6121:
266:
the weak topology and the weak* topology, thereby making redundant the need for many definitions, theorem statements, and proofs. This is also the reason why the weak* topology is also frequently referred to as the "weak topology"; because it is just an instance of the weak topology in the setting of
4643:
3295:
4508:
2031:
1905:
4535:
2940:
2270:
1351:
3115:
3916:
4790:
4414:
1773:
1175:
1046:
6157:
3706:
2852:
1269:
1716:
1114:
5243:
3853:
2982:
2190:
1965:
5131:
3496:
3438:
5314:
2621:
3779:
2895:
2479:
is open in the weak topology if and only if it can be written as a union of (possibly infinitely many) sets, each of which is an intersection of finitely many sets of the form
4882:
4353:
3605:
1837:
2727:
2573:
2447:
3340:
2516:
2414:
6010:
4705:
3567:
3545:
3193:
2694:
2672:
2473:
1677:
1578:
1552:
1232:
968:
943:
918:
827:
756:
721:
611:
501:
432:
375:
340:
310:
234:
196:
4824:
2650:
2056:
1294:
3735:
1799:
1615:" to refer to the original topology since these already have well-known meanings, so using them may cause confusion). We may define a possibly different topology on
1201:
6150:
4298:
4255:
4228:
4155:
4093:
4058:
4023:
3799:
3652:
3632:
3523:
3387:
3149:
3036:
2792:
2757:
2364:
2333:
2217:
2148:
2120:
2083:
1647:
1078:
4180:
holds. In particular, a subset of the continuous dual is weak* compact if and only if it is weak* closed and norm-bounded. This implies, in particular, that when
5673:
4188:
does not contain any weak* neighborhood of 0 (since any such neighborhood is norm-unbounded). Thus, even though norm-closed balls are compact, X* is not weak*
6143:
5836:
4426:
5006:
may carry a variety of different possible topologies. The naming of such topologies depends on the kind of topology one is using on the target space
5963:
5818:
5794:
3001:
5643:
5597:
5567:
5533:
5460:
6404:
5525:
4638:{\displaystyle \int _{\mathbb {R} ^{n}}{\bar {\psi }}_{k}f\,\mathrm {d} \mu \to \int _{\mathbb {R} ^{n}}{\bar {\psi }}f\,\mathrm {d} \mu }
4884:). In an alternative construction of such spaces, one can take the weak dual of a space of test functions inside a Hilbert space such as
6220:
5309:
171:
2903:
5686:
5479:
5775:
5666:
5624:
5506:
5442:
5401:
2222:
1974:
262:, which we now describe. The benefit of this more general construction is that any definition or result proved for it applies to
6287:
6270:
6045:
5294:
2534:
30:
This article is about the weak topology on a normed vector space. For the weak topology induced by a general family of maps, see
5690:
3044:
6260:
1842:
1680:
3874:
4723:
5841:
5559:
4361:
1299:
5897:
1001:
6124:
5846:
5831:
5659:
5861:
142:
did not elevate norm convergence above weak convergence and oftentimes viewed weak convergence as preferable. In 1929,
5299:
6315:
6106:
5866:
4834:
4685:} is bounded). For given test functions, the relevant notion of convergence only corresponds to the topology used in
4026:
3660:
2806:
6060:
5984:
4990:
4852:
4846:
1689:
1087:
6101:
6394:
6336:
5917:
4118:
be locally compact valued field (e.g., the reals, the complex numbers, or any of the p-adic number systems). Let
6320:
5851:
1721:
1123:
6292:
5953:
5754:
5182:
5040:
2336:
1529:
1237:
54:
5826:
3807:
2996:
is equipped with the weak topology, then addition and scalar multiplication remain continuous operations, and
4855:
by forming the strong dual of a space of test functions (such as the compactly supported smooth functions on
2951:
2156:
1931:
6050:
5304:
5276:
5085:
3975:
topological space. However, for infinite-dimensional spaces, the metric cannot be translation-invariant. If
3450:
6265:
6081:
6025:
5989:
5272:
4953:
3738:
3392:
2760:
2590:
6200:
4189:
3940:
3744:
3155:. The uniform and strong topologies are generally different for other spaces of linear maps; see below.
1620:
1588:
211:
70:
5585:
2867:
4858:
4329:
3583:
6299:
6064:
5524:. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY:
4891:
4184:
is an infinite-dimensional normed space then the closed unit ball at the origin in the dual space of
3290:{\displaystyle x\mapsto {\begin{cases}T_{x}:X^{*}\to \mathbb {K} \\T_{x}(\phi )=\phi (x)\end{cases}}}
1365:
984:
888:
3208:
2699:
2545:
2419:
6373:
6230:
6205:
6030:
5968:
5682:
4177:
3303:
3152:
2482:
2380:
2367:
258:
Both the weak topology and the weak* topology are special cases of a more general construction for
203:
139:
111:
74:
66:
65:. The term is most commonly used for the initial topology of a topological vector space (such as a
4688:
3550:
3528:
2677:
2655:
2456:
1660:
1561:
1535:
951:
926:
901:
810:
739:
704:
594:
484:
415:
358:
323:
293:
217:
179:
6250:
6055:
5922:
5551:
5288:
1804:
4802:
6399:
6282:
6035:
5639:
5620:
5603:
5593:
5573:
5563:
5539:
5529:
5502:
5485:
5475:
5456:
5438:
5397:
5017:
4796:
4317:
Consider, for example, the difference between strong and weak convergence of functions in the
3578:
2626:
2540:
1650:
1584:
207:
199:
119:
35:
17:
1206:
6352:
6040:
5958:
5927:
5907:
5892:
5887:
5882:
5719:
5396:, Graduate Texts in Mathematics, vol. 183, New York: Springer-Verlag, pp. xx+596,
5389:
3714:
2305:
1778:
1608:
1470:
1180:
50:
31:
4276:
4233:
4206:
4133:
4071:
4036:
4001:
3784:
3637:
3610:
3501:
3365:
3127:
3014:
2770:
2735:
2342:
2311:
2195:
2126:
2098:
2061:
1625:
1051:
6277:
6170:
5902:
5856:
5804:
5799:
5770:
5651:
4108:
3956:
3355:
1612:
237:
147:
73:. The remainder of this article will deal with this case, which is one of the concepts of
58:
5729:
4230:
is metrizable, in which case the weak* topology is metrizable on norm-bounded subsets of
2036:
1274:
6215:
6091:
5943:
5744:
5319:
4911:
3983:
3980:
3960:
3170:
3164:
2522:
946:
759:
6388:
6238:
6195:
6020:
5749:
5734:
5724:
5519:
4673:
4318:
4061:
143:
131:
62:
6135:
96:, etc.) with respect to the weak topology. Likewise, functions are sometimes called
6086:
5739:
5709:
5562:. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer.
5515:
5016:, IV.7 Topologies of linear maps). There are, in general, a vast array of possible
4663:
4503:{\displaystyle \int _{\mathbb {R} ^{n}}|\psi _{k}-\psi |^{2}\,{\rm {d}}\mu \,\to 0}
4270:
1580:
379:
135:
4261:
has a dual space that is separable (with respect to the dual-norm topology) then
6190:
6166:
6015:
6005:
5912:
5714:
3181:
2151:
1926:
1429:
1081:
921:
631:
287:
259:
253:
241:
146:
introduced weak convergence for normed spaces and also introduced the analogous
98:
93:
42:
6185:
5948:
5788:
5784:
5780:
3972:
3347:
115:
89:
5607:
5577:
5489:
5474:. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press.
5069:
is a normed space, then this topology is defined by the seminorms indexed by
6357:
6243:
6210:
5543:
4065:
3343:
4203:
is separable if and only if the weak* topology on the closed unit ball of
4322:
790:
1372:
then we will assume that they are associated with the canonical pairing
5359:
5357:
5355:
2374:
5353:
5351:
5349:
5347:
5345:
5343:
5341:
5339:
5337:
5335:
3998:
By definition, the weak* topology is weaker than the weak topology on
4099:
is weak*-compact). Moreover, the closed unit ball in a normed space
3440:. In other words, it is the coarsest topology such that the maps
2935:{\displaystyle x_{n}{\overset {\mathrm {w} }{\longrightarrow }}x}
2539:
The weak topology is characterized by the following condition: a
6139:
5655:
3986:
space then the weak* topology on the continuous dual space of
2265:{\displaystyle \langle x,\cdot \rangle :X^{*}\to \mathbb {K} }
2026:{\displaystyle x'=\langle \cdot ,x'\rangle :X\to \mathbb {K} }
138:
made extensive use of weak convergence. The early pioneers of
34:. For the weak topology generated by a cover of a space, see
4520:. Here the notion of convergence corresponds to the norm on
3967:
is a norm-bounded subset of its continuous dual space, then
2377:
for the weak topology is the collection of sets of the form
4169:, all norm-closed balls are compact in the weak* topology.
3283:
2521:
From this point of view, the weak topology is the coarsest
630:
is now automatically defined as described in the article
4944:
is contained in a finite-dimensional vector subspace of
3110:{\displaystyle \|\phi \|=\sup _{\|x\|\leq 1}|\phi (x)|.}
5453:
Real
Analysis: Modern Techniques and Their Applications
1900:{\displaystyle \langle \cdot ,x'\rangle =x'(\cdot )=x'}
3911:{\displaystyle \phi _{n}{\overset {w^{*}}{\to }}\phi }
5315:
Topologies on the set of operators on a
Hilbert space
5185:
5088:
4861:
4805:
4785:{\displaystyle \psi _{k}(x)={\sqrt {2/\pi }}\sin(kx)}
4726:
4691:
4538:
4429:
4364:
4332:
4279:
4236:
4209:
4136:
4074:
4039:
4004:
3877:
3810:
3787:
3747:
3717:
3663:
3640:
3613:
3586:
3553:
3531:
3504:
3453:
3395:
3368:
3306:
3196:
3130:
3047:
3017:
2954:
2906:
2870:
2809:
2773:
2738:
2702:
2680:
2658:
2629:
2593:
2548:
2485:
2459:
2422:
2383:
2345:
2314:
2225:
2198:
2159:
2129:
2101:
2064:
2039:
1977:
1934:
1845:
1807:
1781:
1724:
1692:
1663:
1628:
1564:
1538:
1302:
1277:
1240:
1209:
1183:
1126:
1090:
1054:
1004:
954:
929:
904:
813:
742:
707:
597:
487:
418:
361:
326:
296:
220:
210:
such that addition, multiplication, and division are
182:
5590:
Topological Vector Spaces, Distributions and
Kernels
4918:(i.e. the vector space of all linear functionals on
4409:{\displaystyle \psi _{k}\in L^{2}(\mathbb {R} ^{n})}
4025:. An important fact about the weak* topology is the
1346:{\displaystyle \langle \cdot ,x'\rangle =x'(\cdot )}
6366:
6345:
6329:
6308:
6229:
6178:
6074:
5998:
5977:
5936:
5875:
5817:
5763:
5698:
1041:{\displaystyle (X,Y,\langle \cdot ,\cdot \rangle )}
80:One may call subsets of a topological vector space
6011:Spectral theory of ordinary differential equations
5237:
5125:
4876:
4818:
4784:
4699:
4637:
4502:
4408:
4347:
4292:
4249:
4222:
4149:
4087:
4052:
4017:
3910:
3847:
3793:
3773:
3729:
3700:
3654:in the weak-* topology if it converges pointwise:
3646:
3626:
3599:
3561:
3539:
3517:
3490:
3432:
3381:
3334:
3289:
3143:
3109:
3038:is itself a normed vector space by using the norm
3030:
2976:
2934:
2889:
2846:
2786:
2751:
2721:
2688:
2666:
2644:
2615:
2567:
2510:
2467:
2441:
2408:
2358:
2327:
2292:Weak topology induced by the continuous dual space
2264:
2211:
2184:
2142:
2114:
2077:
2050:
2025:
1959:
1899:
1831:
1793:
1767:
1710:
1671:
1641:
1607:(the reader is cautioned against using the terms "
1572:
1546:
1345:
1288:
1263:
1226:
1195:
1169:
1108:
1072:
1040:
962:
937:
912:
821:
750:
715:
605:
495:
426:
369:
334:
304:
228:
190:
5363:
4273:, the weak* topology is not metrizable on all of
3971:endowed with the weak* (subspace) topology is a
3169:The weak* topology is an important example of a
3061:
5455:(Second ed.). John Wiley & Sons, Inc.
4529:In contrast weak convergence only demands that
4103:is compact in the weak topology if and only if
3701:{\displaystyle \phi _{\lambda }(x)\to \phi (x)}
3120:This norm gives rise to a topology, called the
5470:Narici, Lawrence; Beckenstein, Edward (2011).
2847:{\displaystyle \varphi (x_{n})\to \varphi (x)}
2579:converges in the weak topology to the element
991:(i.e. a vector space of linear functionals on
6151:
5667:
4926:is endowed with the weak topology induced by
3389:is the weak topology induced by the image of
3354:canonical embedding is surjective are called
1711:{\displaystyle \langle \cdot ,\cdot \rangle }
1109:{\displaystyle \langle \cdot ,\cdot \rangle }
8:
5111:
5095:
4890:. Thus one is led to consider the idea of a
3071:
3065:
3054:
3048:
2238:
2226:
2179:
2160:
2006:
1989:
1954:
1935:
1863:
1846:
1742:
1725:
1705:
1693:
1320:
1303:
1258:
1241:
1144:
1127:
1103:
1091:
1032:
1020:
157:
151:
4898:Weak topology induced by the algebraic dual
3931:Weak-* convergence is sometimes called the
1718:is the canonical evaluation map defined by
1488:is a vector space of linear functionals on
122:, etc.) with respect to the weak topology.
6158:
6144:
6136:
5702:
5674:
5660:
5652:
5176:defining the strong topology are given by
5035:, whose naming is not entirely intuitive.
4833:does not exist. On the other hand, by the
4122:be a normed topological vector space over
1768:{\displaystyle \langle x,x'\rangle =x'(x)}
1591:are continuous. We call the topology that
1170:{\displaystyle \langle x,x'\rangle =x'(x)}
634:. However, for clarity, we now repeat it.
290:of vector spaces over a topological field
5238:{\displaystyle p_{q,x}:f\mapsto q(f(x)),}
5190:
5184:
5136:More generally, if a family of seminorms
5114:
5087:
4974:are topological vector spaces, the space
4868:
4864:
4863:
4860:
4810:
4804:
4754:
4749:
4731:
4725:
4693:
4692:
4690:
4627:
4626:
4612:
4611:
4603:
4599:
4598:
4596:
4581:
4580:
4571:
4560:
4559:
4550:
4546:
4545:
4543:
4537:
4493:
4484:
4483:
4482:
4476:
4471:
4458:
4449:
4441:
4437:
4436:
4434:
4428:
4397:
4393:
4392:
4382:
4369:
4363:
4339:
4335:
4334:
4331:
4284:
4278:
4241:
4235:
4214:
4208:
4141:
4135:
4079:
4073:
4044:
4038:
4009:
4003:
3897:
3888:
3882:
3876:
3815:
3809:
3786:
3765:
3752:
3746:
3716:
3668:
3662:
3639:
3618:
3612:
3591:
3585:
3555:
3554:
3552:
3533:
3532:
3530:
3509:
3503:
3458:
3452:
3421:
3394:
3373:
3367:
3323:
3305:
3250:
3238:
3237:
3228:
3215:
3203:
3195:
3135:
3129:
3099:
3082:
3064:
3046:
3022:
3016:
2959:
2953:
2922:
2917:
2911:
2905:
2897:. In this case, it is customary to write
2881:
2869:
2820:
2808:
2778:
2772:
2743:
2737:
2713:
2701:
2682:
2681:
2679:
2660:
2659:
2657:
2628:
2604:
2592:
2556:
2547:
2490:
2484:
2461:
2460:
2458:
2433:
2421:
2388:
2382:
2350:
2344:
2319:
2313:
2258:
2257:
2248:
2224:
2203:
2197:
2192:. That is, it is the weakest topology on
2173:
2158:
2134:
2128:
2106:
2100:
2069:
2063:
2038:
2019:
2018:
1976:
1967:. That is, it is the weakest topology on
1948:
1933:
1844:
1823:
1806:
1780:
1723:
1691:
1665:
1664:
1662:
1633:
1627:
1566:
1565:
1563:
1540:
1539:
1537:
1301:
1276:
1264:{\displaystyle \langle \cdot ,x'\rangle }
1239:
1208:
1182:
1125:
1089:
1053:
1003:
956:
955:
953:
931:
930:
928:
906:
905:
903:
815:
814:
812:
744:
743:
741:
709:
708:
706:
599:
598:
596:
489:
488:
486:
420:
419:
417:
363:
362:
360:
328:
327:
325:
298:
297:
295:
222:
221:
219:
184:
183:
181:
5964:Group algebra of a locally compact group
4126:, compatible with the absolute value in
4033:is normed, then the closed unit ball in
3848:{\displaystyle \phi _{n}(x)\to \phi (x)}
2339:topology on X such that each element of
1440:) with respect to the canonical pairing
5375:
5331:
4799:. In particular, the (strong) limit of
3346:linear mapping, though not necessarily
3011:is a normed space, then the dual space
3002:locally convex topological vector space
2977:{\displaystyle x_{n}\rightharpoonup x.}
2288:We give alternative definitions below.
2185:{\displaystyle \langle X,X^{*}\rangle }
1960:{\displaystyle \langle X,X^{*}\rangle }
979:We now consider the special case where
248:Weak topology with respect to a pairing
5416:
5394:An introduction to Banach space theory
5126:{\displaystyle f\mapsto \|f(x)\|_{Y}.}
5013:
887:. This shows that weak topologies are
5592:. Mineola, N.Y.: Dover Publications.
4837:, the weak limit exists and is zero.
3491:{\displaystyle T_{x}(\phi )=\phi (x)}
1513:
7:
5526:McGraw-Hill Science/Engineering/Math
5291:, a compact set in the weak topology
4176:is a normed space, a version of the
3959:(i.e. has a countable dense subset)
3433:{\displaystyle T:T(X)\subset X^{**}}
2453:is an open subset of the base field
1683:with respect to the given topology.
6221:Topologies on spaces of linear maps
5387:Proposition 2.6.12, p. 226 in
5310:Topologies on spaces of linear maps
4358:. Strong convergence of a sequence
2616:{\displaystyle \phi (x_{\lambda })}
2280:. This topology is also called the
172:Topologies on spaces of linear maps
150:. The weak topology is also called
49:is an alternative term for certain
5615:Willard, Stephen (February 2004).
4930:then the continuous dual space of
4710:For example, in the Hilbert space
4628:
4582:
4485:
3774:{\displaystyle \phi _{n}\in X^{*}}
2923:
25:
5437:(2nd ed.), Springer-Verlag,
2890:{\displaystyle \varphi \in X^{*}}
6120:
6119:
6046:Topological quantum field theory
5295:Weak convergence (Hilbert space)
5012:to define operator convergence (
4877:{\displaystyle \mathbb {R} ^{n}}
4348:{\displaystyle \mathbb {R} ^{n}}
3939:. Indeed, it coincides with the
3600:{\displaystyle \phi _{\lambda }}
2535:Weak convergence (Hilbert space)
1271:is just another way of denoting
435:denote the linear functional on
267:this more general construction.
254:Dual system § Weak topology
5435:A Course in Functional Analysis
4851:One normally obtains spaces of
998:There is a pairing, denoted by
898:We will henceforth assume that
5619:. Courier Dover Publications.
5229:
5226:
5220:
5214:
5208:
5107:
5101:
5092:
4779:
4770:
4743:
4737:
4617:
4589:
4565:
4494:
4472:
4450:
4403:
4388:
4165:-valued linear functionals on
3890:
3842:
3836:
3830:
3827:
3821:
3695:
3689:
3683:
3680:
3674:
3485:
3479:
3470:
3464:
3411:
3405:
3316:
3277:
3271:
3262:
3256:
3234:
3200:
3100:
3096:
3090:
3083:
2965:
2919:
2841:
2835:
2829:
2826:
2813:
2722:{\displaystyle \phi \in X^{*}}
2639:
2633:
2610:
2597:
2568:{\displaystyle (x_{\lambda })}
2562:
2549:
2505:
2499:
2475:. In other words, a subset of
2442:{\displaystyle \phi \in X^{*}}
2403:
2397:
2254:
2015:
1883:
1877:
1762:
1756:
1492:, then the continuous dual of
1340:
1334:
1164:
1158:
1067:
1055:
1035:
1005:
244:with the familiar topologies.
166:The weak and strong topologies
18:Weak topology (polar topology)
1:
5842:Uniform boundedness principle
5364:Narici & Beckenstein 2011
4265:is necessarily separable. If
4157:, the topological dual space
3335:{\displaystyle T:X\to X^{**}}
2511:{\displaystyle \phi ^{-1}(U)}
2409:{\displaystyle \phi ^{-1}(U)}
1520:The weak and weak* topologies
1496:with respect to the topology
656:) is the weakest topology on
546:) is the weakest topology on
130:Starting in the early 1900s,
5554:; Wolff, Manfred P. (1999).
5305:Weak convergence of measures
4717:, the sequence of functions
4700:{\displaystyle \mathbb {C} }
3562:{\displaystyle \mathbb {C} }
3540:{\displaystyle \mathbb {R} }
2689:{\displaystyle \mathbb {C} }
2667:{\displaystyle \mathbb {R} }
2468:{\displaystyle \mathbb {K} }
2335:. In other words, it is the
1672:{\displaystyle \mathbb {K} }
1587:so that vector addition and
1573:{\displaystyle \mathbb {K} }
1547:{\displaystyle \mathbb {K} }
1364:is a vector subspace of the
983:is a vector subspace of the
963:{\displaystyle \mathbb {C} }
938:{\displaystyle \mathbb {R} }
913:{\displaystyle \mathbb {K} }
822:{\displaystyle \mathbb {R} }
789:is induced by the family of
751:{\displaystyle \mathbb {K} }
716:{\displaystyle \mathbb {K} }
606:{\displaystyle \mathbb {K} }
496:{\displaystyle \mathbb {K} }
427:{\displaystyle \mathbb {K} }
370:{\displaystyle \mathbb {K} }
335:{\displaystyle \mathbb {K} }
305:{\displaystyle \mathbb {K} }
236:will be either the field of
229:{\displaystyle \mathbb {K} }
191:{\displaystyle \mathbb {K} }
6405:Topology of function spaces
5300:Weak-star operator topology
4991:continuous linear operators
4948:, every vector subspace of
3994:Properties on normed spaces
3868:. In this case, one writes
2308:with respect to the family
1832:{\displaystyle x'\in X^{*}}
6421:
5985:Invariant subspace problem
5638:(6th ed.), Springer,
4940:, every bounded subset of
4847:distribution (mathematics)
4844:
4095:of a neighborhood of 0 in
3162:
3151:. This is the topology of
2532:
1234:. Note in particular that
251:
169:
29:
6337:Transpose of a linear map
6115:
5705:
5556:Topological Vector Spaces
5472:Topological Vector Spaces
4819:{\displaystyle \psi _{k}}
4658:(or, more typically, all
3180:can be embedded into its
1619:using the topological or
769:, then the weak topology
55:topological vector spaces
5954:Spectrum of a C*-algebra
5433:Conway, John B. (1994),
5140:defines the topology on
5041:strong operator topology
4199:is a normed space, then
4114:In more generality, let
2645:{\displaystyle \phi (x)}
2123:is the weak topology on
1921:is the weak topology on
1649:, which consists of all
1530:topological vector space
1118:canonical evaluation map
6051:Noncommutative geometry
5634:Yosida, Kosaku (1980),
5497:Pedersen, Gert (1989),
5277:weak* operator topology
5271:In particular, see the
4304:is finite-dimensional.
3943:of linear functionals.
1839:, where in particular,
1227:{\displaystyle x'\in Y}
320:are vector spaces over
214:. In most applications
6107:TomitaâTakesaki theory
6082:Approximation property
6026:Calculus of variations
5451:Folland, G.B. (1999).
5273:weak operator topology
5239:
5127:
4954:topological complement
4906:is a vector space and
4878:
4835:RiemannâLebesgue lemma
4820:
4786:
4701:
4639:
4504:
4410:
4349:
4294:
4251:
4224:
4151:
4089:
4054:
4027:BanachâAlaoglu theorem
4019:
3912:
3849:
3795:
3775:
3731:
3730:{\displaystyle x\in X}
3702:
3648:
3628:
3601:
3563:
3541:
3519:
3492:
3434:
3383:
3336:
3291:
3145:
3111:
3032:
2978:
2936:
2891:
2848:
2788:
2753:
2723:
2690:
2668:
2646:
2617:
2569:
2512:
2469:
2443:
2410:
2360:
2329:
2266:
2213:
2186:
2144:
2116:
2079:
2052:
2027:
1961:
1901:
1833:
1795:
1794:{\displaystyle x\in X}
1769:
1712:
1673:
1643:
1574:
1548:
1508:is precisely equal to
1347:
1290:
1265:
1228:
1197:
1196:{\displaystyle x\in X}
1171:
1110:
1074:
1042:
964:
939:
914:
823:
752:
717:
607:
497:
428:
371:
336:
306:
230:
192:
158:
152:
69:) with respect to its
6102:BanachâMazur distance
6065:Generalized functions
5240:
5146:, then the seminorms
5128:
5061:pointwise convergence
4879:
4821:
4787:
4702:
4640:
4505:
4411:
4350:
4295:
4293:{\displaystyle X^{*}}
4252:
4250:{\displaystyle X^{*}}
4225:
4223:{\displaystyle X^{*}}
4152:
4150:{\displaystyle X^{*}}
4090:
4088:{\displaystyle X^{*}}
4064:(more generally, the
4055:
4053:{\displaystyle X^{*}}
4020:
4018:{\displaystyle X^{*}}
3941:pointwise convergence
3937:pointwise convergence
3913:
3850:
3796:
3794:{\displaystyle \phi }
3776:
3732:
3703:
3649:
3647:{\displaystyle \phi }
3629:
3627:{\displaystyle X^{*}}
3602:
3564:
3542:
3520:
3518:{\displaystyle X^{*}}
3493:
3435:
3384:
3382:{\displaystyle X^{*}}
3337:
3292:
3146:
3144:{\displaystyle X^{*}}
3112:
3033:
3031:{\displaystyle X^{*}}
2979:
2937:
2892:
2849:
2789:
2787:{\displaystyle x_{n}}
2754:
2752:{\displaystyle x_{n}}
2724:
2691:
2669:
2647:
2618:
2570:
2533:Further information:
2513:
2470:
2444:
2411:
2361:
2359:{\displaystyle X^{*}}
2330:
2328:{\displaystyle X^{*}}
2267:
2214:
2212:{\displaystyle X^{*}}
2187:
2145:
2143:{\displaystyle X^{*}}
2117:
2115:{\displaystyle X^{*}}
2080:
2078:{\displaystyle X^{*}}
2053:
2028:
1962:
1902:
1834:
1796:
1770:
1713:
1674:
1644:
1642:{\displaystyle X^{*}}
1621:continuous dual space
1589:scalar multiplication
1575:
1549:
1348:
1291:
1266:
1229:
1198:
1172:
1111:
1075:
1073:{\displaystyle (X,Y)}
1043:
965:
940:
915:
824:
753:
718:
626:The weak topology on
608:
498:
458:. Similarly, for all
429:
372:
337:
307:
231:
193:
104:weakly differentiable
5847:Kakutani fixed-point
5832:Riesz representation
5390:Megginson, Robert E.
5183:
5086:
4952:is closed and has a
4892:rigged Hilbert space
4859:
4803:
4724:
4689:
4536:
4427:
4362:
4330:
4277:
4257:. If a normed space
4234:
4207:
4134:
4072:
4037:
4002:
3875:
3808:
3785:
3745:
3715:
3661:
3638:
3611:
3584:
3551:
3529:
3502:
3451:
3393:
3366:
3304:
3194:
3128:
3045:
3015:
2952:
2904:
2868:
2807:
2771:
2736:
2700:
2678:
2656:
2627:
2591:
2546:
2483:
2457:
2420:
2381:
2343:
2312:
2223:
2196:
2157:
2150:with respect to the
2127:
2099:
2062:
2037:
1975:
1932:
1925:with respect to the
1843:
1805:
1779:
1722:
1690:
1661:
1657:into the base field
1626:
1562:
1536:
1366:algebraic dual space
1300:
1275:
1238:
1207:
1181:
1124:
1088:
1052:
1002:
985:algebraic dual space
952:
927:
902:
811:
740:
705:
595:
485:
416:
359:
324:
294:
218:
180:
110:, etc.) if they are
88:, etc.) if they are
61:, for instance on a
6374:Biorthogonal system
6206:Operator topologies
6031:Functional calculus
5990:Mahler's conjecture
5969:Von Neumann algebra
5683:Functional analysis
5636:Functional analysis
5552:Schaefer, Helmut H.
5521:Functional Analysis
5419:, pp. 36, 201.
5366:, pp. 225â273.
5063:. For instance, if
5059:is the topology of
5018:operator topologies
4960:Operator topologies
4676:, if the sequence {
4672:such as a space of
4178:Heine-Borel theorem
3737:. In particular, a
3569:remain continuous.
3153:uniform convergence
2795:converges weakly to
2368:continuous function
2296:Alternatively, the
1084:whose bilinear map
140:functional analysis
75:functional analysis
67:normed vector space
6056:Riemann hypothesis
5755:Topological vector
5289:Eberlein compactum
5235:
5123:
4874:
4816:
4782:
4697:
4648:for all functions
4635:
4500:
4406:
4345:
4290:
4247:
4220:
4147:
4085:
4050:
4015:
3933:simple convergence
3908:
3845:
3791:
3771:
3727:
3698:
3644:
3624:
3597:
3573:Weak-* convergence
3559:
3537:
3525:to the base field
3515:
3488:
3430:
3379:
3350:(spaces for which
3332:
3287:
3282:
3141:
3107:
3081:
3028:
2974:
2932:
2887:
2844:
2784:
2749:
2732:In particular, if
2719:
2686:
2664:
2642:
2613:
2565:
2508:
2465:
2439:
2406:
2356:
2325:
2262:
2209:
2182:
2140:
2112:
2075:
2051:{\displaystyle x'}
2048:
2023:
1957:
1897:
1829:
1791:
1765:
1708:
1669:
1651:linear functionals
1639:
1570:
1544:
1390:In this case, the
1343:
1289:{\displaystyle x'}
1286:
1261:
1224:
1193:
1167:
1106:
1070:
1038:
960:
935:
910:
819:
748:
713:
688:, making all maps
603:
578:, making all maps
493:
424:
367:
332:
302:
226:
188:
159:schwache Topologie
148:weak-* convergence
51:initial topologies
6382:
6381:
6271:in Hilbert spaces
6133:
6132:
6036:Integral operator
5813:
5812:
5645:978-3-540-58654-8
5599:978-0-486-45352-1
5569:978-1-4612-7155-0
5535:978-0-07-054236-5
5462:978-0-471-31716-6
5038:For example, the
4797:orthonormal basis
4762:
4620:
4568:
3903:
3634:is convergent to
3060:
2927:
2152:canonical pairing
2095:weak topology on
1927:canonical pairing
1916:weak topology on
1399:weak topology on
1392:weak topology on
1082:canonical pairing
975:Canonical duality
643:weak topology on
533:weak topology on
200:topological field
99:weakly continuous
36:coherent topology
27:Mathematical term
16:(Redirected from
6412:
6395:General topology
6353:Saturated family
6251:Ultraweak/Weak-*
6160:
6153:
6146:
6137:
6123:
6122:
6041:Jones polynomial
5959:Operator algebra
5703:
5676:
5669:
5662:
5653:
5648:
5630:
5617:General Topology
5611:
5586:Trèves, François
5581:
5547:
5511:
5493:
5466:
5447:
5420:
5414:
5408:
5406:
5385:
5379:
5373:
5367:
5361:
5267:
5257:
5244:
5242:
5241:
5236:
5201:
5200:
5175:
5160:
5145:
5132:
5130:
5129:
5124:
5119:
5118:
5078:
5068:
5058:
5034:
5011:
5005:
4988:
4973:
4967:
4951:
4947:
4943:
4939:
4933:
4925:
4921:
4917:
4905:
4889:
4883:
4881:
4880:
4875:
4873:
4872:
4867:
4832:
4825:
4823:
4822:
4817:
4815:
4814:
4791:
4789:
4788:
4783:
4763:
4758:
4750:
4736:
4735:
4716:
4706:
4704:
4703:
4698:
4696:
4671:
4657:
4644:
4642:
4641:
4636:
4631:
4622:
4621:
4613:
4610:
4609:
4608:
4607:
4602:
4585:
4576:
4575:
4570:
4569:
4561:
4557:
4556:
4555:
4554:
4549:
4525:
4519:
4509:
4507:
4506:
4501:
4489:
4488:
4481:
4480:
4475:
4463:
4462:
4453:
4448:
4447:
4446:
4445:
4440:
4419:
4415:
4413:
4412:
4407:
4402:
4401:
4396:
4387:
4386:
4374:
4373:
4357:
4354:
4352:
4351:
4346:
4344:
4343:
4338:
4303:
4299:
4297:
4296:
4291:
4289:
4288:
4268:
4264:
4260:
4256:
4254:
4253:
4248:
4246:
4245:
4229:
4227:
4226:
4221:
4219:
4218:
4202:
4198:
4187:
4183:
4175:
4168:
4164:
4160:
4156:
4154:
4153:
4148:
4146:
4145:
4129:
4125:
4121:
4117:
4106:
4102:
4098:
4094:
4092:
4091:
4086:
4084:
4083:
4059:
4057:
4056:
4051:
4049:
4048:
4032:
4024:
4022:
4021:
4016:
4014:
4013:
3989:
3978:
3954:
3927:
3917:
3915:
3914:
3909:
3904:
3902:
3901:
3889:
3887:
3886:
3867:
3854:
3852:
3851:
3846:
3820:
3819:
3800:
3798:
3797:
3792:
3780:
3778:
3777:
3772:
3770:
3769:
3757:
3756:
3736:
3734:
3733:
3728:
3707:
3705:
3704:
3699:
3673:
3672:
3653:
3651:
3650:
3645:
3633:
3631:
3630:
3625:
3623:
3622:
3606:
3604:
3603:
3598:
3596:
3595:
3568:
3566:
3565:
3560:
3558:
3546:
3544:
3543:
3538:
3536:
3524:
3522:
3521:
3516:
3514:
3513:
3497:
3495:
3494:
3489:
3463:
3462:
3439:
3437:
3436:
3431:
3429:
3428:
3388:
3386:
3385:
3380:
3378:
3377:
3341:
3339:
3338:
3333:
3331:
3330:
3296:
3294:
3293:
3288:
3286:
3285:
3255:
3254:
3241:
3233:
3232:
3220:
3219:
3179:
3150:
3148:
3147:
3142:
3140:
3139:
3116:
3114:
3113:
3108:
3103:
3086:
3080:
3037:
3035:
3034:
3029:
3027:
3026:
3010:
2999:
2995:
2988:Other properties
2983:
2981:
2980:
2975:
2964:
2963:
2941:
2939:
2938:
2933:
2928:
2926:
2918:
2916:
2915:
2896:
2894:
2893:
2888:
2886:
2885:
2863:
2853:
2851:
2850:
2845:
2825:
2824:
2799:
2793:
2791:
2790:
2785:
2783:
2782:
2766:
2758:
2756:
2755:
2750:
2748:
2747:
2728:
2726:
2725:
2720:
2718:
2717:
2695:
2693:
2692:
2687:
2685:
2673:
2671:
2670:
2665:
2663:
2651:
2649:
2648:
2643:
2622:
2620:
2619:
2614:
2609:
2608:
2586:
2582:
2578:
2574:
2572:
2571:
2566:
2561:
2560:
2529:Weak convergence
2517:
2515:
2514:
2509:
2498:
2497:
2478:
2474:
2472:
2471:
2466:
2464:
2452:
2448:
2446:
2445:
2440:
2438:
2437:
2415:
2413:
2412:
2407:
2396:
2395:
2365:
2363:
2362:
2357:
2355:
2354:
2334:
2332:
2331:
2326:
2324:
2323:
2306:initial topology
2303:
2279:
2275:
2271:
2269:
2268:
2263:
2261:
2253:
2252:
2219:making all maps
2218:
2216:
2215:
2210:
2208:
2207:
2191:
2189:
2188:
2183:
2178:
2177:
2149:
2147:
2146:
2141:
2139:
2138:
2121:
2119:
2118:
2113:
2111:
2110:
2084:
2082:
2081:
2076:
2074:
2073:
2057:
2055:
2054:
2049:
2047:
2032:
2030:
2029:
2024:
2022:
2005:
1985:
1971:making all maps
1970:
1966:
1964:
1963:
1958:
1953:
1952:
1924:
1919:
1906:
1904:
1903:
1898:
1896:
1876:
1862:
1838:
1836:
1835:
1830:
1828:
1827:
1815:
1800:
1798:
1797:
1792:
1774:
1772:
1771:
1766:
1755:
1741:
1717:
1715:
1714:
1709:
1678:
1676:
1675:
1670:
1668:
1656:
1648:
1646:
1645:
1640:
1638:
1637:
1618:
1609:initial topology
1595:starts with the
1594:
1583:equipped with a
1579:
1577:
1576:
1571:
1569:
1557:
1553:
1551:
1550:
1545:
1543:
1527:
1516:, Theorem 3.10)
1511:
1507:
1495:
1491:
1487:
1480:
1477:with respect to
1476:
1471:initial topology
1468:
1453:
1452:
1439:
1435:
1427:
1415:
1395:
1385:
1384:
1371:
1363:
1352:
1350:
1349:
1344:
1333:
1319:
1295:
1293:
1292:
1287:
1285:
1270:
1268:
1267:
1262:
1257:
1233:
1231:
1230:
1225:
1217:
1202:
1200:
1199:
1194:
1176:
1174:
1173:
1168:
1157:
1143:
1115:
1113:
1112:
1107:
1079:
1077:
1076:
1071:
1047:
1045:
1044:
1039:
994:
990:
982:
969:
967:
966:
961:
959:
944:
942:
941:
936:
934:
919:
917:
916:
911:
909:
886:
876:
863:
861:
847:) := |
829:
828:
826:
825:
820:
818:
788:
784:
768:
766:
757:
755:
754:
749:
747:
731:
727:
723:
722:
720:
719:
714:
712:
687:
675:
659:
655:
651:
646:
629:
621:
617:
613:
612:
610:
609:
604:
602:
577:
565:
549:
545:
541:
536:
522:
503:
502:
500:
499:
494:
492:
467:
457:
438:
434:
433:
431:
430:
425:
423:
398:
377:
376:
374:
373:
368:
366:
341:
339:
338:
333:
331:
319:
315:
311:
309:
308:
303:
301:
285:
240:or the field of
235:
233:
232:
227:
225:
197:
195:
194:
189:
187:
161:
155:
153:topologie faible
59:linear operators
32:initial topology
21:
6420:
6419:
6415:
6414:
6413:
6411:
6410:
6409:
6385:
6384:
6383:
6378:
6362:
6341:
6325:
6304:
6225:
6174:
6164:
6134:
6129:
6111:
6075:Advanced topics
6070:
5994:
5973:
5932:
5898:HilbertâSchmidt
5871:
5862:GelfandâNaimark
5809:
5759:
5694:
5680:
5646:
5633:
5627:
5614:
5600:
5584:
5570:
5550:
5536:
5514:
5509:
5496:
5482:
5469:
5463:
5450:
5445:
5432:
5429:
5424:
5423:
5415:
5411:
5404:
5388:
5386:
5382:
5378:, pp. 170.
5374:
5370:
5362:
5333:
5328:
5285:
5259:
5249:
5186:
5181:
5180:
5162:
5159:
5147:
5141:
5110:
5084:
5083:
5070:
5064:
5045:
5021:
5007:
4993:
4975:
4969:
4965:
4962:
4949:
4945:
4941:
4935:
4931:
4923:
4919:
4915:
4903:
4900:
4885:
4862:
4857:
4856:
4849:
4843:
4827:
4806:
4801:
4800:
4727:
4722:
4721:
4711:
4687:
4686:
4684:
4667:
4649:
4597:
4592:
4558:
4544:
4539:
4534:
4533:
4521:
4514:
4470:
4454:
4435:
4430:
4425:
4424:
4417:
4391:
4378:
4365:
4360:
4359:
4333:
4328:
4327:
4321:
4315:
4310:
4301:
4280:
4275:
4274:
4266:
4262:
4258:
4237:
4232:
4231:
4210:
4205:
4204:
4200:
4196:
4190:locally compact
4185:
4181:
4173:
4166:
4162:
4158:
4137:
4132:
4131:
4127:
4123:
4119:
4115:
4104:
4100:
4096:
4075:
4070:
4069:
4040:
4035:
4034:
4030:
4005:
4000:
3999:
3987:
3979:is a separable
3976:
3952:
3949:
3922:
3893:
3878:
3873:
3872:
3859:
3811:
3806:
3805:
3783:
3782:
3761:
3748:
3743:
3742:
3713:
3712:
3664:
3659:
3658:
3636:
3635:
3614:
3609:
3608:
3587:
3582:
3581:
3549:
3548:
3527:
3526:
3505:
3500:
3499:
3454:
3449:
3448:
3445:
3417:
3391:
3390:
3369:
3364:
3363:
3360:weak-* topology
3319:
3302:
3301:
3281:
3280:
3246:
3243:
3242:
3224:
3211:
3204:
3192:
3191:
3177:
3167:
3161:
3159:Weak-* topology
3131:
3126:
3125:
3122:strong topology
3043:
3042:
3018:
3013:
3012:
3008:
2997:
2993:
2990:
2955:
2950:
2949:
2945:or, sometimes,
2907:
2902:
2901:
2877:
2866:
2865:
2858:
2816:
2805:
2804:
2797:
2774:
2769:
2768:
2764:
2739:
2734:
2733:
2709:
2698:
2697:
2676:
2675:
2654:
2653:
2625:
2624:
2600:
2589:
2588:
2587:if and only if
2584:
2580:
2576:
2552:
2544:
2543:
2537:
2531:
2486:
2481:
2480:
2476:
2455:
2454:
2450:
2429:
2418:
2417:
2384:
2379:
2378:
2346:
2341:
2340:
2315:
2310:
2309:
2301:
2294:
2277:
2273:
2272:continuous, as
2244:
2221:
2220:
2199:
2194:
2193:
2169:
2155:
2154:
2130:
2125:
2124:
2102:
2097:
2096:
2065:
2060:
2059:
2040:
2035:
2034:
2033:continuous, as
1998:
1978:
1973:
1972:
1968:
1944:
1930:
1929:
1922:
1917:
1889:
1869:
1855:
1841:
1840:
1819:
1808:
1803:
1802:
1777:
1776:
1748:
1734:
1720:
1719:
1688:
1687:
1659:
1658:
1654:
1629:
1624:
1623:
1616:
1613:strong topology
1592:
1560:
1559:
1555:
1534:
1533:
1525:
1522:
1509:
1497:
1493:
1489:
1485:
1478:
1474:
1458:
1442:
1441:
1437:
1433:
1417:
1405:
1402:
1393:
1374:
1373:
1369:
1361:
1326:
1312:
1298:
1297:
1278:
1273:
1272:
1250:
1236:
1235:
1210:
1205:
1204:
1179:
1178:
1150:
1136:
1122:
1121:
1086:
1085:
1050:
1049:
1000:
999:
992:
988:
980:
977:
950:
949:
947:complex numbers
925:
924:
900:
899:
878:
868:
848:
842:
834:
809:
808:
802:
794:
786:
770:
764:
762:
738:
737:
729:
725:
724:continuous, as
703:
702:
689:
677:
661:
657:
653:
649:
644:
627:
619:
615:
614:continuous, as
593:
592:
579:
567:
551:
547:
543:
539:
534:
505:
483:
482:
469:
459:
440:
436:
414:
413:
400:
390:
357:
356:
343:
322:
321:
317:
313:
292:
291:
271:
256:
250:
238:complex numbers
216:
215:
178:
177:
174:
168:
128:
114:(respectively,
108:weakly analytic
102:(respectively,
92:(respectively,
84:(respectively,
71:continuous dual
39:
28:
23:
22:
15:
12:
11:
5:
6418:
6416:
6408:
6407:
6402:
6397:
6387:
6386:
6380:
6379:
6377:
6376:
6370:
6368:
6367:Other concepts
6364:
6363:
6361:
6360:
6355:
6349:
6347:
6343:
6342:
6340:
6339:
6333:
6331:
6327:
6326:
6324:
6323:
6318:
6316:BanachâAlaoglu
6312:
6310:
6306:
6305:
6303:
6302:
6297:
6296:
6295:
6290:
6288:polar topology
6280:
6275:
6274:
6273:
6268:
6263:
6253:
6248:
6247:
6246:
6235:
6233:
6227:
6226:
6224:
6223:
6218:
6216:Polar topology
6213:
6208:
6203:
6198:
6193:
6188:
6182:
6180:
6179:Basic concepts
6176:
6175:
6169:and spaces of
6165:
6163:
6162:
6155:
6148:
6140:
6131:
6130:
6128:
6127:
6116:
6113:
6112:
6110:
6109:
6104:
6099:
6094:
6092:Choquet theory
6089:
6084:
6078:
6076:
6072:
6071:
6069:
6068:
6058:
6053:
6048:
6043:
6038:
6033:
6028:
6023:
6018:
6013:
6008:
6002:
6000:
5996:
5995:
5993:
5992:
5987:
5981:
5979:
5975:
5974:
5972:
5971:
5966:
5961:
5956:
5951:
5946:
5944:Banach algebra
5940:
5938:
5934:
5933:
5931:
5930:
5925:
5920:
5915:
5910:
5905:
5900:
5895:
5890:
5885:
5879:
5877:
5873:
5872:
5870:
5869:
5867:BanachâAlaoglu
5864:
5859:
5854:
5849:
5844:
5839:
5834:
5829:
5823:
5821:
5815:
5814:
5811:
5810:
5808:
5807:
5802:
5797:
5795:Locally convex
5792:
5778:
5773:
5767:
5765:
5761:
5760:
5758:
5757:
5752:
5747:
5742:
5737:
5732:
5727:
5722:
5717:
5712:
5706:
5700:
5696:
5695:
5681:
5679:
5678:
5671:
5664:
5656:
5650:
5649:
5644:
5631:
5625:
5612:
5598:
5582:
5568:
5548:
5534:
5512:
5507:
5494:
5481:978-1584888666
5480:
5467:
5461:
5448:
5443:
5428:
5425:
5422:
5421:
5409:
5402:
5380:
5368:
5330:
5329:
5327:
5324:
5323:
5322:
5320:Vague topology
5317:
5312:
5307:
5302:
5297:
5292:
5284:
5281:
5246:
5245:
5234:
5231:
5228:
5225:
5222:
5219:
5216:
5213:
5210:
5207:
5204:
5199:
5196:
5193:
5189:
5151:
5134:
5133:
5122:
5117:
5113:
5109:
5106:
5103:
5100:
5097:
5094:
5091:
4961:
4958:
4912:algebraic dual
4899:
4896:
4871:
4866:
4845:Main article:
4842:
4839:
4813:
4809:
4793:
4792:
4781:
4778:
4775:
4772:
4769:
4766:
4761:
4757:
4753:
4748:
4745:
4742:
4739:
4734:
4730:
4695:
4680:
4674:test functions
4646:
4645:
4634:
4630:
4625:
4619:
4616:
4606:
4601:
4595:
4591:
4588:
4584:
4579:
4574:
4567:
4564:
4553:
4548:
4542:
4511:
4510:
4499:
4496:
4492:
4487:
4479:
4474:
4469:
4466:
4461:
4457:
4452:
4444:
4439:
4433:
4416:to an element
4405:
4400:
4395:
4390:
4385:
4381:
4377:
4372:
4368:
4342:
4337:
4314:
4313:Hilbert spaces
4311:
4309:
4306:
4287:
4283:
4244:
4240:
4217:
4213:
4161:of continuous
4144:
4140:
4082:
4078:
4047:
4043:
4012:
4008:
3996:
3995:
3990:is separable.
3984:locally convex
3961:locally convex
3948:
3945:
3919:
3918:
3907:
3900:
3896:
3892:
3885:
3881:
3856:
3855:
3844:
3841:
3838:
3835:
3832:
3829:
3826:
3823:
3818:
3814:
3801:provided that
3790:
3768:
3764:
3760:
3755:
3751:
3726:
3723:
3720:
3709:
3708:
3697:
3694:
3691:
3688:
3685:
3682:
3679:
3676:
3671:
3667:
3643:
3621:
3617:
3594:
3590:
3575:
3574:
3557:
3535:
3512:
3508:
3487:
3484:
3481:
3478:
3475:
3472:
3469:
3466:
3461:
3457:
3443:
3427:
3424:
3420:
3416:
3413:
3410:
3407:
3404:
3401:
3398:
3376:
3372:
3329:
3326:
3322:
3318:
3315:
3312:
3309:
3298:
3297:
3284:
3279:
3276:
3273:
3270:
3267:
3264:
3261:
3258:
3253:
3249:
3245:
3244:
3240:
3236:
3231:
3227:
3223:
3218:
3214:
3210:
3209:
3207:
3202:
3199:
3171:polar topology
3165:Polar topology
3160:
3157:
3138:
3134:
3118:
3117:
3106:
3102:
3098:
3095:
3092:
3089:
3085:
3079:
3076:
3073:
3070:
3067:
3063:
3059:
3056:
3053:
3050:
3025:
3021:
2989:
2986:
2985:
2984:
2973:
2970:
2967:
2962:
2958:
2943:
2942:
2931:
2925:
2921:
2914:
2910:
2884:
2880:
2876:
2873:
2855:
2854:
2843:
2840:
2837:
2834:
2831:
2828:
2823:
2819:
2815:
2812:
2781:
2777:
2746:
2742:
2716:
2712:
2708:
2705:
2684:
2662:
2641:
2638:
2635:
2632:
2612:
2607:
2603:
2599:
2596:
2564:
2559:
2555:
2551:
2530:
2527:
2523:polar topology
2507:
2504:
2501:
2496:
2493:
2489:
2463:
2436:
2432:
2428:
2425:
2405:
2402:
2399:
2394:
2391:
2387:
2353:
2349:
2322:
2318:
2293:
2290:
2286:
2285:
2282:weak* topology
2260:
2256:
2251:
2247:
2243:
2240:
2237:
2234:
2231:
2228:
2206:
2202:
2181:
2176:
2172:
2168:
2165:
2162:
2137:
2133:
2109:
2105:
2087:
2086:
2072:
2068:
2046:
2043:
2021:
2017:
2014:
2011:
2008:
2004:
2001:
1997:
1994:
1991:
1988:
1984:
1981:
1956:
1951:
1947:
1943:
1940:
1937:
1895:
1892:
1888:
1885:
1882:
1879:
1875:
1872:
1868:
1865:
1861:
1858:
1854:
1851:
1848:
1826:
1822:
1818:
1814:
1811:
1790:
1787:
1784:
1764:
1761:
1758:
1754:
1751:
1747:
1744:
1740:
1737:
1733:
1730:
1727:
1707:
1704:
1701:
1698:
1695:
1667:
1636:
1632:
1605:given topology
1568:
1542:
1521:
1518:
1404:), denoted by
1400:
1388:
1387:
1342:
1339:
1336:
1332:
1329:
1325:
1322:
1318:
1315:
1311:
1308:
1305:
1284:
1281:
1260:
1256:
1253:
1249:
1246:
1243:
1223:
1220:
1216:
1213:
1192:
1189:
1186:
1166:
1163:
1160:
1156:
1153:
1149:
1146:
1142:
1139:
1135:
1132:
1129:
1105:
1102:
1099:
1096:
1093:
1069:
1066:
1063:
1060:
1057:
1037:
1034:
1031:
1028:
1025:
1022:
1019:
1016:
1013:
1010:
1007:
976:
973:
972:
971:
958:
933:
920:is either the
908:
889:locally convex
865:
864:
838:
817:
798:
760:absolute value
746:
734:
733:
711:
624:
623:
601:
525:
524:
504:be defined by
491:
422:
365:
330:
300:
252:Main article:
249:
246:
224:
186:
170:Main article:
167:
164:
156:in French and
127:
124:
116:differentiable
86:weakly compact
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
6417:
6406:
6403:
6401:
6398:
6396:
6393:
6392:
6390:
6375:
6372:
6371:
6369:
6365:
6359:
6356:
6354:
6351:
6350:
6348:
6344:
6338:
6335:
6334:
6332:
6328:
6322:
6319:
6317:
6314:
6313:
6311:
6307:
6301:
6298:
6294:
6291:
6289:
6286:
6285:
6284:
6281:
6279:
6276:
6272:
6269:
6267:
6264:
6262:
6259:
6258:
6257:
6254:
6252:
6249:
6245:
6242:
6241:
6240:
6239:Norm topology
6237:
6236:
6234:
6232:
6228:
6222:
6219:
6217:
6214:
6212:
6209:
6207:
6204:
6202:
6199:
6197:
6196:Dual topology
6194:
6192:
6189:
6187:
6184:
6183:
6181:
6177:
6172:
6168:
6161:
6156:
6154:
6149:
6147:
6142:
6141:
6138:
6126:
6118:
6117:
6114:
6108:
6105:
6103:
6100:
6098:
6097:Weak topology
6095:
6093:
6090:
6088:
6085:
6083:
6080:
6079:
6077:
6073:
6066:
6062:
6059:
6057:
6054:
6052:
6049:
6047:
6044:
6042:
6039:
6037:
6034:
6032:
6029:
6027:
6024:
6022:
6021:Index theorem
6019:
6017:
6014:
6012:
6009:
6007:
6004:
6003:
6001:
5997:
5991:
5988:
5986:
5983:
5982:
5980:
5978:Open problems
5976:
5970:
5967:
5965:
5962:
5960:
5957:
5955:
5952:
5950:
5947:
5945:
5942:
5941:
5939:
5935:
5929:
5926:
5924:
5921:
5919:
5916:
5914:
5911:
5909:
5906:
5904:
5901:
5899:
5896:
5894:
5891:
5889:
5886:
5884:
5881:
5880:
5878:
5874:
5868:
5865:
5863:
5860:
5858:
5855:
5853:
5850:
5848:
5845:
5843:
5840:
5838:
5835:
5833:
5830:
5828:
5825:
5824:
5822:
5820:
5816:
5806:
5803:
5801:
5798:
5796:
5793:
5790:
5786:
5782:
5779:
5777:
5774:
5772:
5769:
5768:
5766:
5762:
5756:
5753:
5751:
5748:
5746:
5743:
5741:
5738:
5736:
5733:
5731:
5728:
5726:
5723:
5721:
5718:
5716:
5713:
5711:
5708:
5707:
5704:
5701:
5697:
5692:
5688:
5684:
5677:
5672:
5670:
5665:
5663:
5658:
5657:
5654:
5647:
5641:
5637:
5632:
5628:
5626:9780486434797
5622:
5618:
5613:
5609:
5605:
5601:
5595:
5591:
5587:
5583:
5579:
5575:
5571:
5565:
5561:
5557:
5553:
5549:
5545:
5541:
5537:
5531:
5527:
5523:
5522:
5517:
5516:Rudin, Walter
5513:
5510:
5508:0-387-96788-5
5504:
5500:
5495:
5491:
5487:
5483:
5477:
5473:
5468:
5464:
5458:
5454:
5449:
5446:
5444:0-387-97245-5
5440:
5436:
5431:
5430:
5426:
5418:
5413:
5410:
5405:
5403:0-387-98431-3
5399:
5395:
5391:
5384:
5381:
5377:
5372:
5369:
5365:
5360:
5358:
5356:
5354:
5352:
5350:
5348:
5346:
5344:
5342:
5340:
5338:
5336:
5332:
5325:
5321:
5318:
5316:
5313:
5311:
5308:
5306:
5303:
5301:
5298:
5296:
5293:
5290:
5287:
5286:
5282:
5280:
5278:
5274:
5269:
5266:
5262:
5256:
5252:
5232:
5223:
5217:
5211:
5205:
5202:
5197:
5194:
5191:
5187:
5179:
5178:
5177:
5173:
5169:
5165:
5158:
5154:
5150:
5144:
5139:
5120:
5115:
5104:
5098:
5089:
5082:
5081:
5080:
5077:
5073:
5067:
5062:
5056:
5052:
5048:
5043:
5042:
5036:
5032:
5028:
5024:
5019:
5015:
5010:
5004:
5001: â
5000:
4996:
4992:
4986:
4982:
4978:
4972:
4959:
4957:
4955:
4938:
4929:
4913:
4909:
4902:Suppose that
4897:
4895:
4893:
4888:
4869:
4854:
4853:distributions
4848:
4841:Distributions
4840:
4838:
4836:
4830:
4811:
4807:
4798:
4776:
4773:
4767:
4764:
4759:
4755:
4751:
4746:
4740:
4732:
4728:
4720:
4719:
4718:
4714:
4708:
4683:
4679:
4675:
4670:
4665:
4661:
4656:
4652:
4632:
4623:
4614:
4604:
4593:
4586:
4577:
4572:
4562:
4551:
4540:
4532:
4531:
4530:
4527:
4524:
4517:
4497:
4490:
4477:
4467:
4464:
4459:
4455:
4442:
4431:
4423:
4422:
4421:
4398:
4383:
4379:
4375:
4370:
4366:
4356:
4340:
4325:
4320:
4319:Hilbert space
4312:
4307:
4305:
4285:
4281:
4272:
4242:
4238:
4215:
4211:
4193:
4191:
4179:
4170:
4142:
4138:
4112:
4110:
4080:
4076:
4067:
4063:
4045:
4041:
4028:
4010:
4006:
3993:
3992:
3991:
3985:
3982:
3974:
3970:
3966:
3962:
3958:
3946:
3944:
3942:
3938:
3934:
3929:
3925:
3905:
3898:
3894:
3883:
3879:
3871:
3870:
3869:
3866:
3862:
3839:
3833:
3824:
3816:
3812:
3804:
3803:
3802:
3788:
3781:converges to
3766:
3762:
3758:
3753:
3749:
3740:
3724:
3721:
3718:
3692:
3686:
3677:
3669:
3665:
3657:
3656:
3655:
3641:
3619:
3615:
3592:
3588:
3580:
3572:
3571:
3570:
3510:
3506:
3482:
3476:
3473:
3467:
3459:
3455:
3447:, defined by
3446:
3425:
3422:
3418:
3414:
3408:
3402:
3399:
3396:
3374:
3370:
3361:
3357:
3353:
3349:
3345:
3327:
3324:
3320:
3313:
3310:
3307:
3274:
3268:
3265:
3259:
3251:
3247:
3229:
3225:
3221:
3216:
3212:
3205:
3197:
3190:
3189:
3188:
3186:
3183:
3174:
3172:
3166:
3158:
3156:
3154:
3136:
3132:
3123:
3104:
3093:
3087:
3077:
3074:
3068:
3057:
3051:
3041:
3040:
3039:
3023:
3019:
3005:
3003:
2987:
2971:
2968:
2960:
2956:
2948:
2947:
2946:
2929:
2912:
2908:
2900:
2899:
2898:
2882:
2878:
2874:
2871:
2861:
2838:
2832:
2821:
2817:
2810:
2803:
2802:
2801:
2796:
2779:
2775:
2762:
2744:
2740:
2730:
2714:
2710:
2706:
2703:
2636:
2630:
2623:converges to
2605:
2601:
2594:
2557:
2553:
2542:
2536:
2528:
2526:
2524:
2519:
2502:
2494:
2491:
2487:
2434:
2430:
2426:
2423:
2400:
2392:
2389:
2385:
2376:
2371:
2369:
2351:
2347:
2338:
2320:
2316:
2307:
2299:
2298:weak topology
2291:
2289:
2283:
2249:
2245:
2241:
2235:
2232:
2229:
2204:
2200:
2174:
2170:
2166:
2163:
2153:
2135:
2131:
2122:
2107:
2103:
2092:
2089:
2088:
2070:
2066:
2044:
2041:
2012:
2009:
2002:
1999:
1995:
1992:
1986:
1982:
1979:
1949:
1945:
1941:
1938:
1928:
1920:
1913:
1910:
1909:
1908:
1893:
1890:
1886:
1880:
1873:
1870:
1866:
1859:
1856:
1852:
1849:
1824:
1820:
1816:
1812:
1809:
1788:
1785:
1782:
1759:
1752:
1749:
1745:
1738:
1735:
1731:
1728:
1702:
1699:
1696:
1684:
1682:
1652:
1634:
1630:
1622:
1614:
1610:
1606:
1602:
1598:
1590:
1586:
1582:
1531:
1519:
1517:
1515:
1505:
1501:
1482:
1472:
1466:
1462:
1457:The topology
1455:
1450:
1446:
1431:
1430:weak topology
1425:
1421:
1413:
1409:
1403:
1396:
1382:
1378:
1367:
1359:
1356:
1355:
1354:
1337:
1330:
1327:
1323:
1316:
1313:
1309:
1306:
1282:
1279:
1254:
1251:
1247:
1244:
1221:
1218:
1214:
1211:
1190:
1187:
1184:
1161:
1154:
1151:
1147:
1140:
1137:
1133:
1130:
1120:, defined by
1119:
1100:
1097:
1094:
1083:
1080:, called the
1064:
1061:
1058:
1029:
1026:
1023:
1017:
1014:
1011:
1008:
996:
986:
974:
948:
923:
897:
894:
893:
892:
890:
885:
881:
875:
871:
859:
855:
851:
846:
841:
837:
833:
832:
831:
830:, defined by
806:
801:
797:
792:
782:
778:
774:
761:
736:If the field
700:
696:
692:
685:
681:
673:
669:
665:
660:, denoted by
647:
640:
637:
636:
635:
633:
590:
586:
582:
575:
571:
563:
559:
555:
550:, denoted by
537:
530:
527:
526:
520:
516:
512:
508:
480:
476:
472:
466:
462:
455:
451:
447:
443:
411:
407:
403:
397:
393:
388:
385:
384:
383:
381:
354:
350:
346:
289:
283:
279:
275:
268:
265:
261:
255:
247:
245:
243:
239:
213:
209:
205:
201:
173:
165:
163:
160:
154:
149:
145:
141:
137:
133:
132:David Hilbert
125:
123:
121:
117:
113:
109:
105:
101:
100:
95:
91:
87:
83:
82:weakly closed
78:
76:
72:
68:
64:
63:Hilbert space
60:
57:or spaces of
56:
52:
48:
47:weak topology
44:
37:
33:
19:
6321:MackeyâArens
6309:Main results
6255:
6096:
6087:Balanced set
6061:Distribution
5999:Applications
5852:KreinâMilman
5837:Closed graph
5635:
5616:
5589:
5555:
5520:
5501:, Springer,
5499:Analysis Now
5498:
5471:
5452:
5434:
5427:Bibliography
5412:
5393:
5383:
5376:Folland 1999
5371:
5270:
5264:
5260:
5254:
5250:
5247:
5171:
5167:
5163:
5156:
5152:
5148:
5142:
5137:
5135:
5075:
5071:
5065:
5060:
5054:
5050:
5046:
5039:
5037:
5030:
5026:
5022:
5008:
5002:
4998:
4994:
4984:
4980:
4976:
4970:
4963:
4936:
4927:
4907:
4901:
4886:
4850:
4828:
4794:
4712:
4709:
4681:
4677:
4668:
4664:dense subset
4659:
4654:
4650:
4647:
4528:
4522:
4515:
4512:
4323:
4316:
4271:Banach space
4194:
4171:
4113:
3997:
3968:
3964:
3950:
3936:
3932:
3930:
3923:
3920:
3864:
3860:
3857:
3710:
3576:
3441:
3359:
3351:
3299:
3184:
3175:
3168:
3121:
3119:
3006:
2991:
2944:
2859:
2856:
2794:
2731:
2538:
2520:
2372:
2297:
2295:
2287:
2281:
2276:ranges over
2094:
2090:
2058:ranges over
1915:
1911:
1686:Recall that
1685:
1604:
1600:
1596:
1581:vector space
1523:
1503:
1499:
1483:
1464:
1460:
1456:
1448:
1444:
1423:
1419:
1411:
1407:
1398:
1391:
1389:
1380:
1376:
1357:
1117:
997:
978:
922:real numbers
895:
883:
879:
873:
869:
866:
857:
853:
849:
844:
839:
835:
804:
799:
795:
780:
776:
772:
735:
728:ranges over
698:
697:, â˘) :
694:
690:
683:
679:
671:
667:
663:
642:
638:
625:
618:ranges over
588:
584:
580:
573:
569:
561:
557:
553:
532:
528:
518:
514:
510:
506:
478:
474:
470:
464:
460:
453:
449:
445:
441:
409:
408:, â˘) :
405:
401:
395:
391:
386:
380:bilinear map
352:
348:
344:
281:
277:
273:
269:
263:
257:
242:real numbers
175:
136:Marcel Riesz
129:
107:
103:
97:
85:
81:
79:
46:
40:
6300:Ultrastrong
6283:Strong dual
6191:Dual system
6016:Heat kernel
6006:Hardy space
5913:Trace class
5827:HahnâBanach
5789:Topological
5417:Trèves 2006
5248:indexed by
5014:Yosida 1980
4420:means that
3182:double dual
1912:Definition.
1554:, that is,
1532:(TVS) over
1397:(resp. the
1358:Assumption.
896:Assumption.
648:induced by
639:Definition.
632:Dual system
538:induced by
529:Definition.
439:defined by
202:, namely a
162:in German.
53:, often on
43:mathematics
6389:Categories
6231:Topologies
6186:Dual space
5949:C*-algebra
5764:Properties
5326:References
4130:. Then in
3981:metrizable
3973:metrizable
3963:space and
3947:Properties
3348:surjective
3163:See also:
2366:remains a
2091:Definition
1681:continuous
1514:Rudin 1991
1436:(resp. on
1416:(resp. by
676:or simply
566:or simply
212:continuous
112:continuous
6358:Total set
6244:Dual norm
6211:Polar set
5923:Unbounded
5918:Transpose
5876:Operators
5805:Separable
5800:Reflexive
5785:Algebraic
5771:Barrelled
5608:853623322
5588:(2006) .
5578:840278135
5490:144216834
5209:↦
5112:‖
5096:‖
5093:↦
4914:space of
4808:ψ
4768:
4760:π
4729:ψ
4633:μ
4618:¯
4615:ψ
4594:∫
4590:→
4587:μ
4566:¯
4563:ψ
4541:∫
4495:→
4491:μ
4468:ψ
4465:−
4456:ψ
4432:∫
4376:∈
4367:ψ
4286:∗
4243:∗
4216:∗
4143:∗
4109:reflexive
4081:∗
4060:is weak*-
4046:∗
4011:∗
3957:separable
3906:ϕ
3899:∗
3891:→
3880:ϕ
3834:ϕ
3831:→
3813:ϕ
3789:ϕ
3767:∗
3759:∈
3750:ϕ
3722:∈
3687:ϕ
3684:→
3670:λ
3666:ϕ
3642:ϕ
3620:∗
3593:λ
3589:ϕ
3511:∗
3477:ϕ
3468:ϕ
3426:∗
3423:∗
3415:⊂
3375:∗
3356:reflexive
3344:injective
3328:∗
3325:∗
3317:→
3269:ϕ
3260:ϕ
3235:→
3230:∗
3201:↦
3137:∗
3088:ϕ
3075:≤
3072:‖
3066:‖
3055:‖
3052:ϕ
3049:‖
3024:∗
2966:⇀
2920:⟶
2883:∗
2875:∈
2872:φ
2833:φ
2830:→
2811:φ
2715:∗
2707:∈
2704:ϕ
2631:ϕ
2606:λ
2595:ϕ
2558:λ
2492:−
2488:ϕ
2435:∗
2427:∈
2424:ϕ
2390:−
2386:ϕ
2352:∗
2321:∗
2300:on a TVS
2255:→
2250:∗
2239:⟩
2236:⋅
2227:⟨
2205:∗
2180:⟩
2175:∗
2161:⟨
2136:∗
2108:∗
2071:∗
2016:→
2007:⟩
1993:⋅
1990:⟨
1955:⟩
1950:∗
1936:⟨
1881:⋅
1864:⟩
1850:⋅
1847:⟨
1825:∗
1817:∈
1786:∈
1743:⟩
1726:⟨
1706:⟩
1703:⋅
1697:⋅
1694:⟨
1679:that are
1635:∗
1428:) is the
1338:⋅
1321:⟩
1307:⋅
1304:⟨
1259:⟩
1245:⋅
1242:⟨
1219:∈
1188:∈
1145:⟩
1128:⟨
1104:⟩
1101:⋅
1095:⋅
1092:⟨
1033:⟩
1030:⋅
1024:⋅
1021:⟨
791:seminorms
587:) :
477:) :
387:Notation.
6400:Topology
6293:operator
6266:operator
6125:Category
5937:Algebras
5819:Theorems
5776:Complete
5745:Schwartz
5691:glossary
5544:21163277
5518:(1991).
5392:(1998),
5283:See also
5263:∈
5253:∈
5074:∈
4997: :
4795:form an
4653:∈
4308:Examples
3863:∈
3858:for all
3739:sequence
3711:for all
3176:A space
2864:for all
2761:sequence
2696:for all
2337:coarsest
2045:′
2003:′
1983:′
1894:′
1874:′
1860:′
1813:′
1775:for all
1753:′
1739:′
1601:starting
1597:original
1585:topology
1451:⟩
1443:⟨
1383:⟩
1375:⟨
1331:′
1317:′
1283:′
1255:′
1215:′
1177:for all
1155:′
1141:′
882:∈
872:∈
867:for all
803: :
389:For all
347: :
270:Suppose
260:pairings
208:topology
120:analytic
6346:Subsets
6278:Mackey
6201:Duality
6167:Duality
5928:Unitary
5908:Nuclear
5893:Compact
5888:Bounded
5883:Adjoint
5857:Minâmax
5750:Sobolev
5735:Nuclear
5725:Hilbert
5720:FrĂŠchet
5685: (
4910:is the
4300:unless
4062:compact
3935:or the
3358:). The
2767:, then
2375:subbase
2304:is the
1611:" and "
1469:is the
1116:is the
945:or the
758:has an
288:pairing
206:with a
126:History
94:compact
6171:linear
5903:Normal
5740:Orlicz
5730:HĂślder
5710:Banach
5699:Spaces
5687:topics
5642:
5623:
5606:
5596:
5576:
5566:
5542:
5532:
5505:
5488:
5478:
5459:
5441:
5400:
4922:). If
4678:ψ
4418:ψ
3342:is an
2416:where
2093:: The
862:|
767:|
763:|
468:, let
399:, let
312:(i.e.
144:Banach
90:closed
6261:polar
5715:Besov
4715:(0,Ď)
4662:in a
4269:is a
4066:polar
4029:: if
3955:is a
3498:from
3300:Thus
3124:, on
3000:is a
2759:is a
1653:from
1603:, or
1558:is a
1528:be a
1296:i.e.
652:(and
542:(and
378:is a
286:is a
204:field
198:be a
6330:Maps
6256:Weak
6173:maps
6063:(or
5781:Dual
5640:ISBN
5621:ISBN
5604:OCLC
5594:ISBN
5574:OCLC
5564:ISBN
5540:OCLC
5530:ISBN
5503:ISBN
5486:OCLC
5476:ISBN
5457:ISBN
5439:ISBN
5398:ISBN
5275:and
5258:and
4968:and
3352:this
2449:and
1914:The
1801:and
1524:Let
1203:and
877:and
641:The
583:(â˘,
531:The
473:(â˘,
342:and
316:and
264:both
176:Let
134:and
5560:GTM
5161:on
5044:on
5020:on
4989:of
4964:If
4934:is
4831:â â
4826:as
4765:sin
4666:of
4518:â â
4513:as
4195:If
4172:If
4107:is
4068:in
3951:If
3926:â â
3921:as
3741:of
3607:in
3579:net
3547:or
3362:on
3187:by
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