284:
4283:
4265:
4263:
4262:
4257:
4246:
4229:
4223:
4201:
4199:
4198:
4193:
4191:
4190:
4159:
4157:
4156:
4151:
4149:
4148:
4130:
4128:
4127:
4122:
4117:
4116:
4085:
4083:
4082:
4077:
4065:
4063:
4062:
4057:
4042:
4040:
4039:
4034:
4029:
4018:in the codomain
4017:
4015:
4014:
4009:
3997:
3995:
3994:
3989:
3971:
3969:
3968:
3963:
3961:
3960:
3944:
3942:
3941:
3936:
3921:
3919:
3918:
3913:
3900:
3898:
3897:
3892:
3873:
3871:
3870:
3865:
3852:
3850:
3849:
3844:
3829:
3827:
3826:
3821:
3807:
3805:
3804:
3799:
3784:
3782:
3781:
3776:
3757:
3755:
3754:
3749:
3747:
3715:
3713:
3712:
3707:
3695:
3693:
3692:
3687:
3685:
3669:
3667:
3666:
3661:
3623:
3621:
3620:
3615:
3603:
3601:
3600:
3595:
3583:
3581:
3580:
3575:
3560:
3558:
3557:
3552:
3540:
3538:
3537:
3532:
3511:
3509:
3508:
3503:
3491:
3489:
3488:
3483:
3471:
3469:
3468:
3463:
3439:
3437:
3436:
3431:
3416:
3414:
3413:
3408:
3359:pseudometrizable
3354:be continuous.
3322:
3320:
3319:
3314:
3298:
3296:
3295:
3290:
3278:
3276:
3275:
3270:
3255:
3253:
3252:
3247:
3226:
3224:
3223:
3218:
3206:
3204:
3203:
3198:
3186:
3184:
3183:
3178:
3162:
3160:
3159:
3154:
3142:
3140:
3139:
3134:
3121:seminormed space
3046:
3044:
3043:
3038:
3022:
3020:
3019:
3014:
3002:is equivalent to
2999:
2997:
2996:
2991:
2976:
2974:
2973:
2968:
2944:
2942:
2941:
2936:
2861:) and thus also
2825:
2824:
2813:
2811:
2810:
2805:
2793:
2791:
2790:
2785:
2768:
2767:
2761:
2759:
2758:
2753:
2734:
2732:
2731:
2726:
2705:
2703:
2702:
2697:
2685:
2683:
2682:
2677:
2662:
2660:
2659:
2654:
2642:
2641:
2635:
2633:
2632:
2627:
2608:
2607:
2601:
2599:
2598:
2593:
2551:
2549:
2548:
2543:
2516:
2473:
2471:
2470:
2465:
2453:
2451:
2450:
2445:
2430:
2428:
2427:
2422:
2420:
2416:
2415:
2392:
2390:
2389:
2384:
2360:
2358:
2357:
2352:
2350:
2349:
2333:
2331:
2330:
2325:
2313:
2311:
2310:
2305:
2284:
2282:
2281:
2276:
2254:
2253:
2246:and is called a
2235:
2233:
2232:
2227:
2190:
2188:
2187:
2182:
2136:
2134:
2133:
2128:
2104:
2102:
2101:
2096:
2066:
2064:
2063:
2058:
2016:
2014:
2013:
2008:
1990:
1988:
1987:
1982:
1940:
1938:
1937:
1932:
1875:
1873:
1872:
1867:
1849:
1847:
1846:
1841:
1793:
1791:
1790:
1785:
1773:
1771:
1770:
1765:
1753:
1751:
1750:
1745:
1715:
1693:
1691:
1690:
1685:
1655:
1653:
1652:
1647:
1628:
1626:
1625:
1620:
1598:
1597:
1596:
1594:
1593:
1588:
1572:
1570:
1569:
1564:
1540:
1538:
1537:
1532:
1503:
1501:
1500:
1495:
1483:
1481:
1480:
1475:
1473:
1465:
1459:
1437:
1435:
1434:
1429:
1417:
1415:
1414:
1409:
1407:
1399:
1384:
1382:
1381:
1376:
1374:
1362:
1360:
1359:
1354:
1352:
1336:
1334:
1333:
1328:
1307:
1278:
1276:
1275:
1270:
1259:) then a subset
1257:seminormed space
1224:
1222:
1221:
1216:
1184:
1182:
1181:
1176:
1155:
1153:
1152:
1147:
1131:
1129:
1128:
1123:
1111:
1109:
1108:
1103:
1091:
1089:
1088:
1083:
1068:
1066:
1065:
1060:
1028:
1026: then
1025:
1002:
999:
979:
976:
967:
965:
964:
959:
941:
939:
938:
933:
911:
909:
908:
903:
877:
875:
874:
869:
857:
855:
854:
849:
834:
832:
831:
826:
811:
809:
808:
803:
780:
778:
777:
772:
756:
754:
753:
748:
736:
734:
733:
728:
712:
710:
709:
704:
684:pseudometrizable
681:
679:
678:
673:
653:
651:
650:
645:
625:sequential space
622:
620:
619:
614:
599:
597:
596:
591:
586:
585:
569:
567:
566:
561:
559:
558:
538:
536:
535:
530:
528:
527:
515:
514:
499:
498:
493:
476:
474:
473:
468:
449:
447:
446:
441:
429:
427:
426:
421:
406:
404:
403:
398:
371:
369:
368:
363:
351:
349:
348:
343:
331:
329:
328:
323:
308:
306:
305:
300:
277:
275:
274:
269:
254:
252:
251:
246:
231:
229:
228:
223:
210:
208:
207:
202:
178:
176:
175:
170:
156:
154:
153:
148:
125:
123:
122:
117:
90:Bounded operator
21:
12169:
12168:
12164:
12163:
12162:
12160:
12159:
12158:
12149:Operator theory
12129:
12128:
12127:
12122:
12104:
11866:B-complete/Ptak
11849:
11793:
11757:
11749:
11728:Bounding points
11691:
11633:Densely defined
11579:
11568:Bounded inverse
11514:
11448:
11442:
11412:
11407:
11389:
11353:Advanced topics
11348:
11272:
11251:
11210:
11176:Hilbert–Schmidt
11149:
11140:Gelfand–Naimark
11087:
11037:
10972:
10958:
10928:
10923:
10887:
10865:
10861:
10847:
10844:
10843:
10842:Wiener amalgam
10812:Segal–Bargmann
10788:
10783:
10782:
10778:
10773:
10770:
10769:
10741:
10737:
10735:
10732:
10731:
10702:
10698:
10696:
10693:
10692:
10650:
10646:
10644:
10641:
10640:
10595:Birnbaum–Orlicz
10573:
10564:
10553:
10547:
10544:
10543:
10498:
10495:
10494:
10472:
10428:Bounding points
10401:
10375:
10353:
10310:
10161:Banach manifold
10144:
10068:Gelfand–Naimark
9989:
9963:Spectral theory
9931:Banach algebras
9923:Operator theory
9917:
9878:Pseudo-monotone
9861:Hilbert–Schmidt
9841:Densely defined
9763:
9676:
9590:
9473:
9467:
9424:
9394:
9364:
9337:
9311:
9277:
9242:
9212:
9185:
9151:
9124:
9097:
9087:Springer-Verlag
9059:
9025:
8998:
8988:Springer-Verlag
8978:
8977:
8969:
8965:
8957:
8953:
8945:
8941:
8933:
8929:
8921:
8917:
8909:
8905:
8897:
8886:
8878:
8874:
8866:
8853:
8845:
8841:
8833:
8829:
8821:
8817:
8809:
8784:
8776:
8763:
8758:
8738:
8723:
8714:
8682:
8675:
8647:
8643:
8623:
8620:
8619:
8596:
8576:
8571:
8568:
8567:
8544:
8527:
8522:
8519:
8518:
8497:
8493:
8476:
8473:
8472:
8453:
8450:
8449:
8424:
8416:
8408:
8387:
8383:
8381:
8378:
8377:
8355:
8352:
8351:
8335:
8318:
8310:
8302:
8288:
8268:
8263:
8260:
8259:
8243:
8240:
8239:
8220:
8203:
8191:
8158:
8141:
8121:
8104:
8099:
8096:
8095:
8078:
8061:
8055:
8052:
8051:
8034:
8017:
8005:
7998:
7995:
7994:
7960:
7943:
7932:
7915:
7911: and
7909:
7861:
7858:
7857:
7835:
7832:
7831:
7815:
7801:
7798:
7797:
7770:
7767:
7766:
7765:if and only if
7744:
7736:
7734:
7731:
7730:
7711:
7708:
7707:
7691:
7688:
7687:
7671:
7668:
7667:
7651:
7648:
7647:
7627:
7624:
7623:
7580:
7577:
7576:
7557:
7554:
7553:
7537:
7534:
7533:
7513:
7510:
7509:
7506:
7483:
7480:
7479:
7450:
7447:
7446:
7430:
7427:
7426:
7410:
7407:
7406:
7405:for any subset
7362:
7358:
7331:
7327:
7325:
7322:
7321:
7297:
7276:
7273:
7272:
7252:
7249:
7248:
7232:
7229:
7228:
7212:
7209:
7208:
7185:
7182:
7181:
7158:
7155:
7154:
7132:
7129:
7128:
7118:
7104:(respectively,
7079:
7076:
7075:
7050:
7047:
7046:
7028:
7025:
7024:
7008:
7005:
7004:
6951:
6948:
6947:
6946:the half-space
6928:
6925:
6924:
6872:
6869:
6868:
6867:the half-space
6849:
6846:
6845:
6818:
6815:
6814:
6798:
6795:
6794:
6778:
6775:
6774:
6754:
6751:
6750:
6730:
6727:
6726:
6706:
6703:
6702:
6680:
6677:
6676:
6654:
6651:
6650:
6627:
6624:
6623:
6599:
6596:
6595:
6571:
6568:
6567:
6536:
6533:
6532:
6508:
6505:
6504:
6481:
6478:
6477:
6457:
6454:
6453:
6431:
6428:
6427:
6407:
6404:
6403:
6387:
6384:
6383:
6361:
6358:
6357:
6331:
6328:
6327:
6305:
6302:
6301:
6283:
6280:
6279:
6258:
6250:
6242:
6239:
6238:
6222:
6219:
6218:
6217:In particular,
6191:
6183:
6181:
6178:
6177:
6161:
6158:
6157:
6141:
6138:
6137:
6116:
6113:
6112:
6092:
6089:
6088:
6066:
6063:
6062:
6041:
6038:
6037:
6021:
6018:
6017:
5994:
5991:
5990:
5962:
5958:
5950:
5947:
5946:
5924:
5907:
5895:
5889:
5886:
5885:
5884:the inequality
5866:
5863:
5862:
5838:
5834:
5832:
5829:
5828:
5805:
5802:
5801:
5764:
5761:
5760:
5735:
5732:
5731:
5730:for every real
5709:
5692:
5677:
5671:
5668:
5667:
5643:
5626:
5614:
5608:
5605:
5604:
5588:
5585:
5584:
5558:
5555:
5554:
5532:
5529:
5528:
5502:
5485:
5477:
5473:
5466:
5459:
5456:
5455:
5429:
5420:
5416:
5414:
5411:
5410:
5385:
5382:
5381:
5364:
5347:
5335:
5322:
5319:
5318:
5299:
5296:
5295:
5273:
5270:
5269:
5253:
5250:
5249:
5233:
5216:
5204:
5195:
5187:
5179:
5162:
5147:
5141:
5138:
5137:
5115:
5112:
5111:
5091:
5088:
5087:
5031:
5028:
5027:
5026:if and only if
5011:
5008:
5007:
4991:
4988:
4987:
4961:
4944:
4932:
4925:
4922:
4921:
4902:
4900:
4897:
4896:
4867:
4864:
4863:
4841:
4838:
4837:
4821:
4818:
4817:
4813:of its domain.
4794:
4791:
4790:
4770:
4767:
4766:
4759:) inequalities.
4742:
4739:
4738:
4720:
4717:
4716:
4675:
4672:
4671:
4640:
4632:
4629:
4628:
4611:
4607:
4590:
4587:
4586:
4560:
4543:
4531:
4525:
4522:
4521:
4504:
4500:
4492:
4489:
4488:
4472:
4464:
4461:
4460:
4438:
4435:
4434:
4417:
4413:
4396:
4393:
4392:
4366:
4349:
4337:
4331:
4328:
4327:
4310:
4306:
4304:
4301:
4300:
4279:
4275:
4273:
4270:
4269:
4242:
4225:
4213:
4207:
4204:
4203:
4186:
4182:
4165:
4162:
4161:
4144:
4140:
4138:
4135:
4134:
4112:
4108:
4091:
4088:
4087:
4071:
4068:
4067:
4051:
4048:
4047:
4025:
4023:
4020:
4019:
4003:
4000:
3999:
3977:
3974:
3973:
3956:
3952:
3950:
3947:
3946:
3930:
3927:
3926:
3925:By definition,
3907:
3904:
3903:
3883:
3880:
3879:
3859:
3856:
3855:
3835:
3832:
3831:
3815:
3812:
3811:
3793:
3790:
3789:
3767:
3764:
3763:
3743:
3729:
3726:
3725:
3701:
3698:
3697:
3681:
3679:
3676:
3675:
3655:
3652:
3651:
3648:
3636:
3630:
3609:
3606:
3605:
3589:
3586:
3585:
3566:
3563:
3562:
3546:
3543:
3542:
3517:
3514:
3513:
3497:
3494:
3493:
3477:
3474:
3473:
3445:
3442:
3441:
3425:
3422:
3421:
3390:
3387:
3386:
3308:
3305:
3304:
3299:is necessarily
3284:
3281:
3280:
3264:
3261:
3260:
3235:
3232:
3231:
3212:
3209:
3208:
3192:
3189:
3188:
3172:
3169:
3168:
3148:
3145:
3144:
3128:
3125:
3124:
3109:locally bounded
3069:
3032:
3029:
3028:
3008:
3005:
3004:
2982:
2979:
2978:
2962:
2959:
2958:
2951:TVS-isomorphism
2918:
2915:
2914:
2890:
2847:
2823:locally bounded
2822:
2821:
2799:
2796:
2795:
2779:
2776:
2775:
2765:
2764:
2744:
2741:
2740:
2711:
2708:
2707:
2691:
2688:
2687:
2671:
2668:
2667:
2648:
2645:
2644:
2639:
2638:
2615:
2612:
2611:
2605:
2604:
2575:
2572:
2571:
2568:
2500:
2482:
2479:
2478:
2459:
2456:
2455:
2436:
2433:
2432:
2411:
2407:
2403:
2398:
2395:
2394:
2366:
2363:
2362:
2345:
2341:
2339:
2336:
2335:
2319:
2316:
2315:
2290:
2287:
2286:
2285:of its domain,
2264:
2261:
2260:
2249:
2248:
2209:
2206:
2205:
2202:
2143:
2140:
2139:
2110:
2107:
2106:
2072:
2069:
2068:
2022:
2019:
2018:
1996:
1993:
1992:
1946:
1943:
1942:
1881:
1878:
1877:
1855:
1852:
1851:
1799:
1796:
1795:
1779:
1776:
1775:
1759:
1756:
1755:
1705:
1699:
1696:
1695:
1661:
1658:
1657:
1638:
1635:
1634:
1605:
1602:
1601:
1582:
1579:
1578:
1576:
1575:
1546:
1543:
1542:
1520:
1517:
1516:
1489:
1486:
1485:
1469:
1461:
1449:
1443:
1440:
1439:
1423:
1420:
1419:
1403:
1395:
1393:
1390:
1389:
1370:
1368:
1365:
1364:
1348:
1346:
1343:
1342:
1297:
1291:
1288:
1287:
1286:, meaning that
1264:
1261:
1260:
1245:
1198:
1195:
1194:
1191:
1161:
1158:
1157:
1141:
1138:
1137:
1117:
1114:
1113:
1097:
1094:
1093:
1077:
1074:
1073:
1024:
998:
975:
973:
970:
969:
947:
944:
943:
921:
918:
917:
891:
888:
887:
863:
860:
859:
843:
840:
839:
817:
814:
813:
797:
794:
793:
766:
763:
762:
742:
739:
738:
722:
719:
718:
698:
695:
694:
667:
664:
663:
639:
636:
635:
608:
605:
604:
581:
577:
575:
572:
571:
554:
550:
548:
545:
544:
523:
519:
510:
506:
494:
492:
491:
489:
486:
485:
462:
459:
458:
435:
432:
431:
415:
412:
411:
377:
374:
373:
357:
354:
353:
337:
334:
333:
314:
311:
310:
294:
291:
290:
263:
260:
259:
237:
234:
233:
217:
214:
213:
187:
184:
183:
164:
161:
160:
142:
139:
138:
128:linear operator
99:
96:
95:
92:
86:
81:
71:
23:
22:
15:
12:
11:
5:
12167:
12157:
12156:
12151:
12146:
12141:
12124:
12123:
12121:
12120:
12109:
12106:
12105:
12103:
12102:
12097:
12092:
12087:
12085:Ultrabarrelled
12077:
12071:
12066:
12060:
12055:
12050:
12045:
12040:
12035:
12026:
12020:
12015:
12013:Quasi-complete
12010:
12008:Quasibarrelled
12005:
12000:
11995:
11990:
11985:
11980:
11975:
11970:
11965:
11960:
11955:
11950:
11949:
11948:
11938:
11933:
11928:
11923:
11918:
11913:
11908:
11903:
11898:
11888:
11883:
11873:
11868:
11863:
11857:
11855:
11851:
11850:
11848:
11847:
11837:
11832:
11827:
11822:
11817:
11807:
11801:
11799:
11798:Set operations
11795:
11794:
11792:
11791:
11786:
11781:
11776:
11771:
11766:
11761:
11753:
11745:
11740:
11735:
11730:
11725:
11720:
11715:
11710:
11705:
11699:
11697:
11693:
11692:
11690:
11689:
11684:
11679:
11674:
11669:
11668:
11667:
11662:
11657:
11647:
11642:
11641:
11640:
11635:
11630:
11625:
11620:
11615:
11610:
11600:
11599:
11598:
11587:
11585:
11581:
11580:
11578:
11577:
11572:
11571:
11570:
11560:
11554:
11545:
11540:
11535:
11533:Banach–Alaoglu
11530:
11528:Anderson–Kadec
11524:
11522:
11516:
11515:
11513:
11512:
11507:
11502:
11497:
11492:
11487:
11482:
11477:
11472:
11467:
11462:
11456:
11454:
11453:Basic concepts
11450:
11449:
11441:
11440:
11433:
11426:
11418:
11409:
11408:
11406:
11405:
11394:
11391:
11390:
11388:
11387:
11382:
11377:
11372:
11370:Choquet theory
11367:
11362:
11356:
11354:
11350:
11349:
11347:
11346:
11336:
11331:
11326:
11321:
11316:
11311:
11306:
11301:
11296:
11291:
11286:
11280:
11278:
11274:
11273:
11271:
11270:
11265:
11259:
11257:
11253:
11252:
11250:
11249:
11244:
11239:
11234:
11229:
11224:
11222:Banach algebra
11218:
11216:
11212:
11211:
11209:
11208:
11203:
11198:
11193:
11188:
11183:
11178:
11173:
11168:
11163:
11157:
11155:
11151:
11150:
11148:
11147:
11145:Banach–Alaoglu
11142:
11137:
11132:
11127:
11122:
11117:
11112:
11107:
11101:
11099:
11093:
11092:
11089:
11088:
11086:
11085:
11080:
11075:
11073:Locally convex
11070:
11056:
11051:
11045:
11043:
11039:
11038:
11036:
11035:
11030:
11025:
11020:
11015:
11010:
11005:
11000:
10995:
10990:
10984:
10978:
10974:
10973:
10957:
10956:
10949:
10942:
10934:
10925:
10924:
10922:
10921:
10916:
10911:
10906:
10901:
10895:
10893:
10889:
10888:
10886:
10885:
10873:
10868:
10864:
10860:
10857:
10854:
10851:
10839:
10834:
10833:
10832:
10822:
10820:Sequence space
10817:
10809:
10796:
10791:
10786:
10781:
10777:
10765:
10764:
10763:
10758:
10744:
10740:
10721:
10720:
10719:
10705:
10701:
10682:
10670:
10667:
10664:
10659:
10656:
10653:
10649:
10636:
10628:
10623:
10610:
10605:
10597:
10592:
10580:
10576:
10572:
10567:
10562:
10559:
10556:
10552:
10539:
10531:
10526:
10514:
10511:
10508:
10505:
10502:
10491:
10482:
10480:
10474:
10473:
10471:
10470:
10460:
10455:
10450:
10445:
10440:
10435:
10430:
10425:
10415:
10409:
10407:
10403:
10402:
10400:
10399:
10394:
10389:
10384:
10379:
10371:
10357:
10349:
10344:
10339:
10334:
10329:
10324:
10318:
10316:
10312:
10311:
10309:
10308:
10298:
10297:
10296:
10291:
10286:
10276:
10275:
10274:
10269:
10264:
10254:
10253:
10252:
10247:
10242:
10237:
10235:Gelfand–Pettis
10232:
10227:
10217:
10216:
10215:
10210:
10205:
10200:
10195:
10185:
10180:
10175:
10170:
10169:
10168:
10158:
10152:
10150:
10146:
10145:
10143:
10142:
10137:
10132:
10127:
10122:
10117:
10112:
10107:
10102:
10097:
10092:
10087:
10086:
10085:
10075:
10070:
10065:
10060:
10055:
10050:
10045:
10040:
10035:
10030:
10025:
10020:
10015:
10010:
10008:Banach–Alaoglu
10005:
10003:Anderson–Kadec
9999:
9997:
9991:
9990:
9988:
9987:
9982:
9977:
9976:
9975:
9970:
9960:
9959:
9958:
9953:
9943:
9941:Operator space
9938:
9933:
9927:
9925:
9919:
9918:
9916:
9915:
9910:
9905:
9900:
9895:
9890:
9885:
9880:
9875:
9874:
9873:
9863:
9858:
9857:
9856:
9851:
9843:
9838:
9828:
9827:
9826:
9816:
9811:
9801:
9800:
9799:
9794:
9789:
9779:
9773:
9771:
9765:
9764:
9762:
9761:
9756:
9751:
9750:
9749:
9744:
9734:
9733:
9732:
9727:
9717:
9712:
9707:
9706:
9705:
9695:
9690:
9684:
9682:
9678:
9677:
9675:
9674:
9669:
9664:
9663:
9662:
9652:
9647:
9642:
9641:
9640:
9629:Locally convex
9626:
9625:
9624:
9614:
9609:
9604:
9598:
9596:
9592:
9591:
9589:
9588:
9581:Tensor product
9574:
9568:
9563:
9557:
9552:
9546:
9541:
9536:
9526:
9525:
9524:
9519:
9509:
9504:
9502:Banach lattice
9499:
9498:
9497:
9487:
9481:
9479:
9475:
9474:
9466:
9465:
9458:
9451:
9443:
9437:
9436:
9422:
9406:
9392:
9376:
9362:
9349:
9335:
9315:
9309:
9289:
9276:978-1584888666
9275:
9262:
9240:
9224:
9210:
9197:
9183:
9163:
9149:
9136:
9122:
9109:
9095:
9071:
9057:
9037:
9023:
9010:
8996:
8976:
8975:
8973:, p. 128.
8963:
8951:
8939:
8927:
8915:
8903:
8884:
8872:
8851:
8839:
8837:, p. 476.
8827:
8815:
8782:
8760:
8759:
8757:
8754:
8753:
8752:
8746:
8741:
8732:
8726:
8717:
8708:
8703:
8697:
8691:
8685:
8674:
8671:
8670:
8669:
8658:
8653:
8650:
8646:
8642:
8639:
8636:
8633:
8630:
8627:
8617:
8606:
8603:
8599:
8595:
8592:
8589:
8586:
8583:
8579:
8575:
8565:
8554:
8551:
8547:
8543:
8540:
8537:
8534:
8530:
8526:
8516:
8503:
8500:
8496:
8492:
8489:
8486:
8483:
8480:
8457:
8437:
8434:
8431:
8427:
8423:
8419:
8415:
8411:
8407:
8404:
8401:
8398:
8393:
8390:
8386:
8365:
8362:
8359:
8338:
8334:
8331:
8328:
8325:
8321:
8317:
8313:
8309:
8305:
8298:
8291:
8287:
8284:
8281:
8278:
8275:
8271:
8267:
8247:
8227:
8223:
8219:
8216:
8213:
8210:
8206:
8200:
8197:
8194:
8190:
8183:
8177:
8174:
8171:
8168:
8165:
8161:
8157:
8154:
8151:
8148:
8144:
8140:
8137:
8131:
8124:
8120:
8117:
8114:
8111:
8107:
8103:
8081:
8077:
8074:
8071:
8068:
8064:
8060:
8037:
8033:
8030:
8027:
8024:
8020:
8014:
8011:
8008:
8004:
7982:
7979:
7976:
7973:
7970:
7967:
7963:
7959:
7956:
7953:
7950:
7946:
7942:
7939:
7935:
7931:
7928:
7925:
7922:
7918:
7907:
7904:
7901:
7898:
7895:
7892:
7889:
7886:
7883:
7880:
7877:
7874:
7871:
7868:
7865:
7845:
7842:
7839:
7818:
7814:
7811:
7808:
7805:
7783:
7780:
7777:
7774:
7754:
7751:
7747:
7743:
7739:
7718:
7715:
7695:
7675:
7655:
7631:
7608:
7605:
7602:
7599:
7596:
7593:
7590:
7587:
7584:
7564:
7561:
7541:
7517:
7505:
7502:
7490:
7487:
7463:
7460:
7457:
7454:
7434:
7414:
7394:
7391:
7388:
7385:
7382:
7379:
7376:
7373:
7368:
7365:
7361:
7357:
7354:
7351:
7348:
7345:
7342:
7337:
7334:
7330:
7301:locally convex
7296:
7293:
7280:
7269:bounded subset
7256:
7236:
7216:
7203:
7189:
7162:
7142:
7139:
7136:
7117:
7114:
7089:
7086:
7083:
7063:
7060:
7057:
7054:
7035:
7032:
7012:
7001:
7000:
6988:
6985:
6982:
6979:
6976:
6973:
6970:
6967:
6964:
6961:
6958:
6955:
6935:
6932:
6921:
6909:
6906:
6903:
6900:
6897:
6894:
6891:
6888:
6885:
6882:
6879:
6876:
6856:
6853:
6844:For some real
6842:
6831:
6828:
6825:
6822:
6802:
6782:
6758:
6734:
6723:
6722:
6710:
6700:
6684:
6674:
6658:
6644:locally convex
6631:
6603:
6594:If the domain
6592:
6591:
6575:
6540:
6531:If the domain
6529:
6528:
6512:
6485:
6476:If the domain
6474:
6473:
6472:is continuous.
6461:
6441:
6438:
6435:
6411:
6391:
6380:
6379:
6368:
6365:
6341:
6338:
6335:
6315:
6312:
6309:
6299:
6287:
6276:
6275:
6274:
6261:
6257:
6253:
6249:
6246:
6226:
6204:
6201:
6198:
6194:
6190:
6186:
6165:
6145:
6134:
6123:
6120:
6110:
6096:
6076:
6073:
6070:
6059:
6048:
6045:
6025:
6016:The kernel of
6014:
6013:of its domain.
5998:
5988:
5987:
5986:
5983:duality theory
5970:
5965:
5961:
5957:
5954:
5934:
5931:
5927:
5923:
5920:
5917:
5914:
5910:
5904:
5901:
5898:
5894:
5873:
5870:
5846:
5841:
5837:
5825:
5809:
5789:
5786:
5783:
5780:
5777:
5774:
5771:
5768:
5748:
5745:
5742:
5739:
5719:
5716:
5712:
5708:
5705:
5702:
5699:
5695:
5689:
5686:
5683:
5680:
5676:
5653:
5650:
5646:
5642:
5639:
5636:
5633:
5629:
5623:
5620:
5617:
5613:
5592:
5581:
5580:
5579:
5568:
5565:
5562:
5542:
5539:
5536:
5515:
5512:
5509:
5505:
5501:
5498:
5495:
5492:
5488:
5480:
5476:
5472:
5469:
5465:
5443:
5437:
5434:
5428:
5423:
5419:
5398:
5395:
5392:
5389:
5367:
5363:
5360:
5357:
5354:
5350:
5344:
5341:
5338:
5334:
5329:
5326:
5306:
5303:
5283:
5280:
5277:
5257:
5236:
5232:
5229:
5226:
5223:
5219:
5213:
5210:
5207:
5203:
5198:
5194:
5190:
5186:
5182:
5178:
5175:
5172:
5169:
5165:
5159:
5156:
5153:
5150:
5146:
5119:
5095:
5085:
5084:
5083:
5069:
5057:
5052:
5041:
5038:
5035:
5015:
4995:
4974:
4971:
4968:
4964:
4960:
4957:
4954:
4951:
4947:
4941:
4938:
4935:
4931:
4909:
4905:
4893:bounded subset
4880:
4877:
4874:
4871:
4851:
4848:
4845:
4836:of some point
4825:
4798:
4774:
4764:
4763:
4762:
4761:
4760:
4747:
4725:
4700:
4697:
4694:
4691:
4688:
4685:
4682:
4679:
4659:
4656:
4653:
4650:
4647:
4643:
4639:
4636:
4614:
4610:
4606:
4603:
4600:
4597:
4594:
4585:condition for
4584:
4583:not sufficient
4570:
4567:
4563:
4559:
4556:
4553:
4550:
4546:
4540:
4537:
4534:
4530:
4507:
4503:
4499:
4496:
4475:
4471:
4468:
4448:
4445:
4442:
4420:
4416:
4412:
4409:
4406:
4403:
4400:
4391:condition for
4390:
4376:
4373:
4369:
4365:
4362:
4359:
4356:
4352:
4346:
4343:
4340:
4336:
4313:
4309:
4282:
4278:
4255:
4252:
4249:
4245:
4241:
4238:
4235:
4232:
4228:
4222:
4219:
4216:
4212:
4189:
4185:
4181:
4178:
4175:
4172:
4169:
4147:
4143:
4131:
4120:
4115:
4111:
4107:
4104:
4101:
4098:
4095:
4075:
4055:
4032:
4028:
4007:
3987:
3984:
3981:
3959:
3955:
3934:
3911:
3901:
3890:
3887:
3863:
3853:
3842:
3839:
3819:
3809:
3808:is continuous.
3797:
3774:
3771:
3746:
3742:
3739:
3736:
3733:
3722:locally convex
3705:
3684:
3659:
3647:
3644:
3629:
3626:
3613:
3593:
3573:
3570:
3550:
3530:
3527:
3524:
3521:
3501:
3481:
3461:
3458:
3455:
3452:
3449:
3429:
3406:
3403:
3400:
3397:
3394:
3353:
3329:
3312:
3288:
3268:
3245:
3242:
3239:
3216:
3196:
3176:
3152:
3132:
3110:
3082:
3068:
3065:
3054:
3049:normable space
3036:
3012:
2989:
2986:
2966:
2934:
2931:
2928:
2925:
2922:
2912:
2899:
2889:
2886:
2846:
2843:
2840:
2835:
2826:
2817:
2803:
2783:
2773:
2769:
2751:
2748:
2737:bounded subset
2724:
2721:
2718:
2715:
2695:
2675:
2652:
2643:
2625:
2622:
2619:
2609:
2602:is said to be
2591:
2588:
2585:
2582:
2579:
2541:
2538:
2535:
2532:
2529:
2526:
2523:
2520:
2515:
2512:
2509:
2506:
2503:
2499:
2495:
2492:
2489:
2486:
2463:
2443:
2440:
2419:
2414:
2410:
2406:
2402:
2382:
2379:
2376:
2373:
2370:
2348:
2344:
2323:
2303:
2300:
2297:
2294:
2274:
2271:
2268:
2255:
2245:
2240:is said to be
2225:
2222:
2219:
2216:
2213:
2180:
2177:
2174:
2171:
2168:
2165:
2162:
2159:
2156:
2153:
2150:
2147:
2126:
2123:
2120:
2117:
2114:
2094:
2091:
2088:
2085:
2082:
2079:
2076:
2056:
2053:
2050:
2047:
2044:
2041:
2038:
2035:
2032:
2029:
2026:
2006:
2003:
2000:
1980:
1977:
1974:
1971:
1968:
1965:
1962:
1959:
1956:
1953:
1950:
1930:
1927:
1924:
1921:
1918:
1915:
1912:
1909:
1906:
1903:
1900:
1897:
1894:
1891:
1888:
1885:
1865:
1862:
1859:
1839:
1836:
1833:
1830:
1827:
1824:
1821:
1818:
1815:
1812:
1809:
1806:
1803:
1783:
1763:
1743:
1740:
1737:
1734:
1731:
1728:
1725:
1722:
1719:
1714:
1711:
1708:
1704:
1683:
1680:
1677:
1674:
1671:
1668:
1665:
1645:
1642:
1631:bounded subset
1618:
1615:
1612:
1609:
1599:
1586:
1573:is said to be
1562:
1559:
1556:
1553:
1550:
1541:is a set then
1530:
1527:
1524:
1493:
1472:
1468:
1464:
1458:
1455:
1452:
1448:
1427:
1406:
1402:
1398:
1387:absolute value
1373:
1351:
1340:
1326:
1323:
1320:
1317:
1314:
1311:
1306:
1303:
1300:
1296:
1285:
1268:
1237:Bounded subset
1214:
1211:
1208:
1205:
1202:
1190:
1187:
1186:
1185:
1174:
1171:
1168:
1165:
1145:
1121:
1101:
1081:
1070:
1069:
1058:
1055:
1052:
1049:
1046:
1043:
1040:
1037:
1034:
1031:
1023:
1020:
1017:
1014:
1011:
1008:
1005:
1000: if
997:
994:
991:
988:
985:
982:
957:
954:
951:
931:
928:
925:
901:
898:
895:
867:
847:
836:
835:
824:
821:
801:
770:
759:
758:
746:
726:
702:
671:
660:
659:
643:
612:
601:
600:
589:
584:
580:
557:
553:
541:equicontinuous
526:
522:
518:
513:
509:
505:
502:
497:
466:
439:
419:
408:
407:
396:
393:
390:
387:
384:
381:
361:
341:
321:
318:
298:
280:locally convex
267:
256:
255:
244:
241:
221:
211:
200:
197:
194:
191:
168:
158:
157:is continuous.
146:
115:
112:
109:
106:
103:
85:
82:
70:
67:
9:
6:
4:
3:
2:
12166:
12155:
12152:
12150:
12147:
12145:
12142:
12140:
12137:
12136:
12134:
12119:
12111:
12110:
12107:
12101:
12098:
12096:
12093:
12091:
12088:
12086:
12082:
12078:
12076:) convex
12075:
12072:
12070:
12067:
12065:
12061:
12059:
12056:
12054:
12051:
12049:
12048:Semi-complete
12046:
12044:
12041:
12039:
12036:
12034:
12030:
12027:
12025:
12021:
12019:
12016:
12014:
12011:
12009:
12006:
12004:
12001:
11999:
11996:
11994:
11991:
11989:
11986:
11984:
11981:
11979:
11976:
11974:
11971:
11969:
11966:
11964:
11963:Infrabarreled
11961:
11959:
11956:
11954:
11951:
11947:
11944:
11943:
11942:
11939:
11937:
11934:
11932:
11929:
11927:
11924:
11922:
11921:Distinguished
11919:
11917:
11914:
11912:
11909:
11907:
11904:
11902:
11899:
11897:
11893:
11889:
11887:
11884:
11882:
11878:
11874:
11872:
11869:
11867:
11864:
11862:
11859:
11858:
11856:
11854:Types of TVSs
11852:
11846:
11842:
11838:
11836:
11833:
11831:
11828:
11826:
11823:
11821:
11818:
11816:
11812:
11808:
11806:
11803:
11802:
11800:
11796:
11790:
11787:
11785:
11782:
11780:
11777:
11775:
11774:Prevalent/Shy
11772:
11770:
11767:
11765:
11764:Extreme point
11762:
11760:
11754:
11752:
11746:
11744:
11741:
11739:
11736:
11734:
11731:
11729:
11726:
11724:
11721:
11719:
11716:
11714:
11711:
11709:
11706:
11704:
11701:
11700:
11698:
11696:Types of sets
11694:
11688:
11685:
11683:
11680:
11678:
11675:
11673:
11670:
11666:
11663:
11661:
11658:
11656:
11653:
11652:
11651:
11648:
11646:
11643:
11639:
11638:Discontinuous
11636:
11634:
11631:
11629:
11626:
11624:
11621:
11619:
11616:
11614:
11611:
11609:
11606:
11605:
11604:
11601:
11597:
11594:
11593:
11592:
11589:
11588:
11586:
11582:
11576:
11573:
11569:
11566:
11565:
11564:
11561:
11558:
11555:
11553:
11549:
11546:
11544:
11541:
11539:
11536:
11534:
11531:
11529:
11526:
11525:
11523:
11521:
11517:
11511:
11508:
11506:
11503:
11501:
11498:
11496:
11495:Metrizability
11493:
11491:
11488:
11486:
11483:
11481:
11480:Fréchet space
11478:
11476:
11473:
11471:
11468:
11466:
11463:
11461:
11458:
11457:
11455:
11451:
11446:
11439:
11434:
11432:
11427:
11425:
11420:
11419:
11416:
11404:
11396:
11395:
11392:
11386:
11383:
11381:
11378:
11376:
11375:Weak topology
11373:
11371:
11368:
11366:
11363:
11361:
11358:
11357:
11355:
11351:
11344:
11340:
11337:
11335:
11332:
11330:
11327:
11325:
11322:
11320:
11317:
11315:
11312:
11310:
11307:
11305:
11302:
11300:
11299:Index theorem
11297:
11295:
11292:
11290:
11287:
11285:
11282:
11281:
11279:
11275:
11269:
11266:
11264:
11261:
11260:
11258:
11256:Open problems
11254:
11248:
11245:
11243:
11240:
11238:
11235:
11233:
11230:
11228:
11225:
11223:
11220:
11219:
11217:
11213:
11207:
11204:
11202:
11199:
11197:
11194:
11192:
11189:
11187:
11184:
11182:
11179:
11177:
11174:
11172:
11169:
11167:
11164:
11162:
11159:
11158:
11156:
11152:
11146:
11143:
11141:
11138:
11136:
11133:
11131:
11128:
11126:
11123:
11121:
11118:
11116:
11113:
11111:
11108:
11106:
11103:
11102:
11100:
11098:
11094:
11084:
11081:
11079:
11076:
11074:
11071:
11068:
11064:
11060:
11057:
11055:
11052:
11050:
11047:
11046:
11044:
11040:
11034:
11031:
11029:
11026:
11024:
11021:
11019:
11016:
11014:
11011:
11009:
11006:
11004:
11001:
10999:
10996:
10994:
10991:
10989:
10986:
10985:
10982:
10979:
10975:
10970:
10966:
10962:
10955:
10950:
10948:
10943:
10941:
10936:
10935:
10932:
10920:
10917:
10915:
10912:
10910:
10907:
10905:
10902:
10900:
10897:
10896:
10894:
10890:
10884:
10866:
10862:
10858:
10855:
10849:
10840:
10838:
10835:
10831:
10828:
10827:
10826:
10823:
10821:
10818:
10816:
10815:
10810:
10808:
10794:
10789:
10779:
10775:
10766:
10762:
10759:
10757:
10738:
10729:
10728:
10727:
10726:
10722:
10718:
10699:
10690:
10689:
10688:
10687:
10683:
10681:
10657:
10654:
10651:
10647:
10637:
10635:
10634:
10629:
10627:
10624:
10622:
10620:
10616:
10611:
10609:
10606:
10604:
10603:
10598:
10596:
10593:
10591:
10565:
10560:
10557:
10554:
10550:
10540:
10538:
10537:
10532:
10530:
10527:
10525:
10503:
10500:
10492:
10490:
10489:
10484:
10483:
10481:
10479:
10475:
10469:
10465:
10461:
10459:
10456:
10454:
10451:
10449:
10446:
10444:
10441:
10439:
10438:Extreme point
10436:
10434:
10431:
10429:
10426:
10424:
10420:
10416:
10414:
10411:
10410:
10408:
10404:
10398:
10395:
10393:
10390:
10388:
10385:
10383:
10380:
10378:
10372:
10369:
10365:
10361:
10358:
10356:
10350:
10348:
10345:
10343:
10340:
10338:
10335:
10333:
10330:
10328:
10325:
10323:
10320:
10319:
10317:
10315:Types of sets
10313:
10306:
10302:
10299:
10295:
10292:
10290:
10287:
10285:
10282:
10281:
10280:
10277:
10273:
10270:
10268:
10265:
10263:
10260:
10259:
10258:
10255:
10251:
10248:
10246:
10243:
10241:
10238:
10236:
10233:
10231:
10228:
10226:
10223:
10222:
10221:
10218:
10214:
10211:
10209:
10206:
10204:
10201:
10199:
10196:
10194:
10191:
10190:
10189:
10186:
10184:
10181:
10179:
10178:Convex series
10176:
10174:
10173:Bochner space
10171:
10167:
10164:
10163:
10162:
10159:
10157:
10154:
10153:
10151:
10147:
10141:
10138:
10136:
10133:
10131:
10128:
10126:
10125:Riesz's lemma
10123:
10121:
10118:
10116:
10113:
10111:
10110:Mazur's lemma
10108:
10106:
10103:
10101:
10098:
10096:
10093:
10091:
10088:
10084:
10081:
10080:
10079:
10076:
10074:
10071:
10069:
10066:
10064:
10063:Gelfand–Mazur
10061:
10059:
10056:
10054:
10051:
10049:
10046:
10044:
10041:
10039:
10036:
10034:
10031:
10029:
10026:
10024:
10021:
10019:
10016:
10014:
10011:
10009:
10006:
10004:
10001:
10000:
9998:
9996:
9992:
9986:
9983:
9981:
9978:
9974:
9971:
9969:
9966:
9965:
9964:
9961:
9957:
9954:
9952:
9949:
9948:
9947:
9944:
9942:
9939:
9937:
9934:
9932:
9929:
9928:
9926:
9924:
9920:
9914:
9911:
9909:
9906:
9904:
9901:
9899:
9896:
9894:
9891:
9889:
9886:
9884:
9881:
9879:
9876:
9872:
9869:
9868:
9867:
9864:
9862:
9859:
9855:
9852:
9850:
9847:
9846:
9844:
9842:
9839:
9837:
9833:
9829:
9825:
9822:
9821:
9820:
9817:
9815:
9812:
9810:
9806:
9802:
9798:
9795:
9793:
9790:
9788:
9785:
9784:
9783:
9780:
9778:
9775:
9774:
9772:
9770:
9766:
9760:
9757:
9755:
9752:
9748:
9745:
9743:
9740:
9739:
9738:
9735:
9731:
9728:
9726:
9723:
9722:
9721:
9718:
9716:
9713:
9711:
9708:
9704:
9701:
9700:
9699:
9696:
9694:
9691:
9689:
9686:
9685:
9683:
9679:
9673:
9670:
9668:
9665:
9661:
9658:
9657:
9656:
9653:
9651:
9648:
9646:
9643:
9639:
9635:
9632:
9631:
9630:
9627:
9623:
9620:
9619:
9618:
9615:
9613:
9610:
9608:
9605:
9603:
9600:
9599:
9597:
9593:
9586:
9582:
9578:
9575:
9573:
9569:
9567:
9564:
9562:) convex
9561:
9558:
9556:
9553:
9551:
9547:
9545:
9542:
9540:
9537:
9535:
9531:
9527:
9523:
9520:
9518:
9515:
9514:
9513:
9510:
9508:
9507:Grothendieck
9505:
9503:
9500:
9496:
9493:
9492:
9491:
9488:
9486:
9483:
9482:
9480:
9476:
9471:
9464:
9459:
9457:
9452:
9450:
9445:
9444:
9441:
9433:
9429:
9425:
9419:
9415:
9411:
9407:
9403:
9399:
9395:
9389:
9385:
9381:
9377:
9373:
9369:
9365:
9359:
9355:
9350:
9346:
9342:
9338:
9332:
9328:
9324:
9320:
9316:
9312:
9306:
9301:
9300:
9294:
9293:Rudin, Walter
9290:
9286:
9282:
9278:
9272:
9268:
9263:
9259:
9255:
9251:
9247:
9243:
9237:
9233:
9229:
9225:
9221:
9217:
9213:
9207:
9203:
9198:
9194:
9190:
9186:
9180:
9175:
9174:
9168:
9164:
9160:
9156:
9152:
9146:
9142:
9137:
9133:
9129:
9125:
9123:0-471-60848-3
9119:
9115:
9110:
9106:
9102:
9098:
9092:
9088:
9084:
9080:
9076:
9072:
9068:
9064:
9060:
9058:3-540-13627-4
9054:
9050:
9046:
9042:
9038:
9034:
9030:
9026:
9020:
9016:
9011:
9007:
9003:
8999:
8993:
8989:
8985:
8980:
8979:
8972:
8967:
8961:, p. 50.
8960:
8959:Wilansky 2013
8955:
8949:, p. 55.
8948:
8947:Wilansky 2013
8943:
8936:
8931:
8924:
8919:
8913:, p. 63.
8912:
8911:Wilansky 2013
8907:
8900:
8899:Wilansky 2013
8895:
8893:
8891:
8889:
8881:
8880:Wilansky 2013
8876:
8869:
8864:
8862:
8860:
8858:
8856:
8848:
8847:Wilansky 2013
8843:
8836:
8831:
8825:, p. 54.
8824:
8823:Wilansky 2013
8819:
8812:
8807:
8805:
8803:
8801:
8799:
8797:
8795:
8793:
8791:
8789:
8787:
8779:
8774:
8772:
8770:
8768:
8766:
8761:
8750:
8747:
8745:
8742:
8736:
8733:
8730:
8727:
8721:
8718:
8712:
8709:
8707:
8704:
8701:
8698:
8695:
8692:
8689:
8686:
8680:
8677:
8676:
8656:
8651:
8648:
8644:
8640:
8634:
8631:
8625:
8618:
8604:
8601:
8590:
8587:
8581:
8566:
8552:
8549:
8538:
8532:
8517:
8501:
8498:
8494:
8490:
8484:
8478:
8471:
8470:
8469:
8455:
8432:
8429:
8421:
8413:
8405:
8402:
8396:
8391:
8388:
8384:
8363:
8360:
8357:
8329:
8323:
8307:
8296:
8282:
8279:
8273:
8245:
8225:
8214:
8208:
8198:
8195:
8192:
8181:
8172:
8169:
8166:
8163:
8152:
8146:
8129:
8115:
8109:
8072:
8066:
8028:
8022:
8012:
8009:
8006:
7993:the supremum
7980:
7974:
7971:
7968:
7965:
7954:
7948:
7937:
7926:
7920:
7902:
7899:
7896:
7893:
7887:
7881:
7875:
7869:
7863:
7843:
7840:
7837:
7809:
7806:
7803:
7794:
7781:
7778:
7775:
7772:
7752:
7749:
7741:
7716:
7713:
7693:
7673:
7653:
7645:
7629:
7620:
7603:
7600:
7597:
7591:
7585:
7562:
7559:
7539:
7531:
7528:is a complex
7515:
7501:
7488:
7485:
7477:
7461:
7458:
7455:
7452:
7432:
7412:
7386:
7380:
7377:
7374:
7366:
7363:
7359:
7355:
7352:
7349:
7343:
7335:
7332:
7328:
7318:
7316:
7311:
7309:
7305:
7302:
7292:
7278:
7270:
7254:
7234:
7214:
7206:
7201:
7187:
7178:
7176:
7160:
7140:
7134:
7125:
7123:
7113:
7111:
7110:discontinuous
7107:
7103:
7087:
7084:
7081:
7061:
7058:
7055:
7052:
7033:
7030:
7010:
6983:
6980:
6974:
6968:
6965:
6962:
6959:
6956:
6933:
6930:
6923:For any real
6922:
6904:
6901:
6895:
6889:
6886:
6883:
6880:
6877:
6854:
6851:
6843:
6829:
6826:
6823:
6820:
6800:
6780:
6772:
6770:
6756:
6748:
6732:
6708:
6701:
6698:
6682:
6675:
6672:
6656:
6649:
6647:
6645:
6629:
6621:
6617:
6601:
6589:
6573:
6566:
6564:
6562:
6558:
6557:Fréchet space
6554:
6538:
6526:
6510:
6503:
6501:
6499:
6483:
6459:
6439:
6436:
6433:
6425:
6423:
6409:
6389:
6366:
6363:
6355:
6339:
6336:
6333:
6313:
6310:
6307:
6300:
6285:
6278:The graph of
6277:
6255:
6247:
6244:
6224:
6216:
6215:
6202:
6199:
6196:
6188:
6163:
6143:
6135:
6121:
6118:
6108:
6094:
6074:
6071:
6068:
6060:
6046:
6043:
6036:is closed in
6023:
6015:
6012:
5996:
5989:
5984:
5968:
5963:
5959:
5955:
5952:
5932:
5929:
5918:
5912:
5902:
5899:
5896:
5871:
5868:
5860:
5844:
5839:
5835:
5826:
5823:
5807:
5784:
5781:
5778:
5775:
5772:
5769:
5746:
5743:
5740:
5737:
5717:
5714:
5703:
5697:
5687:
5684:
5681:
5678:
5665:
5664:
5651:
5648:
5637:
5631:
5621:
5618:
5615:
5590:
5582:
5566:
5563:
5560:
5540:
5537:
5534:
5513:
5510:
5507:
5496:
5490:
5478:
5474:
5470:
5467:
5441:
5435:
5432:
5426:
5421:
5417:
5396:
5393:
5390:
5387:
5358:
5352:
5342:
5339:
5336:
5327:
5324:
5304:
5301:
5281:
5278:
5275:
5255:
5227:
5221:
5211:
5208:
5205:
5192:
5184:
5173:
5167:
5157:
5154:
5151:
5148:
5136:The equality
5135:
5134:
5133:
5117:
5109:
5093:
5086:
5081:
5077:
5073:
5067:
5065:
5061:
5055:
5053:
5039:
5036:
5033:
5013:
4993:
4972:
4966:
4955:
4949:
4939:
4936:
4933:
4907:
4894:
4875:
4869:
4849:
4846:
4843:
4823:
4815:
4814:
4812:
4796:
4788:
4772:
4765:
4745:
4723:
4714:
4695:
4692:
4689:
4686:
4680:
4677:
4657:
4654:
4651:
4648:
4645:
4637:
4634:
4612:
4608:
4604:
4598:
4592:
4582:
4568:
4565:
4554:
4548:
4538:
4535:
4532:
4505:
4501:
4497:
4494:
4469:
4466:
4446:
4443:
4440:
4418:
4414:
4410:
4404:
4398:
4389:not necessary
4388:
4374:
4371:
4360:
4354:
4344:
4341:
4338:
4311:
4307:
4298:
4280:
4276:
4267:
4266:
4253:
4250:
4247:
4236:
4230:
4220:
4217:
4214:
4187:
4183:
4179:
4173:
4167:
4145:
4141:
4132:
4118:
4113:
4109:
4105:
4099:
4093:
4073:
4053:
4046:
4030:
4005:
3985:
3982:
3979:
3957:
3953:
3932:
3924:
3923:
3909:
3902:
3888:
3885:
3877:
3861:
3854:
3840:
3837:
3817:
3810:
3795:
3788:
3787:
3786:
3772:
3769:
3761:
3737:
3734:
3731:
3723:
3719:
3703:
3673:
3657:
3643:
3641:
3635:
3625:
3611:
3591:
3571:
3568:
3548:
3525:
3519:
3499:
3479:
3459:
3453:
3450:
3447:
3427:
3420:
3404:
3398:
3395:
3392:
3383:
3382:
3378:
3376:
3372:
3368:
3364:
3361:(such as any
3360:
3355:
3351:
3349:
3345:
3340:
3339:
3335:
3333:
3327:
3324:
3310:
3302:
3286:
3266:
3257:
3243:
3237:
3230:
3214:
3194:
3174:
3166:
3150:
3130:
3122:
3118:
3114:
3108:
3105:
3104:
3100:
3098:
3094:
3090:
3086:
3080:
3078:
3074:
3064:
3062:
3058:
3052:
3050:
3034:
3026:
3010:
3003:
2987:
2984:
2964:
2956:
2952:
2948:
2932:
2926:
2923:
2920:
2910:
2907:
2905:
2903:
2897:
2895:
2885:
2883:
2879:
2875:
2870:
2868:
2864:
2860:
2856:
2852:
2842:
2838:
2833:
2831:
2827:
2820:
2815:
2801:
2781:
2771:
2763:
2749:
2746:
2738:
2719:
2713:
2693:
2673:
2666:
2650:
2637:
2623:
2620:
2617:
2603:
2589:
2583:
2580:
2577:
2567:
2562:
2561:
2557:
2555:
2536:
2527:
2521:
2513:
2510:
2504:
2493:
2487:
2477:
2476:operator norm
2461:
2441:
2438:
2417:
2412:
2408:
2404:
2400:
2380:
2374:
2371:
2368:
2346:
2342:
2321:
2298:
2292:
2272:
2269:
2266:
2259:
2256:if for every
2252:
2247:
2244:
2241:
2239:
2223:
2217:
2214:
2211:
2201:
2196:
2195:
2191:
2178:
2172:
2169:
2163:
2157:
2154:
2151:
2148:
2124:
2118:
2115:
2112:
2086:
2080:
2077:
2051:
2045:
2042:
2039:
2033:
2030:
2024:
2004:
2001:
1998:
1975:
1972:
1969:
1966:
1963:
1960:
1954:
1951:
1948:
1925:
1919:
1916:
1910:
1904:
1901:
1895:
1892:
1889:
1883:
1863:
1860:
1857:
1834:
1831:
1828:
1825:
1822:
1819:
1816:
1810:
1807:
1804:
1801:
1781:
1761:
1754:A linear map
1741:
1735:
1726:
1720:
1712:
1709:
1706:
1675:
1669:
1666:
1643:
1640:
1632:
1613:
1607:
1584:
1574:
1560:
1554:
1551:
1548:
1528:
1525:
1522:
1513:
1512:
1508:
1505:
1491:
1466:
1456:
1453:
1450:
1425:
1400:
1388:
1338:
1324:
1318:
1312:
1304:
1301:
1298:
1283:
1280:
1266:
1258:
1254:
1250:
1244:
1239:
1238:
1234:
1232:
1228:
1212:
1206:
1203:
1200:
1172:
1169:
1166:
1163:
1156:is closed in
1143:
1136:the graph of
1135:
1133:
1119:
1099:
1079:
1056:
1053:
1050:
1044:
1041:
1038:
1035:
1032:
1021:
1018:
1012:
1009:
1006:
995:
992:
989:
986:
983:
980:
955:
952:
949:
929:
926:
923:
915:
913:
896:
885:
881:
865:
845:
822:
819:
799:
792:
790:
788:
784:
768:
744:
724:
716:
700:
693:
691:
689:
685:
669:
657:
641:
634:
632:
630:
626:
610:
587:
578:
551:
542:
520:
507:
503:
500:
495:
484:
480:
464:
457:
455:
453:
437:
417:
394:
391:
388:
385:
382:
379:
359:
339:
319:
316:
296:
289:
285:
283:
281:
265:
242:
239:
219:
212:
198:
195:
192:
189:
182:
166:
159:
144:
137:
136:
135:
133:
129:
113:
107:
104:
101:
94:Suppose that
91:
80:
76:
66:
64:
60:
59:normed spaces
55:
53:
49:
46:
42:
38:
34:
30:
19:
12024:Polynomially
11953:Grothendieck
11946:tame Fréchet
11896:Bornological
11756:Linear cone
11748:Convex cone
11723:Banach disks
11665:Sesquilinear
11617:
11520:Main results
11510:Vector space
11469:
11465:Completeness
11460:Banach space
11365:Balanced set
11339:Distribution
11277:Applications
11130:Krein–Milman
11115:Closed graph
10892:Applications
10813:
10724:
10685:
10632:
10618:
10614:
10601:
10535:
10487:
10374:Linear cone
10367:
10363:
10352:Convex cone
10245:Paley–Wiener
10105:Mackey–Arens
10095:Krein–Milman
10048:Closed range
10043:Closed graph
10013:Banach–Mazur
9893:Self-adjoint
9835:
9797:sesquilinear
9530:Polynomially
9470:Banach space
9413:
9383:
9353:
9322:
9298:
9266:
9231:
9201:
9172:
9140:
9113:
9078:
9075:Conway, John
9044:
9014:
8983:
8966:
8954:
8942:
8930:
8918:
8906:
8875:
8842:
8830:
8818:
7795:
7621:
7530:normed space
7507:
7319:
7315:bounded sets
7312:
7298:
7179:
7126:
7119:
7002:
6747:real numbers
6724:
6593:
6561:normed space
6530:
6475:
6381:
6352:denotes the
5006:is equal to
4045:neighborhood
3998:centered at
3716:need not be
3649:
3637:
3419:normed space
3384:
3380:
3379:
3363:normed space
3356:
3341:
3337:
3336:
3325:
3258:
3106:
3102:
3101:
3097:bornological
3087:valued in a
3070:
3061:automorphism
2908:
2906:
2891:
2882:normed space
2871:
2859:normed space
2848:
2665:neighborhood
2569:
2559:
2558:
2203:
2193:
2192:
1514:
1510:
1509:
1506:
1253:normed space
1246:
1236:
1235:
1193:Throughout,
1192:
1071:
837:
787:normed space
760:
688:Banach space
661:
602:
409:
257:
130:between two
93:
56:
40:
36:
26:
12018:Quasinormed
11931:FK-AK space
11825:Linear span
11820:Convex hull
11805:Affine hull
11608:Almost open
11548:Hahn–Banach
11294:Heat kernel
11284:Hardy space
11191:Trace class
11105:Hahn–Banach
11067:Topological
10613:Continuous
10448:Linear span
10433:Convex hull
10413:Affine hull
10272:holomorphic
10208:holomorphic
10188:Derivatives
10078:Hahn–Banach
10018:Banach–Saks
9936:C*-algebras
9903:Trace class
9866:Functionals
9754:Ultrastrong
9667:Quasinormed
8350:so that if
5072:boundedness
3113:bounded set
1577:bounded on
1385:) with the
785:(such as a
627:(such as a
543:subsets of
33:mathematics
12133:Categories
12058:Stereotype
11916:(DF)-space
11911:Convenient
11650:Functional
11618:Continuous
11603:Linear map
11543:F. Riesz's
11485:Linear map
11227:C*-algebra
11042:Properties
10366:), and (Hw
10267:continuous
10203:functional
9951:C*-algebra
9836:Continuous
9698:Dual space
9672:Stereotype
9650:Metrizable
9577:Projective
8756:References
7476:additivity
7295:Properties
7102:continuous
6999:is closed.
6920:is closed.
6813:such that
6298:is closed.
6176:such that
4862:such that
4713:polar sets
4086:such that
3972:of radius
3724:) and let
3632:See also:
3332:equivalent
3077:equivalent
3027:(which if
2894:continuous
2855:continuous
2839:at a point
2706:such that
2564:See also:
2198:See also:
1850:for every
1241:See also:
1227:linear map
968:such that
916:for every
372:such that
88:See also:
73:See also:
45:continuous
12074:Uniformly
12033:Reflexive
11881:Barrelled
11877:Countably
11789:Symmetric
11687:Transpose
11201:Unbounded
11196:Transpose
11154:Operators
11083:Separable
11078:Reflexive
11063:Algebraic
11049:Barrelled
10825:Sobolev W
10768:Schwartz
10743:∞
10704:∞
10700:ℓ
10666:Ω
10652:λ
10510:Σ
10392:Symmetric
10327:Absorbing
10240:regulated
10220:Integrals
10073:Goldstine
9908:Transpose
9845:Fredholm
9715:Ultraweak
9703:Dual norm
9634:Seminorms
9602:Barrelled
9572:Injective
9560:Uniformly
9534:Reflexive
9432:849801114
9402:853623322
9382:(2006) .
9345:840278135
9285:144216834
9258:840293704
9230:(1983) .
9043:(1987) .
9033:878109401
9006:297140003
8649:≤
8641:⊆
8602:≤
8550:≤
8499:≤
8491:⊆
8430:≤
8406:∈
8389:≤
8196:∈
8170:∈
8010:∈
7972:∈
7900:∈
7841:⊆
7813:→
7776:≤
7750:≤
7607:‖
7601:
7595:‖
7589:‖
7583:‖
7456:∈
7364:−
7333:−
7138:→
7085:
7056:
6981:≤
6960:∈
6902:≤
6881:∈
6824:≤
6437:
6354:real part
6337:
6311:
6197:≤
6111:dense in
5964:∘
5956:∈
5930:≤
5900:∈
5840:∘
5715:≤
5682:∈
5649:≤
5619:∈
5564:≠
5471:∈
5340:∈
5279:≠
5268:and when
5209:∈
5152:∈
4970:∞
4937:∈
4847:∈
4724:≤
4687:−
4605:⊆
4566:≤
4536:∈
4411:⊆
4342:∈
4248:≤
4218:∈
4180:⊆
4106:⊆
3741:→
3718:Hausdorff
3457:→
3402:→
3241:→
2930:→
2621:∈
2587:→
2540:∞
2534:‖
2519:‖
2511:≤
2508:‖
2502:‖
2491:‖
2485:‖
2378:→
2270:⊆
2221:→
2170:≤
2167:‖
2161:‖
2152:∈
2122:→
2090:‖
2087:⋅
2084:‖
2017:(because
2002:≠
1973:∈
1876:(because
1861:∈
1832:∈
1739:∞
1733:‖
1718:‖
1710:∈
1679:‖
1676:⋅
1673:‖
1656:which if
1558:→
1526:⊆
1454:∈
1401:⋅
1322:∞
1316:‖
1310:‖
1302:∈
1233:(TVSs).
1210:→
1167:×
1048:‖
1039:−
1030:‖
1022:δ
1016:‖
1010:−
1004:‖
990:∈
950:δ
900:‖
897:⋅
894:‖
878:are both
583:′
556:′
525:′
517:→
512:′
483:transpose
452:Hausdorff
450:are both
389:≤
383:∘
193:∈
111:→
12118:Category
12069:Strictly
12043:Schwartz
11983:LF-space
11978:LB-space
11936:FK-space
11906:Complete
11886:BK-space
11811:Relative
11758:(subset)
11750:(subset)
11677:Seminorm
11660:Bilinear
11403:Category
11215:Algebras
11097:Theorems
11054:Complete
11023:Schwartz
10969:glossary
10761:weighted
10631:Hilbert
10608:Bs space
10478:Examples
10443:Interior
10419:Relative
10397:Zonotope
10376:(subset)
10354:(subset)
10305:Strongly
10284:Lebesgue
10279:Measures
10149:Analysis
9995:Theorems
9946:Spectrum
9871:positive
9854:operator
9792:operator
9782:Bilinear
9747:operator
9730:operator
9710:Operator
9607:Complete
9555:Strictly
9412:(2013).
9372:24909067
9169:(1973).
9159:30593138
9132:18412261
9105:21195908
9077:(1990).
9067:17499190
8673:See also
8094:because
7644:open map
7445:and any
7308:normable
7180:Suppose
7116:Examples
5409:the set
4297:supremum
3584:so that
3023:being a
2610:a point
2236:between
1229:between
481:and its
288:seminorm
50:between
12083:)
12031:)
11973:K-space
11958:Hilbert
11941:Fréchet
11926:F-space
11901:Brauner
11894:)
11879:)
11861:Asplund
11843:)
11813:)
11733:Bounded
11628:Compact
11613:Bounded
11550: (
11206:Unitary
11186:Nuclear
11171:Compact
11166:Bounded
11161:Adjoint
11135:Min–max
11028:Sobolev
11013:Nuclear
11003:Hilbert
10998:Fréchet
10963: (
10626:Hardy H
10529:c space
10466:)
10421:)
10342:Bounded
10230:Dunford
10225:Bochner
10198:Gateaux
10193:Fréchet
9968:of ODEs
9913:Unitary
9888:Nuclear
9819:Compact
9809:Bounded
9777:Adjoint
9617:Fréchet
9612:F-space
9583: (
9579:)
9532:)
9512:Hilbert
9485:Asplund
9250:0248498
9220:8210342
7686:and if
7106:bounded
6061:Either
5064:bounded
3373:into a
3367:bounded
3350:but to
3348:bounded
3303:, then
3073:bounded
2955:bounded
2909:Example
2902:bounded
2878:bounded
2863:bounded
2762:It is "
2243:bounded
1339:bounded
1284:bounded
12095:Webbed
12081:Quasi-
12003:Montel
11993:Mackey
11892:Ultra-
11871:Banach
11779:Radial
11743:Convex
11713:Affine
11655:Linear
11623:Closed
11447:(TVSs)
11181:Normal
11018:Orlicz
11008:Hölder
10988:Banach
10977:Spaces
10965:topics
10542:Besov
10382:Radial
10347:Convex
10332:Affine
10301:Weakly
10294:Vector
10166:bundle
9956:radius
9883:Normal
9849:kernel
9814:Closed
9737:Strong
9655:Normed
9645:Mackey
9490:Banach
9472:topics
9430:
9420:
9400:
9390:
9370:
9360:
9343:
9333:
9307:
9283:
9273:
9256:
9248:
9238:
9218:
9208:
9193:886098
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9181:
9157:
9147:
9130:
9120:
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9093:
9065:
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9031:
9021:
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8994:
8300:
8294:
8185:
8179:
8133:
8127:
7646:. If
7642:is an
6622:) and
5527:Using
5076:normed
3117:normed
3000:which
2957:, but
2953:) and
2774:point
1255:(or a
880:normed
12053:Smith
12038:Riesz
12029:Semi-
11841:Quasi
11835:Polar
10993:Besov
10617:with
10464:Quasi
10458:Polar
10262:Borel
10213:quasi
9742:polar
9725:polar
9539:Riesz
7729:then
7575:then
7202:every
6669:is a
6614:is a
6586:is a
6559:or a
6496:is a
6009:is a
5294:then
5130:is a
4891:is a
4809:is a
3758:be a
3670:be a
3365:) is
2913:: If
2841:").
2816:every
2735:is a
1629:is a
1225:is a
713:is a
623:is a
539:maps
126:is a
61:is a
43:is a
11672:Norm
11596:form
11584:Maps
11341:(or
11059:Dual
10615:C(K)
10250:weak
9787:form
9720:Weak
9693:Dual
9660:norm
9622:tame
9495:list
9428:OCLC
9418:ISBN
9398:OCLC
9388:ISBN
9368:OCLC
9358:ISBN
9341:OCLC
9331:ISBN
9305:ISBN
9281:OCLC
9271:ISBN
9254:OCLC
9236:ISBN
9216:OCLC
9206:ISBN
9189:OCLC
9179:ISBN
9155:OCLC
9145:ISBN
9128:OCLC
9118:ISBN
9101:OCLC
9091:ISBN
9063:OCLC
9053:ISBN
9029:OCLC
9019:ISBN
9002:OCLC
8992:ISBN
8361:>
7532:and
7100:are
7074:and
6402:and
5782:>
5741:>
5391:>
5066:but
4967:<
4746:<
4487:and
4372:<
3983:>
3650:Let
2904:".
2869:).
2772:some
2537:<
2238:TVSs
1736:<
1319:<
1282:norm
1092:and
1051:<
1019:<
953:>
927:>
858:and
430:and
77:and
35:, a
9832:Dis
9327:GTM
8574:sup
8525:sup
8316:sup
8266:sup
8238:If
8189:sup
8136:sup
8102:sup
8059:sup
8003:sup
7796:If
7508:If
7478:of
7425:of
7306:is
7271:of
7207:on
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6793:on
6695:is
6642:is
6551:is
6523:is
6452:of
6382:If
6356:of
6156:on
6109:not
6107:is
5893:sup
5861:of
5675:sup
5612:sup
5464:sup
5333:sup
5202:sup
5145:sup
5106:is
5078:or
5068:not
5056:not
4930:sup
4895:of
4785:is
4529:sup
4335:sup
4211:sup
4133:If
3878:of
3874:is
3762:on
3720:or
3377:.
3352:not
3119:or
3099:.
3095:or
3053:not
2898:not
2834:not
2739:of
2636:or
2498:sup
2454:if
1703:sup
1633:of
1600:if
1515:If
1447:sup
1363:or
1295:sup
1072:If
882:or
838:If
781:is
761:If
682:is
662:If
654:is
603:If
477:is
410:If
352:on
309:on
278:is
258:If
179:is
39:or
27:In
12135::
10967:–
10602:BV
10536:BK
10488:AC
10370:))
10303:/
9805:Un
9426:.
9396:.
9366:.
9339:.
9325:.
9279:.
9252:.
9246:MR
9244:.
9214:.
9187:.
9153:.
9126:.
9099:.
9089:.
9081:.
9061:.
9047:.
9027:.
9000:.
8990:.
8887:^
8854:^
8785:^
8764:^
8397::=
7938::=
7876::=
7598:Re
7299:A
7082:Im
7053:Re
6434:Im
6334:Re
6308:Re
6248::=
5567:0.
5538::=
5427::=
5328::=
5040:0.
4655:Id
4447:Id
3328:or
3081:or
2965:Id
2921:Id
2494::=
1955::=
1811::=
757:).
54:.
12079:(
12064:B
12062:(
12022:(
11890:(
11875:(
11839:(
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11437:e
11430:t
11423:v
11345:)
11069:)
11065:/
11061:(
10971:)
10953:e
10946:t
10939:v
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10867:p
10863:L
10859:,
10856:X
10853:(
10850:W
10814:F
10795:)
10790:n
10785:R
10780:(
10776:S
10739:L
10725:L
10686:â„“
10669:)
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10648:L
10633:H
10619:K
10579:)
10575:R
10571:(
10566:s
10561:q
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10555:p
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10501:b
10462:(
10417:(
10368:x
10364:x
9834:)
9830:(
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9803:(
9636:/
9587:)
9570:(
9550:B
9548:(
9528:(
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9434:.
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9313:.
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9260:.
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8436:}
8433:r
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8410:F
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8215:u
8212:(
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8176:}
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7978:}
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7962:|
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7873:)
7870:U
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7864:f
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7838:U
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7810:X
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7782:.
7779:p
7773:f
7753:p
7746:|
7742:f
7738:|
7717:,
7714:X
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7674:X
7654:f
7630:X
7604:f
7592:=
7586:f
7563:,
7560:X
7540:f
7516:X
7489:.
7486:F
7462:,
7459:X
7453:x
7433:Y
7413:D
7393:)
7390:)
7387:x
7384:(
7381:F
7378:+
7375:D
7372:(
7367:1
7360:F
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7353:x
7350:+
7347:)
7344:D
7341:(
7336:1
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7279:X
7255:X
7235:X
7215:X
7188:X
7161:X
7141:Y
7135:X
7088:f
7062:,
7059:f
7034:,
7031:f
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6987:}
6984:r
6978:)
6975:x
6972:(
6969:f
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6963:X
6957:x
6954:{
6934:,
6931:r
6908:}
6905:r
6899:)
6896:x
6893:(
6890:f
6887::
6884:X
6878:x
6875:{
6855:,
6852:r
6830:.
6827:p
6821:f
6801:X
6781:p
6757:f
6733:X
6709:f
6683:f
6673:.
6657:f
6630:Y
6602:X
6574:f
6539:X
6511:f
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