4956:
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The argument for the existence of discontinuous linear maps on normed spaces can be generalized to all metrizable topological vector spaces, especially to all Fréchet spaces, but there exist infinite-dimensional locally convex topological vector spaces such that every functional is continuous. On the
3335:
of a topological vector space is the collection of continuous linear maps from the space into the underlying field. Thus the failure of some linear maps to be continuous for infinite-dimensional normed spaces implies that for these spaces, one needs to distinguish the algebraic dual space from the
2644:
The upshot is that the existence of discontinuous linear maps depends on AC; it is consistent with set theory without AC that there are no discontinuous linear maps on complete spaces. In particular, no concrete construction such as the derivative can succeed in defining a discontinuous linear map
3161:
So the natural question to ask about linear operators that are not everywhere-defined is whether they are closable. The answer is, "not necessarily"; indeed, every infinite-dimensional normed space admits linear operators that are not closable. As in the case of discontinuous operators considered
462:
2640:
function is continuous (this is to be understood in the terminology of constructivism, according to which only representable functions are considered to be functions). Such stances are held by only a small minority of working mathematicians.
723:
1680:
2562:
Notice that by using the fact that any set of linearly independent vectors can be completed to a basis, we implicitly used the axiom of choice, which was not needed for the concrete example in the previous section.
864:
3336:
continuous dual space which is then a proper subset. It illustrates the fact that an extra dose of caution is needed in doing analysis on infinite-dimensional spaces as compared to finite-dimensional ones.
3357:. The upshot is that spaces with fewer convex sets have fewer functionals, and in the worst-case scenario, a space may have no functionals at all other than the zero functional. This is the case for the
3619:
2814:
1005:
199:
2621:
297:
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viewpoint. For example, H. G. Garnir, in searching for so-called "dream spaces" (topological vector spaces on which every linear map into a normed space is continuous), was led to adopt ZF +
602:
2604:
Solovay's result shows that it is not necessary to assume that all infinite-dimensional vector spaces admit discontinuous linear maps, and there are schools of analysis which adopt a more
529:
2575:(AC) is used in the general existence theorem of discontinuous linear maps. In fact, there are no constructive examples of discontinuous linear maps with complete domain (for example,
1406:
2178:
This example can be extended into a general theorem about the existence of discontinuous linear maps on any infinite-dimensional normed space (as long as the codomain is not trivial).
3050:
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3349:, which applies to all locally convex spaces, guarantees the existence of many continuous linear functionals, and so a large dual space. In fact, to every convex set, the
2537:
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2017:
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615:
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3099:
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2706:
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2657:, a class of operators which share some of the features of continuous operators. It makes sense to ask which linear operators on a given space are closed. The
1809:
1785:
1587:
1483:
1316:
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743:
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closed operator on a complete domain is continuous, so to obtain a discontinuous closed operator, one must permit operators which are not defined everywhere.
2633:
4277:
4130:
2601:
in which every set of reals is measurable. This implies that there are no discontinuous linear real functions. Clearly AC does not hold in the model.
3518:
3966:
1867:
which assigns to each function its derivative is similarly discontinuous. Note that although the derivative operator is not continuous, it is
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3793:
1488:
208:
4404:
4379:
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The fact that the domain is not complete here is important: discontinuous operators on complete spaces require a little more work.
467:
4829:
4331:
4270:
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2579:). In analysis as it is usually practiced by working mathematicians, the axiom of choice is always employed (it is an axiom of
1329:
936:
4574:
4398:
756:
135:
3624:
One can consider even more general spaces. For example, the existence of a homomorphism between complete separable metric
2445:
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is infinite-dimensional, to show the existence of a linear functional which is not continuous then amounts to constructing
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Examples of discontinuous linear maps are easy to construct in spaces that are not complete; on any Cauchy sequence
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3637:
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3017:
2950:
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4814:
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4393:
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3283:
2586:); thus, to the analyst, all infinite-dimensional topological vector spaces admit discontinuous linear maps.
902:
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1235:
457:{\displaystyle \|f(x)\|=\left\|\sum _{i=1}^{n}x_{i}f(e_{i})\right\|\leq \sum _{i=1}^{n}|x_{i}|\|f(e_{i})\|.}
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grow without bound. In a sense, the linear operators are not continuous because the space has "holes".
1045:
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3162:
above, the proof requires the axiom of choice and so is in general nonconstructive, though again, if
3108:
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873:
2855:
That is, in studying operators that are not everywhere-defined, one may restrict one's attention to
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2186:
Discontinuous linear maps can be proven to exist more generally, even if the space is complete. Let
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real function is linear if and only if it is measurable, so for every such function there is a
25:
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87:
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For example, the weak topology w.r.t. the space of all (algebraically) linear functionals.
2037:
1922:, are linearly independent. One may find a Hamel basis containing them, and define a map
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8:
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of linearly independent vectors which does not have a limit, there is a linear operator
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3441:
from which it follows that these spaces are nonconvex. Note that here is indicated the
3084:
2894:
2711:
2691:
2671:
1894:(note that some authors use this term in a broader sense to mean an algebraic basis of
1794:
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3670:(1970), "A model of set-theory in which every set of reals is Lebesgue measurable",
71:
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1485:
and with real values, is linear, but not continuous. Indeed, consider the sequence
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is not complete here, as must be the case when there is such a constructible map.
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65:
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is infinite-dimensional, this proof will fail as there is no guarantee that the
64:, it is trickier; such maps can be proven to exist, but the proof relies on the
4905:
4854:
4569:
4211:
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3864:
3169:
In fact, there is even an example of a linear operator whose graph has closure
2617:
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61:
45:
718:{\displaystyle \|f(x)\|\leq \left(\sum _{i=1}^{n}|x_{i}|\right)M\leq CM\|x\|.}
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29:
1997:
acts as the identity on the rest of the Hamel basis, and extend to all of
1675:{\displaystyle T(f_{n})={\frac {n^{2}\cos(n^{2}\cdot 0)}{n}}=n\to \infty }
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1584:. This sequence converges uniformly to the constantly zero function, but
17:
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is the zero map which is trivially continuous. In all other cases, when
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21:
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so this provides a sort of maximally discontinuous linear map (confer
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32:
and are often used as approximations to more general functions (see
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2390:
1143:
3640: – A vector space with a topology defined by convex open sets
4767:
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72:
A linear map from a finite-dimensional space is always continuous
2281:, which will imply the existence of a discontinuous linear map
2269:
is not the zero space. We will find a discontinuous linear map
2624:
which states, among other things, that any linear map from an
2559:, and since it is clearly not bounded, it is not continuous.
1169:
is not the zero space, one can find a discontinuous map from
749:
and so is continuous. In fact, to see this, simply note that
3614:{\displaystyle \|f\|=\int _{I}{\frac {|f(x)|}{1+|f(x)|}}dx.}
2653:
Many naturally occurring linear discontinuous operators are
2539:
Complete this sequence of linearly independent vectors to a
3801:
3266:() respectively, and so normed spaces. Define an operator
2809:{\displaystyle T:\operatorname {Dom} (T)\subseteq X\to Y.}
1000:{\displaystyle f(B(x,\delta ))\subseteq B(f(x),\epsilon )}
610:
any two norms on a finite-dimensional space are equivalent
44:), then it makes sense to ask whether all linear maps are
3282:) on to the same function on . As a consequence of the
2580:
3621:
This non-locally convex space has a trivial dual space.
859:{\displaystyle \|f(x)-f(x')\|=\|f(x-x')\|\leq K\|x-x'\|}
3642:
Pages displaying short descriptions of redirect targets
2620:
is a negation of strong AC) as his axioms to prove the
194:{\displaystyle \left(e_{1},e_{2},\ldots ,e_{n}\right)}
52:
topological vector spaces (e.g., infinite-dimensional
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3166:is not complete, there are constructible examples.
2555:so defined will extend uniquely to a linear map on
4131:Spectral theory of ordinary differential equations
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1551:{\displaystyle f_{n}(x)={\frac {\sin(n^{2}x)}{n}}}
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1224:
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283:{\displaystyle f(x)=\sum _{i=1}^{n}x_{i}f(e_{i}),}
282:
193:
108:
3507:Another such example is the space of real-valued
2628:to a TVS is continuous. Going to the extreme of
1767:, as would hold for a continuous map. Note that
597:{\displaystyle \sum _{i=1}^{n}|x_{i}|\leq C\|x\|}
48:. It turns out that for maps defined on infinite-
4972:
2034:be any sequence of rationals which converges to
1346:
478:
2566:
2551:at the other vectors in the basis to be zero.
4271:
3787:
2616:(dependent choice is a weakened form and the
205:which may be taken to be unit vectors. Then,
3528:
3522:
2487:
2474:
2181:
1877:
1339:
1333:
1282:
1269:
1261:
1239:
1153:is the zero space {0}, the only map between
853:
836:
827:
801:
795:
760:
709:
703:
634:
619:
591:
585:
524:{\displaystyle M=\sup _{i}\{\|f(e_{i})\|\},}
515:
512:
490:
487:
448:
426:
316:
301:
3236:the space of polynomial functions from to
2389:which is not bounded. For that, consider a
56:), the answer is generally no: there exist
4278:
4264:
3794:
3780:
3649: – Type of function in linear algebra
1401:{\displaystyle \|f\|=\sup _{x\in }|f(x)|.}
68:and does not provide an explicit example.
28:which preserve the algebraic structure of
3706:
3379:
3326:
3286:, the graph of this operator is dense in
3244:
3218:
2490:
2244:
2216:
2141:
2119:
2005:
1936:
1454:
4084:Group algebra of a locally compact group
3758:Handbook of Analysis and its Foundations
3710:Handbook of Analysis and Its Foundations
3045:{\displaystyle {\overline {\Gamma (T)}}}
2978:{\displaystyle {\overline {\Gamma (T)}}}
2848:{\displaystyle \operatorname {Dom} (T).}
2756:{\displaystyle \operatorname {Dom} (T),}
3666:
3339:
3202:Such an operator is not closable. Let
926:{\displaystyle \delta \leq \epsilon /K}
608:>0 which follows from the fact that
4973:
4417:Uniform boundedness (Banach–Steinhaus)
3504:which do have nontrivial dual spaces.
3398:{\displaystyle L^{p}(\mathbb {R} ,dx)}
2442:, which we normalize. Then, we define
1288:{\displaystyle \|T(e_{i})\|/\|e_{i}\|}
4259:
3775:
3628:can also be shown nonconstructively.
3052:is itself the graph of some operator
2494:{\displaystyle T(e_{n})=n\|e_{n}\|\,}
1787:is real-valued, and so is actually a
1180:
3270:which takes the polynomial function
36:). If the spaces involved are also
24:form an important class of "simple"
3445:on the real line. There are other
2648:
1949:{\displaystyle f:\mathbb {R} \to R}
13:
3741:Constantin Costara, Dumitru Popa,
3024:
2957:
2947:. Otherwise, consider its closure
2869:
2622:Garnir–Wright closed graph theorem
2374:is an arbitrary nonzero vector in
2155:), but not continuous. Note that
1898:vector space). Note that any two
1760:{\displaystyle T(f_{n})\to T(0)=0}
1697:
1669:
60:. If the domain of definition is
14:
5002:
2816:We don't lose much if we replace
1079:{\displaystyle B(f(x),\epsilon )}
4955:
4954:
4240:
4239:
4166:Topological quantum field theory
3743:Exercises in Functional Analysis
3075:{\displaystyle {\overline {T}},}
2645:everywhere on a complete space.
1298:For example, consider the space
4942:With the approximation property
3713:, Academic Press, p. 136,
3125:{\displaystyle {\overline {T}}}
2254:{\displaystyle K=\mathbb {C} .}
2175:relies on the axiom of choice.
892:{\displaystyle \epsilon >0,}
4405:Open mapping (Banach–Schauder)
3727:
3700:
3660:
3638:Finest locally convex topology
3595:
3591:
3585:
3578:
3565:
3561:
3555:
3548:
3392:
3375:
3033:
3027:
2966:
2960:
2878:
2872:
2839:
2833:
2797:
2788:
2782:
2747:
2741:
2465:
2452:
2340:{\displaystyle g(x)=f(x)y_{0}}
2324:
2318:
2309:
2303:
2223:{\displaystyle K=\mathbb {R} }
2088:
2082:
1972:
1966:
1940:
1851:
1748:
1742:
1736:
1733:
1720:
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1666:
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1607:
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1502:
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261:
221:
215:
100:
1:
3962:Uniform boundedness principle
3653:
2532:{\displaystyle n=1,2,\ldots }
1811:(an element of the algebraic
3434:{\displaystyle 0<p<1,}
3251:{\displaystyle \mathbb {R} }
3225:{\displaystyle \mathbb {R} }
3117:
3064:
3037:
2970:
2859:without loss of generality.
2265:is infinite-dimensional and
2148:{\displaystyle \mathbb {R} }
2126:{\displaystyle \mathbb {Q} }
2012:{\displaystyle \mathbb {R} }
1703:{\displaystyle n\to \infty }
1458:{\displaystyle T(f)=f'(0)\,}
1165:is infinite-dimensional and
1086:are the normed balls around
1035:{\displaystyle B(x,\delta )}
866:for some universal constant
7:
4626:Radially convex/Star-shaped
4611:Pre-compact/Totally bounded
3631:
3497:{\displaystyle 0<p<1}
3317:nowhere continuous function
2589:On the other hand, in 1970
2567:Role of the axiom of choice
1886:as a vector space over the
1882:An algebraic basis for the
1135:), which gives continuity.
10:
5007:
4312:Continuous linear operator
4105:Invariant subspace problem
3511:on the unit interval with
3308:{\displaystyle X\times Y,}
3195:{\displaystyle X\times Y.}
3007:{\displaystyle X\times Y.}
2933:{\displaystyle X\times Y,}
2884:{\displaystyle \Gamma (T)}
2100:{\displaystyle f(\pi )=0.}
1987:{\displaystyle f(\pi )=0,}
1322:on the interval with the
4950:
4695:
4657:Algebraic interior (core)
4639:
4537:
4425:
4399:Vector-valued Hahn–Banach
4360:
4294:
4287:Topological vector spaces
4235:
4194:
4118:
4097:
4056:
3995:
3937:
3883:
3825:
3818:
3284:Stone–Weierstrass theorem
3258:. They are subspaces of
2857:densely defined operators
2668:To be more concrete, let
2182:General existence theorem
1878:A nonconstructive example
1232:such that the quantities
753:is linear, and therefore
84:be two normed spaces and
58:discontinuous linear maps
42:topological vector spaces
4487:Topological homomorphism
4347:Topological vector space
4074:Spectrum of a C*-algebra
3760:, Academic Press, 1997.
3707:Schechter, Eric (1996),
3353:associates a continuous
531:and using the fact that
109:{\displaystyle f:X\to Y}
4171:Noncommutative geometry
2427:{\displaystyle n\geq 1}
2171:. The construction of
1577:{\displaystyle n\geq 1}
747:bounded linear operator
4991:Functions and mappings
4545:Absolutely convex/disk
4227:Tomita–Takesaki theory
4202:Approximation property
4146:Calculus of variations
3615:
3498:
3466:
3435:
3399:
3327:Impact for dual spaces
3309:
3252:
3226:
3196:
3153:
3126:
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2979:
2934:
2905:
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2757:
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2702:
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2533:
2495:
2428:
2368:
2341:
2255:
2224:
2149:
2127:
2101:
2048:
2013:
1988:
1950:
1916:
1861:
1860:{\displaystyle X\to X}
1835:
1805:
1781:
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1704:
1676:
1578:
1552:
1479:
1459:
1402:
1312:
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860:
739:
719:
665:
598:
558:
525:
458:
405:
347:
284:
247:
195:
110:
4580:Complemented subspace
4394:hyperplane separation
4222:Banach–Mazur distance
4185:Generalized functions
3673:Annals of Mathematics
3616:
3499:
3467:
3465:{\displaystyle L^{p}}
3436:
3400:
3310:
3253:
3227:
3197:
3154:
3127:
3096:
3077:
3047:
3009:
2980:
2935:
2906:
2886:
2850:
2811:
2758:
2723:
2703:
2683:
2534:
2496:
2429:
2369:
2367:{\displaystyle y_{0}}
2342:
2293:given by the formula
2256:
2225:
2150:
2128:
2102:
2049:
2014:
1989:
1951:
1917:
1862:
1836:
1834:{\displaystyle X^{*}}
1806:
1782:
1762:
1705:
1677:
1579:
1553:
1480:
1460:
1403:
1313:
1290:
1227:
1207:
1205:{\displaystyle e_{i}}
1130:
1101:
1081:
1037:
1002:
928:
894:
861:
740:
720:
645:
599:
538:
526:
459:
385:
327:
285:
227:
196:
111:
4830:Locally convex space
4380:Closed graph theorem
4332:Locally convex space
3967:Kakutani fixed-point
3952:Riesz representation
3519:
3509:measurable functions
3476:
3449:
3410:
3362:
3340:Beyond normed spaces
3290:
3240:
3214:
3208:polynomial functions
3177:
3140:
3109:
3085:
3056:
3018:
2989:
2951:
2915:
2895:
2866:
2824:
2767:
2732:
2712:
2692:
2672:
2659:closed graph theorem
2636:, which states that
2571:As noted above, the
2505:
2446:
2436:linearly independent
2412:
2351:
2297:
2234:
2206:
2137:
2115:
2076:
2047:{\displaystyle \pi }
2038:
2019:by linearity. Let {
2001:
1960:
1926:
1915:{\displaystyle \pi }
1906:
1845:
1818:
1795:
1771:
1714:
1688:
1588:
1562:
1489:
1469:
1419:
1330:
1302:
1236:
1216:
1189:
1128:{\displaystyle f(x)}
1110:
1090:
1046:
1011:
937:
903:
874:
757:
729:
616:
535:
468:
298:
209:
136:
88:
34:linear approximation
4981:Functional analysis
4810:Interpolation space
4342:Operator topologies
4151:Functional calculus
4110:Mahler's conjecture
4089:Von Neumann algebra
3803:Functional analysis
3347:Hahn–Banach theorem
1902:numbers, say 1 and
1841:). The linear map
292:triangle inequality
4840:(Pseudo)Metrizable
4672:Minkowski addition
4524:Sublinear function
4176:Riemann hypothesis
3875:Topological vector
3745:, Springer, 2003.
3668:Solovay, Robert M.
3647:Sublinear function
3611:
3494:
3462:
3431:
3395:
3305:
3248:
3222:
3192:
3152:{\displaystyle T.}
3149:
3122:
3091:
3072:
3042:
3004:
2975:
2930:
2901:
2881:
2845:
2820:by the closure of
2806:
2753:
2718:
2698:
2678:
2663:everywhere-defined
2541:vector space basis
2529:
2491:
2424:
2364:
2337:
2251:
2220:
2145:
2123:
2097:
2044:
2009:
1984:
1946:
1912:
1857:
1831:
1801:
1777:
1757:
1700:
1672:
1574:
1548:
1475:
1455:
1398:
1372:
1308:
1285:
1222:
1202:
1181:A concrete example
1125:
1096:
1076:
1032:
997:
923:
889:
856:
735:
715:
594:
521:
486:
454:
280:
191:
130:finite-dimensional
116:a linear map from
106:
38:topological spaces
4968:
4967:
4687:Relative interior
4433:Bilinear operator
4317:Linear functional
4253:
4252:
4156:Integral operator
3933:
3932:
3756:Schechter, Eric,
3676:, Second Series,
3600:
3355:linear functional
3120:
3094:{\displaystyle T}
3067:
3040:
2973:
2904:{\displaystyle T}
2721:{\displaystyle Y}
2701:{\displaystyle X}
2681:{\displaystyle T}
2591:Robert M. Solovay
2107:By construction,
1804:{\displaystyle X}
1789:linear functional
1780:{\displaystyle T}
1658:
1546:
1478:{\displaystyle X}
1345:
1311:{\displaystyle X}
1225:{\displaystyle T}
1099:{\displaystyle x}
738:{\displaystyle f}
477:
132:, choose a basis
4998:
4958:
4957:
4932:Uniformly smooth
4601:
4593:
4560:Balanced/Circled
4550:Absorbing/Radial
4280:
4273:
4266:
4257:
4256:
4243:
4242:
4161:Jones polynomial
4079:Operator algebra
3823:
3822:
3796:
3789:
3782:
3773:
3772:
3734:
3731:
3725:
3723:
3704:
3698:
3696:
3664:
3643:
3620:
3618:
3617:
3612:
3601:
3599:
3598:
3581:
3569:
3568:
3551:
3545:
3543:
3542:
3503:
3501:
3500:
3495:
3471:
3469:
3468:
3463:
3461:
3460:
3443:Lebesgue measure
3440:
3438:
3437:
3432:
3404:
3402:
3401:
3396:
3382:
3374:
3373:
3345:other hand, the
3314:
3312:
3311:
3306:
3257:
3255:
3254:
3249:
3247:
3231:
3229:
3228:
3223:
3221:
3206:be the space of
3201:
3199:
3198:
3193:
3158:
3156:
3155:
3150:
3131:
3129:
3128:
3123:
3121:
3113:
3100:
3098:
3097:
3092:
3081:
3079:
3078:
3073:
3068:
3060:
3051:
3049:
3048:
3043:
3041:
3036:
3022:
3013:
3011:
3010:
3005:
2984:
2982:
2981:
2976:
2974:
2969:
2955:
2939:
2937:
2936:
2931:
2910:
2908:
2907:
2902:
2890:
2888:
2887:
2882:
2854:
2852:
2851:
2846:
2815:
2813:
2812:
2807:
2762:
2760:
2759:
2754:
2727:
2725:
2724:
2719:
2707:
2705:
2704:
2699:
2687:
2685:
2684:
2679:
2661:asserts that an
2649:Closed operators
2634:Ceitin's theorem
2538:
2536:
2535:
2530:
2500:
2498:
2497:
2492:
2486:
2485:
2464:
2463:
2433:
2431:
2430:
2425:
2373:
2371:
2370:
2365:
2363:
2362:
2346:
2344:
2343:
2338:
2336:
2335:
2260:
2258:
2257:
2252:
2247:
2229:
2227:
2226:
2221:
2219:
2154:
2152:
2151:
2146:
2144:
2132:
2130:
2129:
2124:
2122:
2106:
2104:
2103:
2098:
2053:
2051:
2050:
2045:
2018:
2016:
2015:
2010:
2008:
1993:
1991:
1990:
1985:
1955:
1953:
1952:
1947:
1939:
1921:
1919:
1918:
1913:
1900:noncommensurable
1866:
1864:
1863:
1858:
1840:
1838:
1837:
1832:
1830:
1829:
1810:
1808:
1807:
1802:
1786:
1784:
1783:
1778:
1766:
1764:
1763:
1758:
1732:
1731:
1709:
1707:
1706:
1701:
1681:
1679:
1678:
1673:
1659:
1654:
1644:
1643:
1625:
1624:
1614:
1606:
1605:
1583:
1581:
1580:
1575:
1557:
1555:
1554:
1549:
1547:
1542:
1535:
1534:
1515:
1501:
1500:
1484:
1482:
1481:
1476:
1464:
1462:
1461:
1456:
1444:
1407:
1405:
1404:
1399:
1394:
1377:
1371:
1320:smooth functions
1317:
1315:
1314:
1309:
1294:
1292:
1291:
1286:
1281:
1280:
1268:
1257:
1256:
1231:
1229:
1228:
1223:
1211:
1209:
1208:
1203:
1201:
1200:
1134:
1132:
1131:
1126:
1105:
1103:
1102:
1097:
1085:
1083:
1082:
1077:
1041:
1039:
1038:
1033:
1006:
1004:
1003:
998:
932:
930:
929:
924:
919:
898:
896:
895:
890:
865:
863:
862:
857:
852:
823:
791:
744:
742:
741:
736:
724:
722:
721:
716:
690:
686:
685:
680:
679:
670:
664:
659:
603:
601:
600:
595:
578:
573:
572:
563:
557:
552:
530:
528:
527:
522:
508:
507:
485:
463:
461:
460:
455:
444:
443:
425:
420:
419:
410:
404:
399:
381:
377:
373:
372:
357:
356:
346:
341:
289:
287:
286:
281:
273:
272:
257:
256:
246:
241:
200:
198:
197:
192:
190:
186:
185:
184:
166:
165:
153:
152:
115:
113:
112:
107:
5006:
5005:
5001:
5000:
4999:
4997:
4996:
4995:
4986:Axiom of choice
4971:
4970:
4969:
4964:
4946:
4708:B-complete/Ptak
4691:
4635:
4599:
4591:
4570:Bounding points
4533:
4475:Densely defined
4421:
4410:Bounded inverse
4356:
4290:
4284:
4254:
4249:
4231:
4195:Advanced topics
4190:
4114:
4093:
4052:
4018:Hilbert–Schmidt
3991:
3982:Gelfand–Naimark
3929:
3879:
3814:
3800:
3738:
3737:
3732:
3728:
3721:
3705:
3701:
3686:10.2307/1970696
3665:
3661:
3656:
3641:
3634:
3594:
3577:
3570:
3564:
3547:
3546:
3544:
3538:
3534:
3520:
3517:
3516:
3477:
3474:
3473:
3456:
3452:
3450:
3447:
3446:
3411:
3408:
3407:
3378:
3369:
3365:
3363:
3360:
3359:
3351:Minkowski gauge
3342:
3329:
3291:
3288:
3287:
3243:
3241:
3238:
3237:
3217:
3215:
3212:
3211:
3178:
3175:
3174:
3141:
3138:
3137:
3112:
3110:
3107:
3106:
3086:
3083:
3082:
3059:
3057:
3054:
3053:
3023:
3021:
3019:
3016:
3015:
2990:
2987:
2986:
2956:
2954:
2952:
2949:
2948:
2916:
2913:
2912:
2896:
2893:
2892:
2867:
2864:
2863:
2825:
2822:
2821:
2768:
2765:
2764:
2733:
2730:
2729:
2713:
2710:
2709:
2693:
2690:
2689:
2673:
2670:
2669:
2651:
2573:axiom of choice
2569:
2506:
2503:
2502:
2481:
2477:
2459:
2455:
2447:
2444:
2443:
2413:
2410:
2409:
2407:
2401:
2358:
2354:
2352:
2349:
2348:
2331:
2327:
2298:
2295:
2294:
2243:
2235:
2232:
2231:
2215:
2207:
2204:
2203:
2198:over the field
2184:
2140:
2138:
2135:
2134:
2118:
2116:
2113:
2112:
2111:is linear over
2077:
2074:
2073:
2071:
2059:
2039:
2036:
2035:
2033:
2027:
2004:
2002:
1999:
1998:
1961:
1958:
1957:
1935:
1927:
1924:
1923:
1907:
1904:
1903:
1880:
1846:
1843:
1842:
1825:
1821:
1819:
1816:
1815:
1796:
1793:
1792:
1772:
1769:
1768:
1727:
1723:
1715:
1712:
1711:
1689:
1686:
1685:
1639:
1635:
1620:
1616:
1615:
1613:
1601:
1597:
1589:
1586:
1585:
1563:
1560:
1559:
1530:
1526:
1516:
1514:
1496:
1492:
1490:
1487:
1486:
1470:
1467:
1466:
1437:
1420:
1417:
1416:
1390:
1373:
1349:
1331:
1328:
1327:
1318:of real-valued
1303:
1300:
1299:
1276:
1272:
1264:
1252:
1248:
1237:
1234:
1233:
1217:
1214:
1213:
1196:
1192:
1190:
1187:
1186:
1183:
1111:
1108:
1107:
1091:
1088:
1087:
1047:
1044:
1043:
1012:
1009:
1008:
938:
935:
934:
915:
904:
901:
900:
875:
872:
871:
870:. Thus for any
845:
816:
784:
758:
755:
754:
730:
727:
726:
681:
675:
671:
666:
660:
649:
644:
640:
617:
614:
613:
574:
568:
564:
559:
553:
542:
536:
533:
532:
503:
499:
481:
469:
466:
465:
439:
435:
421:
415:
411:
406:
400:
389:
368:
364:
352:
348:
342:
331:
326:
322:
299:
296:
295:
268:
264:
252:
248:
242:
231:
210:
207:
206:
180:
176:
161:
157:
148:
144:
143:
139:
137:
134:
133:
89:
86:
85:
74:
66:axiom of choice
12:
11:
5:
5004:
4994:
4993:
4988:
4983:
4966:
4965:
4963:
4962:
4951:
4948:
4947:
4945:
4944:
4939:
4934:
4929:
4927:Ultrabarrelled
4919:
4913:
4908:
4902:
4897:
4892:
4887:
4882:
4877:
4868:
4862:
4857:
4855:Quasi-complete
4852:
4850:Quasibarrelled
4847:
4842:
4837:
4832:
4827:
4822:
4817:
4812:
4807:
4802:
4797:
4792:
4791:
4790:
4780:
4775:
4770:
4765:
4760:
4755:
4750:
4745:
4740:
4730:
4725:
4715:
4710:
4705:
4699:
4697:
4693:
4692:
4690:
4689:
4679:
4674:
4669:
4664:
4659:
4649:
4643:
4641:
4640:Set operations
4637:
4636:
4634:
4633:
4628:
4623:
4618:
4613:
4608:
4603:
4595:
4587:
4582:
4577:
4572:
4567:
4562:
4557:
4552:
4547:
4541:
4539:
4535:
4534:
4532:
4531:
4526:
4521:
4516:
4511:
4510:
4509:
4504:
4499:
4489:
4484:
4483:
4482:
4477:
4472:
4467:
4462:
4457:
4452:
4442:
4441:
4440:
4429:
4427:
4423:
4422:
4420:
4419:
4414:
4413:
4412:
4402:
4396:
4387:
4382:
4377:
4375:Banach–Alaoglu
4372:
4370:Anderson–Kadec
4366:
4364:
4358:
4357:
4355:
4354:
4349:
4344:
4339:
4334:
4329:
4324:
4319:
4314:
4309:
4304:
4298:
4296:
4295:Basic concepts
4292:
4291:
4283:
4282:
4275:
4268:
4260:
4251:
4250:
4248:
4247:
4236:
4233:
4232:
4230:
4229:
4224:
4219:
4214:
4212:Choquet theory
4209:
4204:
4198:
4196:
4192:
4191:
4189:
4188:
4178:
4173:
4168:
4163:
4158:
4153:
4148:
4143:
4138:
4133:
4128:
4122:
4120:
4116:
4115:
4113:
4112:
4107:
4101:
4099:
4095:
4094:
4092:
4091:
4086:
4081:
4076:
4071:
4066:
4064:Banach algebra
4060:
4058:
4054:
4053:
4051:
4050:
4045:
4040:
4035:
4030:
4025:
4020:
4015:
4010:
4005:
3999:
3997:
3993:
3992:
3990:
3989:
3987:Banach–Alaoglu
3984:
3979:
3974:
3969:
3964:
3959:
3954:
3949:
3943:
3941:
3935:
3934:
3931:
3930:
3928:
3927:
3922:
3917:
3915:Locally convex
3912:
3898:
3893:
3887:
3885:
3881:
3880:
3878:
3877:
3872:
3867:
3862:
3857:
3852:
3847:
3842:
3837:
3832:
3826:
3820:
3816:
3815:
3799:
3798:
3791:
3784:
3776:
3770:
3769:
3754:
3736:
3735:
3726:
3719:
3699:
3658:
3657:
3655:
3652:
3651:
3650:
3644:
3633:
3630:
3610:
3607:
3604:
3597:
3593:
3590:
3587:
3584:
3580:
3576:
3573:
3567:
3563:
3560:
3557:
3554:
3550:
3541:
3537:
3533:
3530:
3527:
3524:
3493:
3490:
3487:
3484:
3481:
3459:
3455:
3430:
3427:
3424:
3421:
3418:
3415:
3394:
3391:
3388:
3385:
3381:
3377:
3372:
3368:
3341:
3338:
3328:
3325:
3319:). Note that
3304:
3301:
3298:
3295:
3246:
3220:
3191:
3188:
3185:
3182:
3148:
3145:
3132:is called the
3119:
3116:
3090:
3071:
3066:
3063:
3039:
3035:
3032:
3029:
3026:
3003:
3000:
2997:
2994:
2972:
2968:
2965:
2962:
2959:
2929:
2926:
2923:
2920:
2900:
2880:
2877:
2874:
2871:
2844:
2841:
2838:
2835:
2832:
2829:
2805:
2802:
2799:
2796:
2793:
2790:
2787:
2784:
2781:
2778:
2775:
2772:
2752:
2749:
2746:
2743:
2740:
2737:
2717:
2697:
2688:be a map from
2677:
2650:
2647:
2630:constructivism
2618:Baire property
2606:constructivist
2568:
2565:
2528:
2525:
2522:
2519:
2516:
2513:
2510:
2489:
2484:
2480:
2476:
2473:
2470:
2467:
2462:
2458:
2454:
2451:
2423:
2420:
2417:
2403:
2397:
2361:
2357:
2334:
2330:
2326:
2323:
2320:
2317:
2314:
2311:
2308:
2305:
2302:
2250:
2246:
2242:
2239:
2218:
2214:
2211:
2183:
2180:
2143:
2121:
2096:
2093:
2090:
2087:
2084:
2081:
2067:
2055:
2043:
2029:
2023:
2007:
1983:
1980:
1977:
1974:
1971:
1968:
1965:
1945:
1942:
1938:
1934:
1931:
1911:
1890:is known as a
1879:
1876:
1856:
1853:
1850:
1828:
1824:
1800:
1776:
1756:
1753:
1750:
1747:
1744:
1741:
1738:
1735:
1730:
1726:
1722:
1719:
1699:
1696:
1693:
1671:
1668:
1665:
1662:
1657:
1653:
1650:
1647:
1642:
1638:
1634:
1631:
1628:
1623:
1619:
1612:
1609:
1604:
1600:
1596:
1593:
1573:
1570:
1567:
1545:
1541:
1538:
1533:
1529:
1525:
1522:
1519:
1513:
1510:
1507:
1504:
1499:
1495:
1474:
1453:
1450:
1447:
1443:
1440:
1436:
1433:
1430:
1427:
1424:
1415:map, given by
1397:
1393:
1389:
1386:
1383:
1380:
1376:
1370:
1367:
1364:
1361:
1358:
1355:
1352:
1348:
1344:
1341:
1338:
1335:
1307:
1284:
1279:
1275:
1271:
1267:
1263:
1260:
1255:
1251:
1247:
1244:
1241:
1221:
1199:
1195:
1182:
1179:
1124:
1121:
1118:
1115:
1095:
1075:
1072:
1069:
1066:
1063:
1060:
1057:
1054:
1051:
1031:
1028:
1025:
1022:
1019:
1016:
996:
993:
990:
987:
984:
981:
978:
975:
972:
969:
966:
963:
960:
957:
954:
951:
948:
945:
942:
922:
918:
914:
911:
908:
899:we can choose
888:
885:
882:
879:
855:
851:
848:
844:
841:
838:
835:
832:
829:
826:
822:
819:
815:
812:
809:
806:
803:
800:
797:
794:
790:
787:
783:
780:
777:
774:
771:
768:
765:
762:
734:
714:
711:
708:
705:
702:
699:
696:
693:
689:
684:
678:
674:
669:
663:
658:
655:
652:
648:
643:
639:
636:
633:
630:
627:
624:
621:
593:
590:
587:
584:
581:
577:
571:
567:
562:
556:
551:
548:
545:
541:
520:
517:
514:
511:
506:
502:
498:
495:
492:
489:
484:
480:
476:
473:
453:
450:
447:
442:
438:
434:
431:
428:
424:
418:
414:
409:
403:
398:
395:
392:
388:
384:
380:
376:
371:
367:
363:
360:
355:
351:
345:
340:
337:
334:
330:
325:
321:
318:
315:
312:
309:
306:
303:
290:and so by the
279:
276:
271:
267:
263:
260:
255:
251:
245:
240:
237:
234:
230:
226:
223:
220:
217:
214:
189:
183:
179:
175:
172:
169:
164:
160:
156:
151:
147:
142:
105:
102:
99:
96:
93:
73:
70:
9:
6:
4:
3:
2:
5003:
4992:
4989:
4987:
4984:
4982:
4979:
4978:
4976:
4961:
4953:
4952:
4949:
4943:
4940:
4938:
4935:
4933:
4930:
4928:
4924:
4920:
4918:) convex
4917:
4914:
4912:
4909:
4907:
4903:
4901:
4898:
4896:
4893:
4891:
4890:Semi-complete
4888:
4886:
4883:
4881:
4878:
4876:
4872:
4869:
4867:
4863:
4861:
4858:
4856:
4853:
4851:
4848:
4846:
4843:
4841:
4838:
4836:
4833:
4831:
4828:
4826:
4823:
4821:
4818:
4816:
4813:
4811:
4808:
4806:
4805:Infrabarreled
4803:
4801:
4798:
4796:
4793:
4789:
4786:
4785:
4784:
4781:
4779:
4776:
4774:
4771:
4769:
4766:
4764:
4763:Distinguished
4761:
4759:
4756:
4754:
4751:
4749:
4746:
4744:
4741:
4739:
4735:
4731:
4729:
4726:
4724:
4720:
4716:
4714:
4711:
4709:
4706:
4704:
4701:
4700:
4698:
4696:Types of TVSs
4694:
4688:
4684:
4680:
4678:
4675:
4673:
4670:
4668:
4665:
4663:
4660:
4658:
4654:
4650:
4648:
4645:
4644:
4642:
4638:
4632:
4629:
4627:
4624:
4622:
4619:
4617:
4616:Prevalent/Shy
4614:
4612:
4609:
4607:
4606:Extreme point
4604:
4602:
4596:
4594:
4588:
4586:
4583:
4581:
4578:
4576:
4573:
4571:
4568:
4566:
4563:
4561:
4558:
4556:
4553:
4551:
4548:
4546:
4543:
4542:
4540:
4538:Types of sets
4536:
4530:
4527:
4525:
4522:
4520:
4517:
4515:
4512:
4508:
4505:
4503:
4500:
4498:
4495:
4494:
4493:
4490:
4488:
4485:
4481:
4480:Discontinuous
4478:
4476:
4473:
4471:
4468:
4466:
4463:
4461:
4458:
4456:
4453:
4451:
4448:
4447:
4446:
4443:
4439:
4436:
4435:
4434:
4431:
4430:
4428:
4424:
4418:
4415:
4411:
4408:
4407:
4406:
4403:
4400:
4397:
4395:
4391:
4388:
4386:
4383:
4381:
4378:
4376:
4373:
4371:
4368:
4367:
4365:
4363:
4359:
4353:
4350:
4348:
4345:
4343:
4340:
4338:
4337:Metrizability
4335:
4333:
4330:
4328:
4325:
4323:
4322:Fréchet space
4320:
4318:
4315:
4313:
4310:
4308:
4305:
4303:
4300:
4299:
4297:
4293:
4288:
4281:
4276:
4274:
4269:
4267:
4262:
4261:
4258:
4246:
4238:
4237:
4234:
4228:
4225:
4223:
4220:
4218:
4217:Weak topology
4215:
4213:
4210:
4208:
4205:
4203:
4200:
4199:
4197:
4193:
4186:
4182:
4179:
4177:
4174:
4172:
4169:
4167:
4164:
4162:
4159:
4157:
4154:
4152:
4149:
4147:
4144:
4142:
4141:Index theorem
4139:
4137:
4134:
4132:
4129:
4127:
4124:
4123:
4121:
4117:
4111:
4108:
4106:
4103:
4102:
4100:
4098:Open problems
4096:
4090:
4087:
4085:
4082:
4080:
4077:
4075:
4072:
4070:
4067:
4065:
4062:
4061:
4059:
4055:
4049:
4046:
4044:
4041:
4039:
4036:
4034:
4031:
4029:
4026:
4024:
4021:
4019:
4016:
4014:
4011:
4009:
4006:
4004:
4001:
4000:
3998:
3994:
3988:
3985:
3983:
3980:
3978:
3975:
3973:
3970:
3968:
3965:
3963:
3960:
3958:
3955:
3953:
3950:
3948:
3945:
3944:
3942:
3940:
3936:
3926:
3923:
3921:
3918:
3916:
3913:
3910:
3906:
3902:
3899:
3897:
3894:
3892:
3889:
3888:
3886:
3882:
3876:
3873:
3871:
3868:
3866:
3863:
3861:
3858:
3856:
3853:
3851:
3848:
3846:
3843:
3841:
3838:
3836:
3833:
3831:
3828:
3827:
3824:
3821:
3817:
3812:
3808:
3804:
3797:
3792:
3790:
3785:
3783:
3778:
3777:
3774:
3767:
3766:0-12-622760-8
3763:
3759:
3755:
3752:
3751:1-4020-1560-7
3748:
3744:
3740:
3739:
3730:
3722:
3720:9780080532998
3716:
3712:
3711:
3703:
3695:
3691:
3687:
3683:
3679:
3675:
3674:
3669:
3663:
3659:
3648:
3645:
3639:
3636:
3635:
3629:
3627:
3622:
3608:
3605:
3602:
3588:
3582:
3574:
3571:
3558:
3552:
3539:
3535:
3531:
3525:
3514:
3510:
3505:
3491:
3488:
3485:
3482:
3479:
3457:
3453:
3444:
3428:
3425:
3422:
3419:
3416:
3413:
3405:
3389:
3386:
3383:
3370:
3366:
3356:
3352:
3348:
3337:
3334:
3324:
3322:
3318:
3302:
3299:
3296:
3293:
3285:
3281:
3277:
3273:
3269:
3265:
3261:
3235:
3209:
3205:
3189:
3186:
3183:
3180:
3172:
3167:
3165:
3159:
3146:
3143:
3135:
3114:
3104:
3088:
3069:
3061:
3030:
3001:
2998:
2995:
2992:
2963:
2946:
2943:
2927:
2924:
2921:
2918:
2911:is closed in
2898:
2875:
2862:If the graph
2860:
2858:
2842:
2836:
2830:
2827:
2819:
2803:
2800:
2794:
2791:
2785:
2779:
2776:
2773:
2770:
2750:
2744:
2738:
2735:
2715:
2695:
2675:
2666:
2664:
2660:
2656:
2646:
2642:
2639:
2635:
2631:
2627:
2623:
2619:
2615:
2611:
2607:
2602:
2600:
2596:
2592:
2587:
2585:
2582:
2578:
2577:Banach spaces
2574:
2564:
2560:
2558:
2554:
2550:
2546:
2542:
2526:
2523:
2520:
2517:
2514:
2511:
2508:
2482:
2478:
2471:
2468:
2460:
2456:
2449:
2441:
2437:
2421:
2418:
2415:
2406:
2400:
2396:
2392:
2388:
2384:
2379:
2377:
2359:
2355:
2332:
2328:
2321:
2315:
2312:
2306:
2300:
2292:
2288:
2284:
2280:
2276:
2272:
2268:
2264:
2248:
2240:
2237:
2212:
2209:
2201:
2197:
2196:normed spaces
2193:
2189:
2179:
2176:
2174:
2170:
2166:
2162:
2158:
2110:
2094:
2091:
2085:
2079:
2070:
2066:
2062:
2058:
2041:
2032:
2026:
2022:
1996:
1981:
1978:
1975:
1969:
1963:
1943:
1932:
1929:
1909:
1901:
1897:
1893:
1889:
1885:
1875:
1872:
1870:
1854:
1848:
1826:
1822:
1814:
1798:
1790:
1774:
1754:
1751:
1745:
1739:
1728:
1724:
1717:
1691:
1682:
1663:
1660:
1655:
1648:
1645:
1640:
1636:
1629:
1626:
1621:
1617:
1610:
1602:
1598:
1591:
1571:
1568:
1565:
1543:
1536:
1531:
1527:
1520:
1517:
1511:
1505:
1497:
1493:
1472:
1448:
1441:
1438:
1434:
1428:
1422:
1414:
1412:
1395:
1384:
1378:
1365:
1362:
1359:
1353:
1350:
1342:
1336:
1325:
1321:
1305:
1296:
1277:
1273:
1265:
1253:
1249:
1242:
1219:
1197:
1193:
1178:
1176:
1172:
1168:
1164:
1160:
1156:
1152:
1148:
1145:
1141:
1136:
1119:
1113:
1093:
1070:
1067:
1061:
1055:
1049:
1026:
1023:
1020:
1014:
991:
988:
982:
976:
970:
967:
958:
955:
952:
946:
940:
920:
916:
912:
909:
906:
886:
883:
880:
877:
869:
849:
846:
842:
839:
833:
830:
820:
817:
813:
810:
804:
798:
788:
785:
778:
775:
769:
763:
752:
748:
732:
712:
706:
700:
697:
694:
691:
687:
676:
672:
661:
656:
653:
650:
646:
641:
637:
628:
622:
612:, one finds
611:
607:
588:
582:
579:
569:
565:
554:
549:
546:
543:
539:
518:
504:
500:
493:
482:
474:
471:
451:
440:
436:
429:
416:
412:
401:
396:
393:
390:
386:
382:
369:
365:
358:
353:
349:
343:
338:
335:
332:
328:
319:
310:
304:
293:
277:
269:
265:
258:
253:
249:
243:
238:
235:
232:
228:
224:
218:
212:
204:
187:
181:
177:
173:
170:
167:
162:
158:
154:
149:
145:
140:
131:
127:
123:
119:
103:
97:
94:
91:
83:
79:
69:
67:
63:
59:
55:
54:normed spaces
51:
47:
43:
39:
35:
31:
30:linear spaces
27:
23:
19:
4866:Polynomially
4795:Grothendieck
4788:tame Fréchet
4738:Bornological
4598:Linear cone
4590:Convex cone
4565:Banach disks
4507:Sesquilinear
4479:
4362:Main results
4352:Vector space
4307:Completeness
4302:Banach space
4207:Balanced set
4181:Distribution
4119:Applications
3972:Krein–Milman
3957:Closed graph
3757:
3742:
3729:
3709:
3702:
3677:
3671:
3662:
3623:
3506:
3472:spaces with
3406:spaces with
3343:
3330:
3320:
3279:
3275:
3271:
3267:
3263:
3259:
3233:
3203:
3170:
3168:
3163:
3160:
3133:
3102:
2944:
2941:
2861:
2817:
2728:with domain
2667:
2662:
2652:
2643:
2637:
2603:
2593:exhibited a
2588:
2570:
2561:
2556:
2552:
2548:
2547:by defining
2544:
2439:
2404:
2398:
2394:
2386:
2382:
2380:
2375:
2290:
2286:
2282:
2278:
2274:
2270:
2266:
2262:
2261:Assume that
2199:
2191:
2187:
2185:
2177:
2172:
2159:is also not
2156:
2108:
2068:
2064:
2060:
2056:
2030:
2024:
2020:
1994:
1895:
1884:real numbers
1881:
1873:
1683:
1409:
1326:, that is,
1324:uniform norm
1297:
1184:
1174:
1170:
1166:
1162:
1158:
1154:
1150:
1149:exists. If
1146:
1139:
1137:
867:
750:
605:
202:
125:
121:
117:
81:
77:
75:
57:
15:
4860:Quasinormed
4773:FK-AK space
4667:Linear span
4662:Convex hull
4647:Affine hull
4450:Almost open
4390:Hahn–Banach
4136:Heat kernel
4126:Hardy space
4033:Trace class
3947:Hahn–Banach
3909:Topological
2632:, there is
2438:vectors in
2072:) = π, but
2054:. Then lim
1892:Hamel basis
1710:instead of
1465:defined on
1413:-at-a-point
50:dimensional
22:linear maps
18:mathematics
4975:Categories
4900:Stereotype
4758:(DF)-space
4753:Convenient
4492:Functional
4460:Continuous
4445:Linear map
4385:F. Riesz's
4327:Linear map
4069:C*-algebra
3884:Properties
3654:References
3515:given by
3333:dual space
3101:is called
2599:set theory
2584:set theory
2169:Vitali set
2161:measurable
2133:(not over
1813:dual space
1411:derivative
46:continuous
40:(that is,
4916:Uniformly
4875:Reflexive
4723:Barrelled
4719:Countably
4631:Symmetric
4529:Transpose
4043:Unbounded
4038:Transpose
3996:Operators
3925:Separable
3920:Reflexive
3905:Algebraic
3891:Barrelled
3536:∫
3529:‖
3523:‖
3513:quasinorm
3297:×
3210:from to
3184:×
3118:¯
3065:¯
3038:¯
3025:Γ
2996:×
2971:¯
2958:Γ
2922:×
2870:Γ
2831:
2798:→
2792:⊆
2780:
2739:
2527:…
2501:for each
2488:‖
2475:‖
2419:≥
2086:π
2042:π
1970:π
1941:→
1910:π
1888:rationals
1852:→
1827:∗
1737:→
1698:∞
1695:→
1670:∞
1667:→
1646:⋅
1630:
1569:≥
1521:
1354:∈
1340:‖
1334:‖
1283:‖
1270:‖
1262:‖
1240:‖
1071:ϵ
1027:δ
992:ϵ
968:⊆
959:δ
913:ϵ
910:≤
907:δ
878:ϵ
854:‖
843:−
837:‖
831:≤
828:‖
814:−
802:‖
796:‖
776:−
761:‖
710:‖
704:‖
695:≤
647:∑
638:≤
635:‖
620:‖
604:for some
592:‖
586:‖
580:≤
540:∑
513:‖
491:‖
464:Letting
449:‖
427:‖
387:∑
383:≤
329:∑
317:‖
302:‖
229:∑
171:…
101:→
26:functions
4960:Category
4911:Strictly
4885:Schwartz
4825:LF-space
4820:LB-space
4778:FK-space
4748:Complete
4728:BK-space
4653:Relative
4600:(subset)
4592:(subset)
4519:Seminorm
4502:Bilinear
4245:Category
4057:Algebras
3939:Theorems
3896:Complete
3865:Schwartz
3811:glossary
3680:: 1–56,
3632:See also
3103:closable
2940:we call
2763:written
2391:sequence
2165:additive
1956:so that
1442:′
1144:supremum
933:so that
850:′
821:′
789:′
379:‖
324:‖
62:complete
4925:)
4873:)
4815:K-space
4800:Hilbert
4783:Fréchet
4768:F-space
4743:Brauner
4736:)
4721:)
4703:Asplund
4685:)
4655:)
4575:Bounded
4470:Compact
4455:Bounded
4392: (
4048:Unitary
4028:Nuclear
4013:Compact
4008:Bounded
4003:Adjoint
3977:Min–max
3870:Sobolev
3855:Nuclear
3845:Hilbert
3840:Fréchet
3805: (
3694:0265151
3262:() and
3134:closure
2626:F-space
4937:Webbed
4923:Quasi-
4845:Montel
4835:Mackey
4734:Ultra-
4713:Banach
4621:Radial
4585:Convex
4555:Affine
4497:Linear
4465:Closed
4289:(TVSs)
4023:Normal
3860:Orlicz
3850:Hölder
3830:Banach
3819:Spaces
3807:topics
3764:
3749:
3717:
3692:
3626:groups
3105:, and
2945:closed
2655:closed
2347:where
2202:where
1869:closed
725:Thus,
4895:Smith
4880:Riesz
4871:Semi-
4683:Quasi
4677:Polar
3835:Besov
2638:every
2595:model
2434:) of
2285:from
2273:from
2163:; an
745:is a
124:. If
4514:Norm
4438:form
4426:Maps
4183:(or
3901:Dual
3762:ISBN
3747:ISBN
3715:ISBN
3489:<
3483:<
3423:<
3417:<
3331:The
3232:and
2190:and
1558:for
1408:The
1157:and
1106:and
1042:and
881:>
80:and
76:Let
3682:doi
3173:of
3171:all
3136:of
3014:If
2985:in
2891:of
2828:Dom
2777:Dom
2736:Dom
2708:to
2597:of
2581:ZFC
2543:of
2381:If
2289:to
2277:to
2230:or
2194:be
1896:any
1791:on
1684:as
1627:cos
1518:sin
1347:sup
1173:to
1138:If
479:sup
294:,
201:in
128:is
120:to
16:In
4977::
3809:–
3690:MR
3688:,
3678:92
3274:↦
2614:BP
2612:+
2610:DC
2378:.
2095:0.
1871:.
1177:.
20:,
4921:(
4906:B
4904:(
4864:(
4732:(
4717:(
4681:(
4651:(
4401:)
4279:e
4272:t
4265:v
4187:)
3911:)
3907:/
3903:(
3813:)
3795:e
3788:t
3781:v
3768:.
3753:.
3724:.
3697:.
3684::
3609:.
3606:x
3603:d
3596:|
3592:)
3589:x
3586:(
3583:f
3579:|
3575:+
3572:1
3566:|
3562:)
3559:x
3556:(
3553:f
3549:|
3540:I
3532:=
3526:f
3492:1
3486:p
3480:0
3458:p
3454:L
3429:,
3426:1
3420:p
3414:0
3393:)
3390:x
3387:d
3384:,
3380:R
3376:(
3371:p
3367:L
3321:X
3303:,
3300:Y
3294:X
3280:x
3278:(
3276:p
3272:x
3268:T
3264:C
3260:C
3245:R
3234:Y
3219:R
3204:X
3190:.
3187:Y
3181:X
3164:X
3147:.
3144:T
3115:T
3089:T
3070:,
3062:T
3034:)
3031:T
3028:(
3002:.
2999:Y
2993:X
2967:)
2964:T
2961:(
2942:T
2928:,
2925:Y
2919:X
2899:T
2879:)
2876:T
2873:(
2843:.
2840:)
2837:T
2834:(
2818:X
2804:.
2801:Y
2795:X
2789:)
2786:T
2783:(
2774::
2771:T
2751:,
2748:)
2745:T
2742:(
2716:Y
2696:X
2676:T
2557:X
2553:T
2549:T
2545:X
2524:,
2521:2
2518:,
2515:1
2512:=
2509:n
2483:n
2479:e
2472:n
2469:=
2466:)
2461:n
2457:e
2453:(
2450:T
2440:X
2422:1
2416:n
2408:(
2405:n
2402:)
2399:n
2395:e
2393:(
2387:f
2383:X
2376:Y
2360:0
2356:y
2333:0
2329:y
2325:)
2322:x
2319:(
2316:f
2313:=
2310:)
2307:x
2304:(
2301:g
2291:Y
2287:X
2283:g
2279:K
2275:X
2271:f
2267:Y
2263:X
2249:.
2245:C
2241:=
2238:K
2217:R
2213:=
2210:K
2200:K
2192:Y
2188:X
2173:f
2157:f
2142:R
2120:Q
2109:f
2092:=
2089:)
2083:(
2080:f
2069:n
2065:r
2063:(
2061:f
2057:n
2031:n
2028:}
2025:n
2021:r
2006:R
1995:f
1982:,
1979:0
1976:=
1973:)
1967:(
1964:f
1944:R
1937:R
1933::
1930:f
1855:X
1849:X
1823:X
1799:X
1775:T
1755:0
1752:=
1749:)
1746:0
1743:(
1740:T
1734:)
1729:n
1725:f
1721:(
1718:T
1692:n
1664:n
1661:=
1656:n
1652:)
1649:0
1641:2
1637:n
1633:(
1622:2
1618:n
1611:=
1608:)
1603:n
1599:f
1595:(
1592:T
1572:1
1566:n
1544:n
1540:)
1537:x
1532:2
1528:n
1524:(
1512:=
1509:)
1506:x
1503:(
1498:n
1494:f
1473:X
1452:)
1449:0
1446:(
1439:f
1435:=
1432:)
1429:f
1426:(
1423:T
1396:.
1392:|
1388:)
1385:x
1382:(
1379:f
1375:|
1369:]
1366:1
1363:,
1360:0
1357:[
1351:x
1343:=
1337:f
1306:X
1278:i
1274:e
1266:/
1259:)
1254:i
1250:e
1246:(
1243:T
1220:T
1198:i
1194:e
1175:Y
1171:X
1167:Y
1163:X
1159:Y
1155:X
1151:Y
1147:M
1140:X
1123:)
1120:x
1117:(
1114:f
1094:x
1074:)
1068:,
1065:)
1062:x
1059:(
1056:f
1053:(
1050:B
1030:)
1024:,
1021:x
1018:(
1015:B
1007:(
995:)
989:,
986:)
983:x
980:(
977:f
974:(
971:B
965:)
962:)
956:,
953:x
950:(
947:B
944:(
941:f
921:K
917:/
887:,
884:0
868:K
847:x
840:x
834:K
825:)
818:x
811:x
808:(
805:f
799:=
793:)
786:x
782:(
779:f
773:)
770:x
767:(
764:f
751:f
733:f
713:.
707:x
701:M
698:C
692:M
688:)
683:|
677:i
673:x
668:|
662:n
657:1
654:=
651:i
642:(
632:)
629:x
626:(
623:f
606:C
589:x
583:C
576:|
570:i
566:x
561:|
555:n
550:1
547:=
544:i
519:,
516:}
510:)
505:i
501:e
497:(
494:f
488:{
483:i
475:=
472:M
452:.
446:)
441:i
437:e
433:(
430:f
423:|
417:i
413:x
408:|
402:n
397:1
394:=
391:i
375:)
370:i
366:e
362:(
359:f
354:i
350:x
344:n
339:1
336:=
333:i
320:=
314:)
311:x
308:(
305:f
278:,
275:)
270:i
266:e
262:(
259:f
254:i
250:x
244:n
239:1
236:=
233:i
225:=
222:)
219:x
216:(
213:f
203:X
188:)
182:n
178:e
174:,
168:,
163:2
159:e
155:,
150:1
146:e
141:(
126:X
122:Y
118:X
104:Y
98:X
95::
92:f
82:Y
78:X
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