15660:
18938:
15072:
18409:
15655:{\displaystyle {\begin{alignedat}{9}U_{B_{1}}&\,{\stackrel {\scriptscriptstyle {\text{def}}}{=}}\,{\Big \{}~~\;~~\;~~\;~~f\ \in \mathbb {K} ^{X}~~\;~~:\sup _{u\in U}|f(u)|\leq 1{\Big \}}\\&={\big \{}~~\;~~\;~~\;~~f\,\in \mathbb {K} ^{X}~~\;~~:f(u)\in B_{1}{\text{ for all }}u\in U{\big \}}\\&={\Big \{}\left(f_{x}\right)_{x\in X}\in \prod _{x\in X}\mathbb {K} \,~:~\;~f_{u}~\in B_{1}{\text{ for all }}u\in U{\Big \}}\\&=\prod _{x\in X}C_{x}\quad {\text{ where }}\quad C_{x}~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~{\begin{cases}B_{1}&{\text{ if }}x\in U\\\mathbb {K} &{\text{ if }}x\not \in U\\\end{cases}}\\\end{alignedat}}}
18933:{\displaystyle {\begin{alignedat}{4}M\left(f_{\bullet }(x)\right){\stackrel {\scriptscriptstyle {\text{def}}}{=}}&~M\circ f_{\bullet }(x)&&{\text{ by definition of notation }}\\=&~\left(M\left(f_{i}(x)\right)\right)_{i\in I}~~~&&{\text{ because }}f_{\bullet }(x)=\left(f_{i}(x)\right)_{i\in I}:I\to \mathbb {K} \\=&~\left(sf_{i}(x)\right)_{i\in I}&&M\left(f_{i}(x)\right)~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~sf_{i}(x)\\=&~\left(f_{i}(sx)\right)_{i\in I}&&{\text{ by linearity of }}f_{i}\\=&~f_{\bullet }(sx)&&{\text{ notation }}\end{alignedat}}}
8477:
32213:
31498:
20381:
8226:
8647:
20033:
22496:
8472:{\displaystyle {\begin{alignedat}{4}H:\;&&\prod _{x\in X}\mathbb {K} &&\;\to \;&\left(\prod _{u\in U}\mathbb {K} \right)\times \prod _{x\in X\setminus U}\mathbb {K} \\&&\left(f_{x}\right)_{x\in X}&&\;\mapsto \;&\left(\left(f_{u}\right)_{u\in U},\;\left(f_{x}\right)_{x\in X\setminus U}\right)\\\end{alignedat}}}
8486:
21654:
10407:
14862:
29189:
13669:
22306:
24340:
12922:
27112:
18003:
7545:
20376:{\displaystyle A\left(z_{\bullet }\right)~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~A\circ z_{\bullet }=\left(A\left(z_{i}\right)\right)_{i\in I}=\left(A\left(f_{i}(x),f_{i}(y)\right)\right)_{i\in I}=\left(f_{i}(x)+f_{i}(y)\right)_{i\in I}=\left(f_{i}(x+y)\right)_{i\in I}=f_{\bullet }(x+y)}
22068:
9434:
29794:
23478:
21483:
14642:
12125:
13469:
22868:
10217:
8642:{\displaystyle {\begin{alignedat}{4}H:\;&&\mathbb {K} ^{X}&&\;\to \;&\mathbb {K} ^{U}\times \mathbb {K} ^{X\setminus U}\\&&f&&\;\mapsto \;&\left(f{\big \vert }_{U},\;f{\big \vert }_{X\setminus U}\right)\\\end{alignedat}}.}
21361:
28993:
21866:
17542:
16199:
15077:
27886:
25530:
13457:
24226:
12756:
28252:
14548:
6436:
19389:
26963:
21230:
11223:
437:
24648:
17816:
11936:
7425:
21950:
9309:
30249:
7962:
6968:
22562:
19285:
2721:
502:
23321:
26649:
28062:
2544:
1204:
363:
11649:
11337:
22491:{\displaystyle \operatorname {Box} _{P}~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\left\{\prod B_{R_{\bullet }}~:~R_{\bullet }\in T_{P}\right\}~=~\left\{\prod B_{R_{\bullet }}~:~P\subseteq \prod B_{R_{\bullet }}\right\},}
7687:
10129:
15950:
6549:
29689:
11821:
11597:
29918:
2898:
21945:
21737:
19826:
4375:
29856:
26412:
24449:
7191:
3979:
30115:
29311:
11991:
1913:
18179:
28988:
19613:
22218:
19514:
24742:
13977:
19753:
9598:
5676:
5296:
3391:
1985:
1771:
1488:
21073:
20894:
14134:
9277:
5573:
3907:
2089:
1840:
23285:
28834:
13878:
22697:
15717:
7298:
3695:
3072:
73:
This theorem has applications in physics when one describes the set of states of an algebra of observables, namely that any state can be written as a convex linear combination of so-called pure states.
21004:
19923:
9947:
24028:
16984:
15063:
13791:
9762:
9211:
2390:
856:
29225:
21478:
18348:
17116:
8718:
7054:
6707:
3272:
2787:
2607:
2291:
21649:{\displaystyle {\begin{alignedat}{4}U^{\circ }&=U^{\#}&&\\&=X^{\#}&&\cap \prod _{x\in X}B_{m_{x}}\\&=X^{\prime }&&\cap \prod _{x\in X}B_{m_{x}}\\\end{alignedat}}}
17440:
25391:
16542:
14067:
10514:
26123:
25422:
10402:{\textstyle \Pr {}_{z}\left(\left(f_{i}\right)_{i\in I}\right)~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\left(\Pr {}_{z}\left(f_{i}\right)\right)_{i\in I}=\left(f_{i}(z)\right)_{i\in I},}
2964:
2341:
27224:
1416:
23722:
13025:
9651:
3020:
2138:
764:
25030:
22765:
17387:
5474:
5170:
5018:
4941:
4895:
4735:
4508:
1296:
26278:
25262:
19709:
10001:
26073:
20456:
16626:
14638:
13167:
6059:, but is usually not a neighborhood of the origin in the weak-* topology, as it has empty interior in the weak* topology, unless the space is finite-dimensional. In fact, it is a result of
4163:
4064:
28490:
27770:
24508:
18095:
11483:
11420:
982:
29341:
28327:
23640:
23510:
22592:
21259:
14857:{\displaystyle {\frac {1}{r_{x}}}|f(x)|=\left|{\frac {1}{r_{x}}}f(x)\right|=\left|f\left({\frac {1}{r_{x}}}x\right)\right|=\left|f\left(u_{x}\right)\right|\leq \sup _{u\in U}|f(u)|\leq 1.}
11546:
1692:
20822:
20522:
13068:
9844:
9702:
7242:
5369:
5094:
3148:
688:
29982:
29560:
28127:
20658:
19676:
16315:
15802:
11135:
10722:
10590:
9150:
7870:
4582:
611:
368:
29184:{\displaystyle \cap \operatorname {Box} _{P}=\bigcap _{R_{\bullet }\in T_{P}}\prod _{x\in X}B_{R_{x}}=\prod _{x\in X}\bigcap _{R_{\bullet }\in T_{P}}B_{R_{x}}=\prod _{x\in X}B_{m_{x}}.}
24559:
23593:
Due to the constructive nature of its proof (as opposed to the general case, which is based on the axiom of choice), the sequential Banach–Alaoglu theorem is often used in the field of
23034:
18998:
11679:
The essence of the Banach–Alaoglu theorem can be found in the next proposition, from which the Banach–Alaoglu theorem follows. Unlike the Banach–Alaoglu theorem, this proposition does
11105:
8221:
4019:
24820:
22622:
19055:
17764:
11041:
10879:
6212:
29485:
29427:
29372:
27690:
22728:
22279:
12314:
8012:
7336:
7092:
6623:
30310:
28762:
21775:
18250:
17658:
16104:
14283:
24077:
20028:
13722:
13664:{\displaystyle U^{\#}~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~{\Big \{}f\in X^{\#}~:~\sup _{u\in U}|f(u)|\leq 1{\Big \}}~=~\left\{f\in X^{\#}~:~f(U)\subseteq B_{1}\right\}}
10212:
6307:
29265:
27775:
27365:
26906:
23909:
22108:
17055:
13356:
13347:
8047:
25308:
24682:
8777:
7608:
7126:
6828:
27320:
27287:
24201:
14971:
14335:
12968:
12724:
8873:
6767:
26536:
25196:
24775:
24375:
20590:
19202:
17607:
17575:
17195:
15834:
14919:
11000:
10838:
10791:
9499:
8096:
6168:
4662:
3811:
2174:
28173:
27145:
16788:
16439:
13267:
9019:
7365:
6857:
6736:
6140:
4296:
4260:
3552:
29397:
28367:
27494:
15978:
25971:
1100:
30330:
30053:
24839:
22752:
21093:
20610:
18404:
18275:
17812:
16727:
15854:
15742:
14468:
13287:
10539:
10154:
6578:
6349:
5724:
3769:
1598:
1545:
28876:
26835:
25907:
24165:
23199:
23060:
22968:
22245:
21681:
21127:
20945:
19945:
19290:
19130:
19077:
18059:
17680:
17415:
17231:
16931:
16909:
16887:
16865:
16072:
16046:
14225:
12650:
12227:
11982:
11773:
10928:
10748:
9304:
9082:
8900:
8153:
5504:
4822:
4769:
4542:
4405:
3632:
3574:
3472:
3427:
1570:
1326:
1092:
883:
571:
243:
29655:
28673:
28607:
26958:
26867:
26439:
26336:
26305:
26025:
25592:
25164:
24965:
24938:
24903:
24231:
23865:
23537:
21132:
16401:
14404:
13101:
12459:
8491:
8231:
7613:
6038:
6011:
5960:
5921:
5840:
5224:
5197:
4849:
4097:
3211:
2205:
2020:
715:
30801:
27172:
16707:
16267:
14368:
12751:
12611:
12188:
8815:
7819:
7574:
5703:
5604:
5428:
5323:
5048:
4972:
4689:
4462:
4432:
3838:
1353:
27636:
17306:
16014:
11849:
11245:
27546:
27461:
23317:
21770:
14437:
9466:
9055:
3610:
30120:
29684:
29622:
29593:
28905:
28702:
28640:
28574:
28519:
28396:
28159:
27254:
26760:
24335:{\displaystyle {\begin{alignedat}{4}F:\;&&B_{1}^{\,\prime }&&\;\to \;&D\\&&f&&\;\mapsto \;&(f(x))_{x\in X}.\\\end{alignedat}}}
23953:
22137:
21395:
19418:
17709:
17337:
16468:
14163:
13916:
12578:
12156:
10957:
10688:
10440:
10037:
8125:
5399:
5124:
3501:
3178:
30008:
28545:
28089:
27913:
27573:
26720:
25417:
24554:
20756:
20721:
20689:
17143:
16823:
16759:
15010:
14463:
14189:
13819:
12917:{\displaystyle U^{\circ }~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~{\Big \{}f\in X^{\prime }~:~\sup _{u\in U}|f(u)|\leq 1{\Big \}}~=~U^{\#}\cap X^{\prime }.}
12677:
12353:
12254:
8926:
7875:
7420:
6254:
3299:
3102:
1640:
642:
270:
29512:
29458:
28283:
26740:
26476:
25685:
24108:
16342:
16226:
11364:
10906:
9891:
8980:
8953:
6886:
1014:
27593:
27385:
25718:
25645:
19207:
6991:
6792:
3724:
25556:
22501:
2614:
442:
129:
25331:
23811:
23768:
23745:
23663:
23560:
23107:
22302:
17787:
16669:
16365:
16099:
13210:
12549:
12522:
11844:
11131:
10653:
9522:
8842:
7730:
6345:
4629:
28854:
28436:
28416:
27514:
27425:
27405:
26926:
26800:
26780:
26693:
26669:
26541:
26496:
25991:
25927:
25874:
25854:
25216:
25125:
25101:
25081:
25057:
24872:
24528:
24221:
24132:
23831:
23788:
23580:
23219:
23169:
23149:
23080:
22916:
22892:
21488:
21417:
21254:
20776:
19846:
18414:
18199:
18115:
17435:
17271:
17251:
17163:
17004:
16843:
16646:
16562:
14568:
13997:
13230:
13187:
12499:
12479:
12428:
12401:
12377:
11960:
11751:
11731:
11701:
11669:
10630:
10610:
9864:
9790:
9102:
8738:
8175:
7790:
7770:
7750:
7707:
6877:
6643:
6476:
6456:
5980:
5894:
5860:
5813:
5793:
5773:
5749:
4789:
4602:
4223:
4203:
4183:
3319:
2918:
2743:
2564:
2432:
2412:
2245:
2225:
1520:
1373:
1247:
1227:
1047:
923:
903:
788:
533:
294:
221:
27918:
27107:{\displaystyle U^{\circ ,\sigma }~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~{\Big \{}f\in (X,\sigma )^{\prime }~:~\sup _{u\in U}|f(u)|\leq 1{\Big \}}}
2440:
305:
17998:{\displaystyle f_{\bullet }(x)\to f(x),\quad f_{\bullet }(y)\to f(y),\quad f_{\bullet }(x+y)\to f(x+y),\quad {\text{ and }}\quad f_{\bullet }(sx)\to f(sx).}
12359:
Before proving the proposition above, it is first shown how the Banach–Alaoglu theorem follows from it (unlike the proposition, Banach–Alaoglu assumes that
11605:
11254:
7540:{\displaystyle \Pr {}_{z}:\prod _{x\in X}\mathbb {K} \to \mathbb {K} \quad {\text{ defined by }}\quad s_{\bullet }=\left(s_{x}\right)_{x\in X}\mapsto s_{z}}
23912:
25604:
In a
Hilbert space, every bounded and closed set is weakly relatively compact, hence every bounded net has a weakly convergent subnet (Hilbert spaces are
22063:{\displaystyle T_{P}~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\left\{R_{\bullet }\in \mathbb {R} ^{X}~:~P\subseteq \prod B_{R_{\bullet }}\right\}}
9429:{\displaystyle F\left(x_{\bullet }\right)=\left(F\left(x_{i}\right)\right)_{i\in I}~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~F\circ x_{\bullet },}
31534:
24687:
10042:
30742:
29789:{\displaystyle P~\subseteq ~\left(\cap \operatorname {Box} _{P}\right)\cap X^{\prime }~\subseteq ~\left(\cap \operatorname {Box} _{P}\right)\cap X^{\#}}
15862:
31387:
6481:
23116:
A special case of the Banach–Alaoglu theorem is the sequential version of the theorem, which asserts that the closed unit ball of the dual space of a
2343:
it is in general not guaranteed to be a dual system. Throughout, unless stated otherwise, all polar sets will be taken with respect to the canonical
11778:
11551:
29861:
23473:{\displaystyle \rho (x,y)=\sum _{n=1}^{\infty }\,2^{-n}\,{\frac {\left|\langle x-y,x_{n}\rangle \right|}{1+\left|\langle x-y,x_{n}\rangle \right|}}}
2794:
21871:
21686:
19758:
7250:
4301:
30794:
29799:
26345:
24387:
7131:
3912:
31223:
30058:
29270:
6040:
has its usual norm topology. This is because the unit ball in the norm topology is compact if and only if the space is finite-dimensional (cf.
1849:
30484:. Grundlehren der mathematischen Wissenschaften. Vol. 159. Translated by Garling, D.J.H. New York: Springer Science & Business Media.
23958:
18120:
32044:
28910:
31050:
19519:
6086:
and requires only basic concepts from set theory, topology, and functional analysis. What is needed from topology is a working knowledge of
22141:
19423:
6102:
25810:
is equivalent to the following weak version of the Banach–Alaoglu theorem for normed space in which the conclusion of compactness (in the
2247:
in the other topology (the conclusion follows because two topologies are equal if and only if they have the exact same convergent nets).
150:. According to Pietsch , there are at least twelve mathematicians who can lay claim to this theorem or an important predecessor to it.
13921:
31661:
31636:
31213:
30787:
19714:
9527:
5609:
5229:
3324:
1918:
1704:
1421:
21009:
20830:
14072:
9216:
5509:
3843:
2025:
1776:
23224:
28767:
1987:
Specifically, this proof will use the fact that a subset of a complete
Hausdorff space is compact if (and only if) it is closed and
32242:
31618:
31340:
31195:
13827:
12120:{\displaystyle U^{\#}~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~{\Big \{}f\in X^{\#}~:~\sup _{u\in U}|f(u)|\leq 1{\Big \}}.}
8483:
when these products are endowed with their product topologies. In terms of function spaces, this bijection could be expressed as
27367:
is inconsequential and can be ignored because it does not have any effect on the resulting set of linear functionals. However, if
15669:
32086:
31588:
31527:
31171:
22629:
20903:
3637:
3028:
1698:
25734:
19851:
31831:
31655:
20953:
13730:
9896:
9155:
4738:
2349:
797:
29194:
25083:
has a weakly convergent subsequence. (This follows by applying the Banach–Alaoglu theorem to a weakly metrizable subspace of
21422:
18280:
17060:
16936:
15952:
is closed can also be reached by applying the following more general result, this time proved using nets, to the special case
15015:
13466:). Because a closed subset of a compact space is compact, the proof of the proposition will be complete once it is shown that
9714:
8662:
6998:
6651:
3216:
2749:
2569:
2253:
30755:
30720:
30674:
30644:
30614:
30579:
30489:
24384:
It will now be shown that the image of the above map is closed, which will complete the proof of the theorem. Given a point
4944:
4511:
1988:
25336:
24033:
16473:
10445:
30571:
26082:
14006:
22863:{\displaystyle m_{\bullet }~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\left(m_{x}\right)_{x\in X}:X\to [0,\infty )}
2923:
2300:
32096:
31593:
31563:
30864:
27177:
1378:
23668:
12974:
9603:
2969:
2094:
720:
32247:
32216:
31867:
31520:
31063:
30543:
24989:
17342:
5433:
5129:
4977:
4900:
4854:
4694:
4467:
1255:
27692:
is compact only requires the so-called "Tychonoff's theorem for compact
Hausdorff spaces," which is equivalent to the
26243:
25221:
21356:{\displaystyle U^{\#}~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\left\{f\in X^{\#}:f(U)\subseteq B_{1}\right\}}
19681:
9952:
9436:
depending on whichever notation is cleanest or most clearly communicates the intended information. In particular, if
32004:
31152:
31043:
30524:
26030:
20897:
20386:
16567:
14577:
13106:
12321:
9769:
5576:
4102:
4024:
1843:
767:
28441:
27712:
24454:
18064:
11425:
11369:
928:
31909:
31422:
30931:
30914:
29316:
28292:
25775:): in this case there actually exists a constructive proof. In the general case of an arbitrary normed space, the
23747:
use the sequential Banach–Alaoglu theorem to extract a subsequence that converges in the weak* topology to a limit
23604:
23483:
21399:
As a side note, with the help of the above elementary proof, it may be shown (see this footnote) that there exist
11488:
4021:
holds because every continuous linear functional is (in particular) a linear functional. For the reverse inclusion
1648:
90:— echos throughout functional analysis.” In 1912, Helly proved that the unit ball of the continuous dual space of
22567:
20781:
20461:
13030:
9798:
9656:
7196:
5328:
5053:
3107:
647:
31067:
29923:
29517:
28094:
25744:
23124:
in the weak-* topology. In fact, the weak* topology on the closed unit ball of the dual of a separable space is
20625:
19618:
16272:
15747:
10692:
10547:
9107:
7827:
4551:
576:
26185:
25104:
22973:
21861:{\displaystyle \prod B_{R_{\bullet }}~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\prod _{x\in X}B_{R_{x}}}
18943:
17537:{\displaystyle f_{\bullet }(z)~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\left(f_{i}(z)\right)_{i\in I}.}
16194:{\displaystyle U_{B}~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\left\{f\in Y^{X}:f(U)\subseteq B\right\}}
11049:
8180:
6709:
but, since tuples are technically just functions from an indexing set, it can also be identified with the space
3984:
31939:
30904:
24780:
23594:
22597:
19003:
17714:
1250:
29463:
29433:) arbitrary intersections and also under finite unions of at least one set). The elementary proof showed that
29405:
29350:
27881:{\displaystyle U_{B_{1}}~=~{\Big (}\prod _{u\in U}B_{1}{\Big )}\times \prod _{x\in X\setminus U}\mathbb {K} .}
27645:
25525:{\displaystyle \int f_{n_{k}}g\,d\mu \to \int fg\,d\mu \qquad {\text{ for all }}g\in L^{q}(\mu )=X^{\prime }.}
22706:
22257:
13452:{\displaystyle B_{r_{x}}~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\{s\in \mathbb {K} :|s|\leq r_{x}\}}
12269:
11005:
10843:
7967:
6176:
190:
32237:
32071:
31673:
31650:
31218:
30708:
30606:
30254:
28726:
18204:
17612:
14230:
7303:
7059:
6590:
19950:
10162:
32122:
31501:
31274:
31208:
31036:
29230:
27331:
26872:
26141:
26076:
23870:
22073:
17025:
13674:
13233:
9306:
is a sequence). In the proofs below, this resulting net may be denoted by any of the following notations
8017:
6259:
6106:
6098:
82:
According to
Lawrence Narici and Edward Beckenstein, the Alaoglu theorem is a “very important result—maybe
23587:
13302:
10657:
This proof will also use the fact that the topology of pointwise convergence is preserved when passing to
32252:
31943:
31238:
28247:{\displaystyle m_{\bullet }~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\left(m_{x}\right)_{x\in X}}
25267:
24653:
14571:
8743:
7579:
7097:
6799:
27292:
27259:
25615:), norm-closures of convex bounded sets in Hilbert spaces or reflexive Banach spaces are weakly compact.
24170:
14924:
14288:
12930:
12686:
8847:
6741:
6044:). This theorem is one example of the utility of having different topologies on the same vector space.
32179:
31716:
31626:
31568:
31483:
31437:
31361:
26501:
25169:
24747:
24347:
20527:
19139:
17580:
17547:
17168:
15807:
14869:
9471:
8052:
6145:
4634:
4226:
3776:
2143:
1609:
27117:
25779:, which is strictly weaker than the axiom of choice and equivalent to Tychonoff's theorem for compact
16764:
16406:
14543:{\displaystyle \,u_{x}~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~{\frac {1}{r_{x}}}\,x\in U.\,}
13239:
10962:
10800:
10753:
8985:
7341:
6833:
6712:
6431:{\displaystyle B_{r}~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\{c\in \mathbb {K} :|c|\leq r\}}
6116:
4265:
4232:
3506:
31975:
31785:
31478:
31294:
30980:
30570:. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY:
29377:
28332:
27466:
26165:
19384:{\displaystyle z_{i}~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\left(f_{i}(x),f_{i}(y)\right)}
15955:
7368:
25952:
25691:
operators). Hence bounded sequences of operators have a weak accumulation point. As a consequence,
21225:{\displaystyle X=(0,\infty )U~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\{ru:r>0,u\in U\}}
15587:
11218:{\displaystyle \sup _{u\in U}|f(u)|\leq r\qquad {\text{ if and only if }}\qquad f(U)\subseteq B_{r}}
5755:, then the polar of a neighborhood is closed and norm-bounded in the dual space. In particular, if
432:{\displaystyle \left\langle x,f\right\rangle ~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~f(x)}
31748:
31743:
31736:
31731:
31603:
31543:
31330:
31228:
31131:
30964:
30936:
30315:
30013:
26220:
26170:
24824:
24643:{\displaystyle \lim _{i}\left(f_{i}(x)\right)_{x\in X}\to \lambda _{\bullet }\quad {\text{ in }}D,}
23583:
22733:
21078:
20906:
20595:
18353:
18255:
17792:
16712:
15839:
15722:
13297:
13272:
12431:
12380:
11931:{\textstyle B_{r}~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\{s\in \mathbb {K} :|s|\leq r\}}
10519:
10134:
6554:
5709:
3729:
1578:
1525:
1026:
536:
28859:
26805:
25882:
24140:
23174:
23038:
22921:
22223:
21659:
21106:
20928:
19928:
19082:
19060:
18011:
17663:
17393:
17200:
16914:
16892:
16870:
16848:
16051:
16025:
14194:
12616:
12193:
11965:
11756:
10911:
10731:
9282:
9060:
8878:
8132:
5479:
4797:
4744:
4517:
4380:
3615:
3557:
3432:
3396:
1553:
1301:
1067:
861:
546:
226:
32009:
31990:
31666:
31646:
31427:
31203:
29627:
28645:
28579:
26931:
26840:
26417:
26314:
26283:
26003:
25807:
25791:
25721:
25612:
25561:
25133:
24943:
24916:
24881:
23916:
23843:
23515:
16370:
14373:
13073:
12437:
6113:
for details). Also required is a proper understanding of the technical details of how the space
6016:
5989:
5938:
5899:
5818:
5202:
5175:
4827:
4069:
3183:
2183:
1998:
1207:
693:
30244:{\displaystyle f(u)=\Pr {}_{u}(f)\in \Pr {}_{u}\left(\prod _{x\in X}B_{m_{x}}\right)=B_{m_{u}},}
27150:
16674:
16239:
14340:
12729:
12583:
12160:
8785:
7795:
7550:
5681:
5582:
5406:
5301:
5026:
4950:
4667:
4440:
4410:
3816:
1331:
32198:
32188:
32172:
31872:
31821:
31721:
31706:
31458:
31402:
31366:
30909:
27607:
26180:
25802:) but is not equivalent to it (said differently, Banach–Alaoglu is also strictly stronger than
25783:
spaces, suffices for the proof of the Banach–Alaoglu theorem, and is in fact equivalent to it.
25740:
25648:
25060:
24111:
23598:
23117:
20661:
17276:
15983:
13350:
11228:
7957:{\displaystyle \prod _{x\in X}B_{r_{x}}\subseteq \prod _{x\in X}\mathbb {K} =\mathbb {K} ^{X},}
7372:
143:
136:
67:
27519:
27434:
23290:
21742:
14409:
9439:
9028:
6963:{\displaystyle s_{\bullet }~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~(s(x))_{x\in X}.}
3579:
32167:
31854:
31836:
31801:
31641:
31165:
30844:
29660:
29598:
29565:
28881:
28836:
holds (the intersection on the left is a closed, rather than open, disk − possibly of radius
28678:
28612:
28550:
28495:
28372:
28134:
27229:
26745:
26217: – On strongly convergent combinations of a weakly convergent sequence in a Banach space
25877:
24972:
23929:
22557:{\textstyle \prod B_{m_{\bullet }}=\cap \operatorname {Box} _{P}\in \operatorname {Box} _{P}}
22113:
21370:
19394:
19280:{\displaystyle z_{\bullet }=\left(z_{i}\right)_{i\in I}:I\to \mathbb {K} \times \mathbb {K} }
17685:
17313:
16444:
14139:
13892:
12554:
12132:
10933:
10664:
10416:
10013:
8101:
6215:
5374:
5099:
3477:
3153:
1062:
540:
31161:
29987:
28524:
28067:
27891:
27551:
26698:
25396:
24533:
20734:
20699:
20667:
17121:
16801:
16737:
14988:
14442:
14370:
was defined in the proposition's statement as being any positive real number that satisfies
14168:
13797:
12655:
12331:
12232:
8905:
7399:
6232:
3277:
3080:
2716:{\displaystyle U^{\circ \circ }=\left\{x\in X~:~\sup _{f\in U^{\circ }}|f(x)|\leq 1\right\}}
1618:
620:
497:{\displaystyle \left\langle X,X^{\#},\left\langle \cdot ,\cdot \right\rangle \right\rangle }
248:
58:. A common proof identifies the unit ball with the weak-* topology as a closed subset of a
32183:
32127:
32106:
31441:
30943:
30662:
30499:
29490:
29436:
29430:
28261:
26725:
26461:
26205:
25658:
24086:
23121:
22895:
20729:
16796:
16320:
16204:
14983:
12260:
11342:
10884:
10725:
10658:
9869:
9022:
8958:
8931:
6227:
6110:
987:
273:
166:
31028:
27578:
27370:
26644:{\displaystyle U^{\circ }={\Big \{}f\in X^{\prime }~:~\sup _{u\in U}|f(u)|\leq 1{\Big \}}}
25694:
25621:
6976:
6777:
3700:
8:
32066:
32061:
32019:
31598:
31407:
31345:
31059:
31017:
30874:
30849:
30712:
28057:{\displaystyle U^{\#}~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~X^{\#}\cap \left.}
25535:
20947:
is a complete
Hausdorff locally convex topological vector space, the same is true of the
5752:
93:
47:
20:
25313:
23793:
23750:
23727:
23645:
23542:
23089:
22284:
17769:
16651:
16347:
16081:
13192:
12531:
12504:
11826:
11113:
10635:
9504:
8824:
7712:
6327:
4611:
32051:
31994:
31928:
31913:
31780:
31770:
31432:
31299:
30894:
30598:
30432:
28839:
28421:
28401:
27499:
27410:
27390:
26911:
26785:
26765:
26678:
26654:
26481:
25976:
25912:
25859:
25839:
25652:
25201:
25110:
25086:
25066:
25042:
24857:
24513:
24206:
24117:
23816:
23773:
23565:
23204:
23154:
23134:
23083:
23065:
22901:
22877:
21402:
21239:
20761:
19831:
18184:
18100:
17420:
17256:
17236:
17148:
16989:
16828:
16631:
16547:
14553:
13982:
13215:
13172:
12484:
12464:
12413:
12386:
12362:
11985:
11945:
11736:
11716:
11686:
11654:
10615:
10595:
9849:
9775:
9087:
8723:
8160:
8155:
7775:
7755:
7735:
7692:
6862:
6628:
6461:
6441:
5965:
5879:
5845:
5798:
5778:
5758:
5734:
4774:
4587:
4208:
4188:
4168:
3304:
2903:
2728:
2549:
2539:{\displaystyle U^{\circ }=\left\{f\in X^{\prime }~:~\sup _{u\in U}|f(u)|\leq 1\right\}}
2417:
2397:
2230:
2210:
1915:
may fail to be a complete space, which is the reason why this proof involves the space
1505:
1358:
1232:
1212:
1199:{\displaystyle U^{\circ }=\left\{f\in X^{\prime }~:~\sup _{u\in U}|f(u)|\leq 1\right\}}
1032:
908:
888:
773:
518:
358:{\displaystyle \left\langle \cdot ,\cdot \right\rangle :X\times X^{\#}\to \mathbb {K} }
279:
206:
39:
30477:
11644:{\displaystyle f~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\operatorname {Id} }
11332:{\displaystyle f(U)~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\{f(u):u\in U\}.}
10750:
then the topology of pointwise convergence (or equivalently, the product topology) on
31763:
31689:
31412:
30926:
30761:
30751:
30726:
30716:
30680:
30670:
30650:
30640:
30620:
30610:
30585:
30575:
30565:
30549:
30539:
30520:
30503:
30485:
27693:
26339:
26280:
is said to be "compact (resp. totally bounded, etc.) in the weak-* topology" if when
26196:
26191:
26175:
26152:
in terms of the FIP, except that it only involves those closed subsets that are also
25795:
25787:
25776:
25033:
20825:
11939:
10794:
9793:
8657:
6223:
6171:
6094:
6087:
2177:
1992:
1615:
To start the proof, some definitions and readily verified results are recalled. When
791:
59:
30779:
7682:{\displaystyle \Pr {}_{z}(s)~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~s(z).}
6055:. This is because the closed unit ball is only a neighborhood of the origin in the
32156:
31726:
31711:
31512:
31417:
31335:
31304:
31284:
31269:
31264:
31259:
30996:
30609:. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer.
30447:
26214:
24080:
15663:
12683:. The proof of the Banach–Alaoglu theorem will be complete once it is shown that
12317:
10124:{\displaystyle \Pr {}_{z}\left(\left(f_{i}\right)_{i\in I}\right)\to \Pr {}_{z}(f)}
9765:
7376:
6219:
6079:
6041:
5298:
This fact, together with (3) and the definition of "totally bounded", implies that
158:
147:
63:
32039:
31578:
31096:
15945:{\displaystyle U_{B_{1}}=\left\{f\in \mathbb {K} ^{X}:f(U)\subseteq B_{1}\right\}}
1600:
This proof will use some of the basic properties that are listed in the articles:
32131:
31979:
31279:
31233:
31181:
31176:
31147:
30921:
30814:
30495:
28130:
27697:
27639:
26446:
26308:
25997:
25930:
25811:
25768:
25752:
25605:
24983:
24378:
23834:
23287:
be a countable dense subset. Then the following defines a metric, where for any
21364:
20900:
20619:
15066:
13460:
12680:
12325:
6544:{\displaystyle rU~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\{ru:u\in U\}}
6090:
6067:
6064:
6056:
6052:
6047:
It should be cautioned that despite appearances, the Banach–Alaoglu theorem does
5928:
4545:
1695:
1643:
1054:
614:
87:
31106:
21103:
The above elementary proof of the Banach–Alaoglu theorem actually shows that if
20824:
of pointwise convergence (also known as the weak-* topology) then the resulting
7247:
This is the reason why many authors write, often without comment, the equality
32162:
32111:
31826:
31468:
31320:
31121:
30859:
30700:
30632:
29400:
26137:
11816:{\displaystyle \mathbb {K} =\mathbb {R} {\text{ or }}\mathbb {K} =\mathbb {C} }
11592:{\displaystyle X~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\mathbb {K} }
10003:
7383:), also comes equipped with associated maps that are known as its (coordinate)
6083:
1573:
1058:
55:
29913:{\displaystyle f\in P~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~U^{\#}}
25814:
of the closed unit ball of the dual space) is replaced with the conclusion of
22281:
denotes the set of all such products of closed balls containing the polar set
6309:
An explanation of these details is now given for readers who are interested.
2893:{\displaystyle U^{\#}=\left\{f\in X^{\#}~:~\sup _{u\in U}|f(u)|\leq 1\right\}}
146:). The proof for the general case was published in 1940 by the mathematician
32231:
32146:
32056:
31999:
31959:
31887:
31862:
31806:
31758:
31694:
31473:
31397:
31126:
31111:
31101:
30899:
30882:
30839:
30684:
30654:
30624:
30553:
30538:. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press.
30507:
29344:
28286:
28255:
26149:
26129:
22700:
21940:{\displaystyle R_{\bullet }=\left(R_{x}\right)_{x\in X}\in \mathbb {R} ^{X},}
21732:{\displaystyle P~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~U^{\circ }}
21233:
20948:
13822:
13463:
12525:
12264:
12257:
8480:
8479:
canonically identifies these two
Cartesian products; moreover, this map is a
7380:
5924:
5863:
4605:
162:
132:
51:
30765:
30730:
30589:
30451:
21006:
A closed subset of a complete space is complete, so by the lemma, the space
19821:{\displaystyle A(x,y)~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~x+y.}
6060:
4370:{\displaystyle U^{\#}\cap X^{\prime }=U^{\circ }\cap X^{\prime }=U^{\circ }}
32193:
32141:
32101:
32091:
31969:
31816:
31811:
31608:
31558:
31463:
31116:
31086:
30738:
30561:
29851:{\displaystyle f\in \left(\cap \operatorname {Box} _{P}\right)\cap X^{\#}.}
26407:{\displaystyle \left(X^{\prime },\sigma \left(X^{\prime },X\right)\right),}
26208: – On when a space equals the closed convex hull of its extreme points
26188: – Relates three different kinds of weak compactness in a Banach space
24979:
24875:
24444:{\displaystyle \lambda _{\bullet }=\left(\lambda _{x}\right)_{x\in X}\in D}
22759:
12652:
Thus the hypotheses of the above proposition are satisfied, and so the set
11600:
8780:
7186:{\displaystyle s(x)~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~s_{x}}
3974:{\displaystyle \left(X^{\prime },\sigma \left(X^{\prime },X\right)\right):}
297:
139:
30110:{\displaystyle f\in \cap \operatorname {Box} _{P}=\prod B_{m_{\bullet }},}
29306:{\displaystyle \cap \operatorname {Box} _{P}\in \operatorname {Box} _{P},}
1908:{\displaystyle \left(X^{\prime },\sigma \left(X^{\prime },X\right)\right)}
32151:
32136:
32029:
31923:
31918:
31903:
31882:
31846:
31753:
31573:
31392:
31382:
31289:
31091:
30834:
30810:
28064:
Rewriting the definition in this way helps make it apparent that the set
26442:
25688:
18174:{\displaystyle M(c)~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~sc.}
5896:
is a normed space then the closed unit ball in the continuous dual space
4791:
2344:
2294:
1605:
1548:
505:
194:
24:
28983:{\displaystyle m_{x}=\inf _{}\left\{R_{x}:R_{\bullet }\in T_{P}\right\}}
31964:
31877:
31841:
31701:
31583:
31325:
31157:
30829:
26153:
25728:
23125:
22755:
19608:{\displaystyle f_{\bullet }(y)=\left(f_{i}(y)\right)_{i\in I}\to f(y),}
16075:
11339:
As a side note, this characterization does not hold if the closed ball
9057:
is any function then the net (or sequence) that results from "plugging
43:
36:
30387:
30385:
30383:
30381:
30379:
30377:
30364:
30362:
30360:
25724:, if equipped with either the weak operator or the ultraweak topology.
22213:{\displaystyle m_{x}=\inf \left\{R_{x}:R_{\bullet }\in T_{P}\right\},}
19509:{\displaystyle f_{\bullet }(x)=\left(f_{i}(x)\right)_{i\in I}\to f(x)}
32116:
31933:
31001:
30887:
30854:
24906:
2724:
1601:
1095:
198:
135:
proved that the closed unit ball in the continuous dual space of any
25655:
which is in turn the weak-* topology with respect to the predual of
32081:
32076:
32034:
32014:
31984:
31775:
30374:
30357:
25128:
24737:{\displaystyle g(x)=\lambda _{x}\qquad {\text{ for every }}x\in X,}
23911:
is the space of continuous functions vanishing at infinity, by the
22871:
22754:
this may be used as an alternative definition of this (necessarily
17789:
it follows that each of the following nets of scalars converges in
13972:{\textstyle \Pr {}_{z}:\prod _{x\in X}\mathbb {K} \to \mathbb {K} }
11248:
8818:
24381:. This map's inverse, defined on its image, is also continuous.
23128:, and thus compactness and sequential compactness are equivalent.
21683:
can also be chosen to be "minimal" in the following sense: using
14000:
12355:
is compact in the weak-* topology or "weak-* compact" for short).
6317:
Primer on product/function spaces, nets, and pointwise convergence
32024:
24967:
is infinite dimensional then its closed unit ball is necessarily
22248:
19748:{\displaystyle A:\mathbb {K} \times \mathbb {K} \to \mathbb {K} }
9768:. It is well known that the product topology is identical to the
9593:{\displaystyle \left(F\left(x_{i}\right)\right)_{i\in I}\to F(x)}
5671:{\displaystyle \left(X^{\#},\sigma \left(X^{\#},X\right)\right),}
5291:{\displaystyle \left(X^{\#},\sigma \left(X^{\#},X\right)\right).}
3386:{\displaystyle \left(X^{\#},\sigma \left(X^{\#},X\right)\right).}
1980:{\displaystyle \left(X^{\#},\sigma \left(X^{\#},X\right)\right).}
1766:{\displaystyle \left(X^{\#},\sigma \left(X^{\#},X\right)\right).}
1483:{\displaystyle \left(X^{\#},\sigma \left(X^{\#},X\right)\right).}
23867:
is the space of finite Radon measures on the real line (so that
23665:
one common strategy is to first construct a minimizing sequence
21068:{\displaystyle \left(X^{\#},\sigma \left(X^{\#},X\right)\right)}
20889:{\displaystyle \left(X^{\#},\sigma \left(X^{\#},X\right)\right)}
14129:{\displaystyle \Pr {}_{x}\left(U^{\#}\right)\subseteq B_{r_{x}}}
9272:{\displaystyle \left(F\left(x_{i}\right)\right)_{i=1}^{\infty }}
5568:{\displaystyle \left(X^{\#},\sigma \left(X^{\#},X\right)\right)}
3902:{\displaystyle \left(X^{\#},\sigma \left(X^{\#},X\right)\right)}
2084:{\displaystyle \left(X^{\#},\sigma \left(X^{\#},X\right)\right)}
1835:{\displaystyle \left(X^{\#},\sigma \left(X^{\#},X\right)\right)}
23280:{\displaystyle x_{\bullet }=\left(x_{n}\right)_{n=1}^{\infty }}
8982:
may also be denoted by the usual function parentheses notation
7964:
where under the above identification of tuples with functions,
5706:
2227:
in one of these topologies if and only if it also converges to
30750:. Pure and Applied Mathematics. New York: Wiley-Interscience.
28829:{\displaystyle \cap \left\{B_{a}:a\in A\right\}=B_{\inf _{}A}}
23915:), the sequential Banach–Alaoglu theorem is equivalent to the
16671:
so too must this net's limit belong to this closed set; thus
15069:
for readers who are not familiar with this result). The set
13873:{\displaystyle \prod _{x\in X}\mathbb {K} =\mathbb {K} ^{X}.}
6646:
26136:
because a topological space is compact if and only if every
22692:{\textstyle \prod B_{m_{\bullet }}=\prod _{x\in X}B_{m_{x}}}
16986:
in the topology of pointwise convergence. (The vector space
15712:{\displaystyle \prod _{x\in X}\mathbb {K} =\mathbb {K} ^{X}}
12407:
Proof that Banach–Alaoglu follows from the proposition above
7367:(or conversely). However, the Cartesian product, being the
7293:{\displaystyle \mathbb {K} ^{X}=\prod _{x\in X}\mathbb {K} }
5606:
is a closed (by (2)) and totally bounded (by (4)) subset of
70:, this product, and hence the unit ball within, is compact.
31058:
15644:
3690:{\displaystyle \left\{s\in \mathbb {K} :|s|\leq 1\right\},}
3067:{\displaystyle U^{\circ \circ \circ }\subseteq U^{\circ }.}
20999:{\textstyle \mathbb {K} ^{X}=\prod _{x\in X}\mathbb {K} .}
19918:{\displaystyle A\left(z_{\bullet }\right)\to A(f(x),f(y))}
9942:{\textstyle \mathbb {K} ^{X}=\prod _{x\in X}\mathbb {K} ,}
7193:; this function's "tuple of values" is the original tuple
24023:{\displaystyle D_{x}=\{c\in \mathbb {C} :|c|\leq \|x\|\}}
16979:{\textstyle \mathbb {K} ^{X}=\prod _{x\in X}\mathbb {K} }
15058:{\textstyle \mathbb {K} ^{X}=\prod _{x\in X}\mathbb {K} }
13786:{\displaystyle U^{\#}\subseteq \prod _{x\in X}B_{r_{x}}.}
11110:
An important fact used by the proof is that for any real
9757:{\textstyle \mathbb {K} ^{X}=\prod _{x\in X}\mathbb {K} }
9206:{\displaystyle \left(F\left(x_{i}\right)\right)_{i\in I}}
4298:
as desired. Using (1) and the fact that the intersection
2385:{\displaystyle \left\langle X,X^{\prime }\right\rangle .}
851:{\displaystyle f_{\bullet }=\left(f_{i}\right)_{i\in I},}
29220:{\displaystyle P\subseteq \cap \operatorname {Box} _{P}}
25735:
Krein–Milman theorem § Relation to other statements
21473:{\displaystyle m_{\bullet }=\left(m_{x}\right)_{x\in X}}
18343:{\displaystyle M\left(f_{\bullet }(x)\right)\to M(f(x))}
17111:{\displaystyle f_{\bullet }=\left(f_{i}\right)_{i\in I}}
8713:{\displaystyle x_{\bullet }=\left(x_{i}\right)_{i\in I}}
7338:
is sometimes taken as the definition of the set of maps
7049:{\displaystyle s_{\bullet }=\left(s_{x}\right)_{x\in X}}
6702:{\displaystyle s_{\bullet }=\left(s_{x}\right)_{x\in X}}
3267:{\displaystyle f_{\bullet }=\left(f_{i}\right)_{i\in I}}
2782:{\displaystyle \left\langle X,X^{\prime }\right\rangle }
2602:{\displaystyle \left\langle X,X^{\prime }\right\rangle }
2286:{\displaystyle \left\langle X,X^{\prime }\right\rangle }
296:
and these two spaces are henceforth associated with the
30515:
Meise, Reinhold; Vogt, Dietmar (1997). "Theorem 23.5".
26210:
Pages displaying short descriptions of redirect targets
23601:. For instance, if one wants to minimize a functional
10612:
in the product topology if and only if it converges to
2140:
This can be readily verified by showing that given any
142:
is sequentially weak-* compact (Banach only considered
29886:
28438:
can be associated with this unique (minimum) function
28199:
27944:
26995:
25772:
25386:{\displaystyle \left(f_{n_{k}}\right)_{k=1}^{\infty }}
22791:
22632:
22570:
22504:
22335:
21976:
21811:
21705:
21285:
21172:
20956:
20070:
19792:
19316:
18773:
18458:
18148:
17475:
16939:
16537:{\displaystyle \left(f_{i}(u)\right)_{i\in I}\to f(u)}
16130:
15568:
15111:
15018:
14495:
14062:{\textstyle U^{\#}\subseteq \prod _{x\in X}B_{r_{x}},}
14009:
13924:
13677:
13495:
13389:
13305:
12782:
12017:
11875:
11852:
11624:
11570:
11282:
11008:
10965:
10846:
10803:
10756:
10509:{\displaystyle \left(f_{i}(z)\right)_{i\in I}\to f(z)}
10284:
10220:
9899:
9717:
9393:
8875:
is also a net. As with sequences, the value of a net
7650:
7306:
7159:
7062:
6912:
6593:
6503:
6375:
6262:
6179:
6070:
topological vector spaces must be finite-dimensional.
5866:). Consequently, this theorem can be specialized to:
403:
30809:
30318:
30257:
30123:
30061:
30016:
29990:
29926:
29864:
29802:
29692:
29663:
29630:
29601:
29568:
29520:
29493:
29487:
are not empty and moreover, it also even showed that
29466:
29439:
29408:
29380:
29353:
29319:
29273:
29233:
29197:
28996:
28913:
28884:
28862:
28856:− because it is an intersection of closed subsets of
28842:
28770:
28729:
28681:
28648:
28615:
28582:
28553:
28527:
28498:
28444:
28424:
28404:
28375:
28335:
28295:
28264:
28176:
28137:
28097:
28070:
27921:
27894:
27778:
27715:
27648:
27610:
27581:
27554:
27522:
27502:
27469:
27437:
27413:
27393:
27373:
27334:
27295:
27262:
27232:
27180:
27153:
27120:
26966:
26934:
26914:
26875:
26843:
26808:
26788:
26768:
26748:
26728:
26701:
26681:
26657:
26544:
26504:
26484:
26464:
26420:
26348:
26317:
26286:
26246:
26118:{\displaystyle B\cap \bigcap _{C\in {\mathcal {C}}}C}
26085:
26033:
26006:
25979:
25955:
25915:
25885:
25862:
25842:
25697:
25661:
25624:
25564:
25538:
25425:
25399:
25339:
25316:
25270:
25224:
25204:
25172:
25136:
25113:
25089:
25069:
25045:
24992:
24946:
24919:
24884:
24860:
24827:
24783:
24750:
24690:
24656:
24562:
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24516:
24457:
24390:
24350:
24229:
24209:
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23293:
23227:
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22880:
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22709:
22600:
22309:
22287:
22260:
22226:
22144:
22116:
22076:
21953:
21874:
21778:
21745:
21689:
21662:
21486:
21425:
21405:
21373:
21262:
21242:
21135:
21109:
21081:
21012:
20931:
20833:
20784:
20764:
20737:
20702:
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20628:
20622:
actually also follows from its corollary below since
20598:
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19953:
19931:
19854:
19834:
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19717:
19684:
19621:
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19397:
19293:
19210:
19142:
19085:
19063:
19006:
18946:
18412:
18356:
18283:
18258:
18207:
18187:
18123:
18103:
18067:
18014:
17819:
17795:
17772:
17717:
17688:
17666:
17615:
17583:
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17443:
17423:
17396:
17345:
17316:
17279:
17259:
17239:
17203:
17171:
17151:
17124:
17063:
17028:
16992:
16917:
16895:
16873:
16851:
16831:
16804:
16767:
16740:
16715:
16677:
16654:
16634:
16570:
16550:
16476:
16447:
16409:
16373:
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16323:
16275:
16242:
16207:
16107:
16084:
16054:
16028:
15986:
15958:
15865:
15842:
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15672:
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14991:
14927:
14872:
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13175:
13109:
13076:
13033:
12977:
12933:
12759:
12732:
12689:
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12586:
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12507:
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11994:
11968:
11948:
11829:
11781:
11759:
11739:
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11689:
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11554:
11548:
will not change this; for counter-examples, consider
11491:
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9285:
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7695:
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7402:
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5992:
5968:
5941:
5902:
5882:
5848:
5821:
5815:
is the closed unit ball in the continuous dual space
5801:
5781:
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5684:
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5436:
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1103:
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209:
96:
31542:
30667:
Topological Vector Spaces, Distributions and
Kernels
26201:
Pages displaying wikidata descriptions as a fallback
25729:
Relation to the axiom of choice and other statements
23111:
13724:
The following statements guarantee this conclusion:
8652:
Notation for nets and function composition with nets
2959:{\displaystyle \left\langle X,X^{\#}\right\rangle .}
2336:{\displaystyle \left\langle X,X^{\#}\right\rangle ,}
27219:{\displaystyle U^{\circ ,\tau }=U^{\circ ,\sigma }}
26742:can be ignored. To clarify what is meant, suppose
8844:which by definition is just a function of the form
5982:is an infinite dimensional normed space then it is
1411:{\displaystyle \left\langle X,X^{\#}\right\rangle }
31388:Spectral theory of ordinary differential equations
30426:
30424:
30324:
30304:
30243:
30109:
30047:
30002:
29976:
29912:
29850:
29796:is immediate; to prove the reverse inclusion, let
29788:
29678:
29649:
29616:
29587:
29554:
29506:
29479:
29452:
29421:
29391:
29366:
29335:
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29259:
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29183:
28982:
28899:
28870:
28848:
28828:
28756:
28696:
28667:
28634:
28601:
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28539:
28513:
28484:
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28390:
28361:
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27907:
27880:
27764:
27684:
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27540:
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27399:
27379:
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27314:
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27218:
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27106:
26952:
26920:
26900:
26861:
26829:
26794:
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26687:
26663:
26643:
26530:
26490:
26470:
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26406:
26330:
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26067:
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25985:
25965:
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25901:
25868:
25848:
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25679:
25639:
25598:
25586:
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25411:
25385:
25325:
25302:
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25210:
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25158:
25119:
25095:
25075:
25051:
25024:
24959:
24932:
24897:
24866:
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24814:
24769:
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24676:
24642:
24548:
24522:
24502:
24443:
24369:
24334:
24215:
24195:
24159:
24126:
24102:
24071:
24022:
23947:
23903:
23859:
23825:
23805:
23782:
23762:
23739:
23717:{\displaystyle x_{1},x_{2},\ldots \in X^{\prime }}
23716:
23657:
23634:
23574:
23554:
23531:
23504:
23472:
23311:
23279:
23213:
23193:
23163:
23143:
23101:
23074:
23054:
23028:
22962:
22910:
22886:
22862:
22746:
22722:
22691:
22616:
22594:denotes the intersection of all sets belonging to
22586:
22556:
22490:
22296:
22273:
22239:
22212:
22131:
22102:
22062:
21939:
21860:
21764:
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21648:
21472:
21411:
21389:
21355:
21248:
21224:
21121:
21087:
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20939:
20888:
20816:
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20683:
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20604:
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19607:
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19383:
19279:
19196:
19124:
19071:
19049:
18992:
18932:
18398:
18342:
18269:
18244:
18193:
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18109:
18089:
18053:
17997:
17806:
17781:
17758:
17703:
17674:
17652:
17601:
17569:
17536:
17429:
17409:
17381:
17331:
17300:
17265:
17245:
17225:
17189:
17157:
17137:
17110:
17049:
16998:
16978:
16925:
16903:
16881:
16859:
16837:
16817:
16782:
16753:
16721:
16701:
16663:
16640:
16620:
16556:
16536:
16462:
16433:
16395:
16359:
16336:
16309:
16261:
16220:
16193:
16093:
16066:
16040:
16008:
15972:
15944:
15848:
15828:
15796:
15736:
15711:
15654:
15057:
15004:
14965:
14913:
14856:
14632:
14562:
14542:
14457:
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14398:
14362:
14329:
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14183:
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13971:
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13716:
13663:
13451:
13341:
13281:
13261:
13224:
13204:
13181:
13161:
13095:
13062:
13020:{\displaystyle U^{\circ }=U^{\#}\cap X^{\prime },}
13019:
12962:
12916:
12745:
12718:
12671:
12644:
12605:
12572:
12543:
12516:
12493:
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10604:
10584:
10533:
10508:
10434:
10401:
10206:
10148:
10123:
10031:
9995:
9941:
9885:
9858:
9838:
9784:
9756:
9696:
9646:{\displaystyle F\left(x_{\bullet }\right)\to F(x)}
9645:
9592:
9516:
9493:
9460:
9428:
9298:
9271:
9205:
9144:
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3495:
3466:
3421:
3385:
3313:
3293:
3266:
3205:
3172:
3142:
3096:
3066:
3015:{\displaystyle U^{\circ }=U^{\#}\cap X^{\prime }.}
3014:
2958:
2912:
2892:
2781:
2737:
2715:
2601:
2558:
2538:
2426:
2406:
2384:
2335:
2285:
2239:
2219:
2199:
2168:
2133:{\displaystyle \sigma \left(X^{\prime },X\right).}
2132:
2083:
2014:
1979:
1907:
1834:
1765:
1686:
1634:
1592:
1564:
1539:
1514:
1482:
1410:
1367:
1347:
1320:
1290:
1241:
1221:
1198:
1086:
1041:
1008:
976:
917:
897:
877:
850:
782:
759:{\displaystyle \sigma \left(X^{\prime },X\right).}
758:
709:
682:
636:
605:
565:
527:
496:
431:
357:
288:
264:
237:
215:
123:
30533:
30519:. Oxford, England: Clarendon Press. p. 264.
30391:
30368:
28011:
27978:
27840:
27807:
27099:
27011:
26636:
26560:
25759:+ the axiom of choice, which is often denoted by
25755:. Most mainstream functional analysis relies on
25025:{\displaystyle \sigma \left(X,X^{\prime }\right)}
24849:
17609:which has the topology of pointwise convergence,
17382:{\displaystyle f_{\bullet }(z):I\to \mathbb {K} }
15498:
15382:
15240:
15125:
13587:
13511:
12874:
12798:
12109:
12033:
5469:{\displaystyle \sigma \left(X^{\prime },X\right)}
5165:{\displaystyle \sigma \left(X^{\prime },X\right)}
5013:{\displaystyle \sigma \left(X^{\prime },X\right)}
4936:{\displaystyle \sigma \left(X^{\prime },X\right)}
4890:{\displaystyle \sigma \left(X^{\prime },X\right)}
4730:{\displaystyle \sigma \left(X^{\prime },X\right)}
4503:{\displaystyle \sigma \left(X^{\prime },X\right)}
1493:
1291:{\displaystyle \sigma \left(X^{\prime },X\right)}
32229:
30163:
30139:
29928:
28928:
28815:
27054:
26591:
26498:is (originally) endowed with, then the equality
26273:{\displaystyle B^{\prime }\subseteq X^{\prime }}
25786:The Banach–Alaoglu theorem is equivalent to the
25257:{\displaystyle {\frac {1}{p}}+{\frac {1}{q}}=1.}
24564:
24344:This map is injective and it is continuous when
23586:similar to the one employed in the proof of the
22158:
19704:{\displaystyle \mathbb {K} \times \mathbb {K} .}
15195:
14811:
14583:
14234:
14076:
13925:
13542:
13169:which states exactly that the linear functional
13112:
12829:
12064:
11494:
11431:
11233:
11140:
11054:
10304:
10221:
10167:
10100:
10046:
9996:{\displaystyle \left(f_{i}\right)_{i\in I}\to f}
7799:
7617:
7554:
7429:
4165:which states exactly that the linear functional
4108:
2842:
2658:
2488:
1148:
905:in this topology if and only if for every point
30421:
26068:{\displaystyle \{B\cap C:C\in {\mathcal {C}}\}}
25651:(the weak operator topology is weaker than the
25611:As norm-closed, convex sets are weakly closed (
23642:on the dual of a separable normed vector space
20451:{\displaystyle f_{\bullet }(x+y)\to f(x)+f(y).}
16621:{\displaystyle f_{i}(u)\in f_{i}(U)\subseteq B}
14633:{\displaystyle \;\sup _{u\in U}|f(u)|\leq 1,\,}
13162:{\displaystyle \;\sup _{u\in U}|f(u)|\leq 1,\,}
4664:it is possible to prove that this implies that
4158:{\displaystyle \;\sup _{u\in U}|f(u)|\leq 1,\,}
4059:{\displaystyle \,U^{\#}\subseteq U^{\circ },\,}
157:is a generalization of the original theorem by
30534:Narici, Lawrence; Beckenstein, Edward (2011).
28485:{\displaystyle m_{\bullet }:X\to [0,\infty ).}
27765:{\displaystyle U_{B_{1}}=\prod _{x\in X}C_{x}}
26223: – Vector space with a notion of nearness
24503:{\displaystyle \left(f_{i}(x)\right)_{x\in X}}
20664:and any subset of such a space (in particular
18090:{\displaystyle M:\mathbb {K} \to \mathbb {K} }
14069:it is sufficient (and necessary) to show that
11478:{\displaystyle \;\sup _{u\in U}|f(u)|\leq r\;}
11415:{\displaystyle \{c\in \mathbb {K} :|c|<r\}}
10661:. This means, for example, that if for every
3429:it is sufficient (and necessary) to show that
977:{\displaystyle \left(f_{i}(x)\right)_{i\in I}}
31528:
31044:
30795:
30597:
29336:{\displaystyle \cap \operatorname {Box} _{P}}
28322:{\displaystyle R_{\bullet }\leq S_{\bullet }}
25929:has the following property, which is called (
23635:{\displaystyle F:X^{\prime }\to \mathbb {R} }
23505:{\displaystyle \langle \cdot ,\cdot \rangle }
22587:{\textstyle \bigcap \operatorname {Box} _{P}}
15365:
15257:
11541:{\displaystyle \;\sup _{u\in U}|f(u)|<r\;}
8610:
8589:
7393:canonical projection of the Cartesian product
1687:{\displaystyle \sigma \left(X^{\#},X\right),}
26062:
26034:
24017:
24014:
24008:
23975:
23499:
23487:
23459:
23434:
23414:
23389:
21219:
21186:
20817:{\displaystyle \sigma \left(X^{\#},X\right)}
20691:) is closed if and only if it is complete.
20517:{\displaystyle f_{\bullet }(x+y)\to f(x+y),}
15804:is an intersection of two closed subsets of
13446:
13403:
13063:{\displaystyle U^{\#}\subseteq X^{\prime }.}
11925:
11889:
11409:
11373:
11323:
11296:
9839:{\displaystyle \left(f_{i}\right)_{i\in I},}
9697:{\displaystyle F\circ x_{\bullet }\to F(x).}
7237:{\displaystyle \left(s_{x}\right)_{x\in X}.}
6538:
6517:
6425:
6389:
5364:{\displaystyle \sigma \left(X^{\#},X\right)}
5089:{\displaystyle \sigma \left(X^{\#},X\right)}
3143:{\displaystyle \sigma \left(X^{\#},X\right)}
683:{\displaystyle \sigma \left(X^{\#},X\right)}
30430:
29977:{\displaystyle \sup _{u\in U}|f(u)|\leq 1,}
29555:{\displaystyle \left(r_{x}\right)_{x\in X}}
28122:{\displaystyle \prod _{x\in X}\mathbb {K} }
25751:) axiomatic framework is equivalent to the
21772:as in the proof) and defining the notation
20653:{\displaystyle \prod _{x\in X}\mathbb {K} }
19671:{\displaystyle z_{\bullet }\to (f(x),f(y))}
16310:{\displaystyle \left(f_{i}\right)_{i\in I}}
15797:{\displaystyle U_{B_{1}}\cap X^{\#}=U^{\#}}
15719:since it is a product of closed subsets of
11962:need not be endowed with any topology, but
10717:{\displaystyle S_{x}\subseteq \mathbb {K} }
10585:{\displaystyle \left(f_{i}\right)_{i\in I}}
9145:{\displaystyle F\circ x_{\bullet }:I\to Y,}
7865:{\displaystyle \left(r_{x}\right)_{x\in X}}
5199:is identical to the subspace topology that
4577:{\displaystyle U\subseteq U^{\circ \circ }}
3025:A well known fact about polar sets is that
606:{\displaystyle X^{\prime }\subseteq X^{\#}}
31535:
31521:
31051:
31037:
30802:
30788:
28990:so that the previous set equality implies
28285:with respect to natural induced pointwise
26802:is (also) a neighborhood of the origin in
25739:The Banach–Alaoglu may be proven by using
24289:
24285:
24267:
24263:
24240:
24110:is a compact subset of the complex plane,
23029:{\displaystyle U^{\circ }=\left^{\circ },}
18993:{\displaystyle f_{\bullet }(sx)\to sf(x).}
15452:
15314:
15282:
15275:
15268:
15184:
15150:
15143:
15136:
14581:
13110:
12481:is a neighborhood of the origin. Because
11537:
11492:
11474:
11429:
11100:{\displaystyle \sup _{u\in U}|f(u)|\leq r}
10203:
10166:
8603:
8575:
8571:
8523:
8519:
8500:
8422:
8380:
8376:
8272:
8268:
8240:
8216:{\displaystyle X=U\,\cup \,(X\setminus U)}
4106:
4014:{\displaystyle U^{\circ }\subseteq U^{\#}}
2920:with respect to the canonical dual system
191:Topological vector space § Dual space
30711:. Vol. 96 (2nd ed.). New York:
30631:
30514:
30433:"A Geometric Form of the Axiom of Choice"
30415:
29388:
29381:
28878:and so must itself be closed). For every
28864:
28115:
28042:
27871:
25468:
25449:
24815:{\displaystyle F(g)=\lambda _{\bullet }.}
24761:
24670:
24361:
24254:
24184:
23985:
23894:
23628:
23381:
23367:
22737:
22617:{\displaystyle \operatorname {Box} _{P}.}
22010:
21924:
20989:
20959:
20933:
20646:
19933:
19741:
19733:
19725:
19694:
19686:
19273:
19265:
19065:
19050:{\displaystyle f_{\bullet }(sx)\to f(sx)}
18665:
18260:
18083:
18075:
17797:
17759:{\displaystyle x,y,sx,{\text{ and }}x+y,}
17668:
17586:
17375:
17174:
17037:
16972:
16942:
16919:
16897:
16875:
16853:
16770:
16228:in the topology of pointwise convergence.
15966:
15899:
15813:
15727:
15699:
15690:
15621:
15442:
15438:
15298:
15292:
15168:
15122:
15101:
15051:
15021:
14629:
14539:
14526:
14472:
14259:
14255:
13965:
13957:
13857:
13848:
13413:
13158:
11970:
11899:
11809:
11801:
11791:
11783:
11761:
11585:
11383:
11236:
11232:
11036:{\textstyle \prod _{x\in X}\mathbb {K} .}
11026:
10916:
10874:{\textstyle \prod _{x\in X}\mathbb {K} .}
10864:
10736:
10710:
10524:
10139:
9932:
9902:
9750:
9720:
8852:
8543:
8528:
8506:
8332:
8297:
8261:
8197:
8193:
8029:
7941:
7932:
7596:
7469:
7461:
7347:
7324:
7286:
7256:
7114:
7080:
6839:
6816:
6752:
6718:
6611:
6399:
6207:{\textstyle \prod _{x\in X}\mathbb {K} ,}
6197:
6156:
6122:
4154:
4055:
4028:
3653:
3620:
3562:
1583:
1558:
1530:
351:
231:
31341:Group algebra of a locally compact group
30637:Handbook of Analysis and Its Foundations
29480:{\displaystyle \operatorname {Box} _{P}}
29422:{\displaystyle \operatorname {Box} _{P}}
29367:{\displaystyle \operatorname {Box} _{P}}
27685:{\displaystyle \prod _{x\in X}B_{r_{x}}}
27226:because both of these sets are equal to
22723:{\displaystyle \operatorname {Box} _{P}}
22626:This implies (among other things) that
22274:{\displaystyle \operatorname {Box} _{P}}
17006:need not be endowed with any topology).
12328:in functional analysis, this means that
12309:{\displaystyle \prod _{x\in X}B_{r_{x}}}
9524:then the conclusion commonly written as
8007:{\displaystyle \prod _{x\in X}B_{r_{x}}}
7331:{\textstyle \prod _{x\in X}\mathbb {K} }
7087:{\textstyle \prod _{x\in X}\mathbb {K} }
6625:is usually thought of as the set of all
6618:{\textstyle \prod _{x\in X}\mathbb {K} }
30305:{\displaystyle |f(u)|\leq m_{u}\leq 1,}
28757:{\displaystyle A\subseteq [0,\infty ),}
26148:is similar to this characterization of
26145:
26144:(FIP) has non-empty intersection. The
26133:
25103:; or, more succinctly, by applying the
24986:if and only if its closed unit ball is
18245:{\displaystyle f_{\bullet }(x)\to f(x)}
17653:{\displaystyle f_{\bullet }(z)\to f(z)}
14278:{\displaystyle \Pr {}_{x}(f)\,=\,f(x),}
10006:in the product topology if and only if
6583:Identification of functions with tuples
1701:topological vector space is denoted by
766:The weak-* topology is also called the
131:is countably weak-* compact. In 1932,
32230:
31674:Uniform boundedness (Banach–Steinhaus)
30699:
30661:
25310:be a bounded sequence of functions in
24072:{\displaystyle D=\prod _{x\in X}D_{x}}
20023:{\displaystyle A(f(x),f(y))=f(x)+f(y)}
13717:{\textstyle \prod _{x\in X}B_{r_{x}}.}
10207:{\displaystyle \;\Pr {}_{z}(f)=f(z)\;}
9152:although this is typically denoted by
8955:; however, for this proof, this value
7547:where under the above identification,
6438:will denote the closed ball of radius
6302:{\textstyle \prod _{x\in X}B_{r_{x}}.}
4377:is closed in the subspace topology on
2566:with respect to the canonical pairing
31516:
31032:
30783:
30669:. Mineola, N.Y.: Dover Publications.
30560:
30476:
30403:
30351:
29260:{\displaystyle m_{\bullet }\in T_{P}}
27360:{\displaystyle (X,\sigma )^{\prime }}
26901:{\displaystyle (X,\sigma )^{\prime }}
25767:rely upon the axiom of choice in the
23904:{\displaystyle X=C_{0}(\mathbb {R} )}
22251:formula can be used to define them.
22103:{\displaystyle m_{\bullet }\in T_{P}}
18504: by definition of notation
17050:{\displaystyle f\in \mathbb {K} ^{X}}
13342:{\textstyle \prod _{x\in X}B_{r_{x}}}
8042:{\displaystyle s\in \mathbb {K} ^{X}}
7872:are non-negative real numbers. Then
6097:and familiarity with the fact that a
5775:is the open (or closed) unit ball in
1375:with respect to the canonical system
177:and it is commonly called simply the
30572:McGraw-Hill Science/Engineering/Math
27915:may thus equivalently be defined by
26837:Denote the continuous dual space of
26722:) and that the rest of the topology
24878:and endow its continuous dual space
30865:Topologies on spaces of linear maps
30737:
30517:Introduction to Functional Analysis
25876:denote the closed unit ball of its
25647:are precompact with respect to the
25303:{\displaystyle f_{1},f_{2},\ldots }
24975:(despite it being weak-* compact).
24677:{\displaystyle g:X\to \mathbb {C} }
21419:-indexed non-negative real numbers
12501:is a neighborhood of the origin in
12403:is a neighborhood of the origin).
8772:{\displaystyle x_{\bullet }:I\to X}
7603:{\displaystyle s:X\to \mathbb {K} }
7121:{\displaystyle s:X\to \mathbb {K} }
6823:{\displaystyle s:X\to \mathbb {K} }
6226:they induce on subsets such as the
6105:of the origin (see the articles on
6073:
2414:be a neighborhood of the origin in
1418:and it is also a compact subset of
169:. This theorem is also called the
13:
30693:
29905:
29840:
29781:
29736:
28745:
28473:
28143:
28076:
27963:
27927:
27900:
27560:
27352:
27315:{\displaystyle U^{\circ ,\sigma }}
27282:{\displaystyle U^{\circ ,\sigma }}
27238:
27039:
26893:
26707:
26576:
26523:
26426:
26380:
26359:
26323:
26292:
26265:
26252:
26156:(rather than all closed subsets).
26105:
26057:
26012:
25958:
25891:
25514:
25378:
25185:
25012:
24952:
24925:
24890:
24762:
24362:
24255:
24196:{\displaystyle B_{1}^{\,\prime },}
24185:
24149:
23852:
23709:
23619:
23524:
23362:
23272:
23183:
22955:
22930:
22854:
21757:
21596:
21535:
21513:
21379:
21315:
21268:
21151:
21044:
21023:
20865:
20844:
20798:
20743:
20708:
20676:
17215:
17130:
16810:
16746:
15789:
15776:
14997:
14966:{\displaystyle f(x)\in B_{r_{x}},}
14330:{\displaystyle f(x)\in B_{r_{x}}.}
14209:
14097:
14015:
13806:
13739:
13617:
13527:
13478:
13254:
13088:
13052:
13039:
13009:
12996:
12963:{\displaystyle U^{\circ }=U^{\#}:}
12952:
12906:
12893:
12814:
12719:{\displaystyle U^{\#}=U^{\circ },}
12695:
12664:
12446:
12340:
12241:
12049:
12000:
9764:is assumed to be endowed with the
9264:
8868:{\displaystyle \mathbb {N} \to X,}
6762:{\displaystyle X\to \mathbb {K} ,}
6738:of all functions having prototype
6256:and products of subspaces such as
6241:
6051:imply that the weak-* topology is
6025:
5998:
5947:
5908:
5827:
5644:
5623:
5544:
5523:
5488:
5450:
5383:
5345:
5264:
5243:
5211:
5184:
5146:
5108:
5070:
4994:
4917:
4871:
4836:
4806:
4753:
4711:
4526:
4484:
4389:
4349:
4323:
4310:
4247:
4084:
4034:
4006:
3947:
3926:
3878:
3857:
3785:
3411:
3359:
3338:
3286:
3198:
3162:
3124:
3089:
3004:
2991:
2943:
2827:
2803:
2769:
2589:
2473:
2369:
2320:
2273:
2192:
2158:
2111:
2060:
2039:
2007:
1953:
1932:
1884:
1863:
1811:
1790:
1739:
1718:
1665:
1627:
1502:Denote by the underlying field of
1456:
1435:
1398:
1310:
1272:
1133:
1076:
737:
702:
690:and denote the weak-* topology on
664:
629:
598:
585:
555:
462:
342:
257:
14:
32264:
30639:. San Diego, CA: Academic Press.
30431:Bell, J.; Fremlin, David (1972).
28033:
27862:
26531:{\displaystyle U^{\circ }=U^{\#}}
25191:{\displaystyle 1<p<\infty }
25063:, then every bounded sequence in
24770:{\displaystyle B_{1}^{\,\prime }}
24370:{\displaystyle B_{1}^{\,\prime }}
24203:can be identified as a subset of
23597:to construct solutions to PDE or
23582:in this metric can be shown by a
23112:Sequential Banach–Alaoglu theorem
20585:{\displaystyle f(x+y)=f(x)+f(y),}
19197:{\displaystyle f(x+y)=f(x)+f(y):}
17602:{\displaystyle \mathbb {K} ^{X},}
17570:{\displaystyle f_{\bullet }\to f}
17190:{\displaystyle \mathbb {K} ^{X}.}
15829:{\displaystyle \mathbb {K} ^{X},}
14914:{\displaystyle |f(x)|\leq r_{x},}
14439:would be a valid choice for each
12322:topology of pointwise convergence
10995:{\textstyle \prod _{x\in X}S_{x}}
10833:{\textstyle \prod _{x\in X}S_{x}}
10786:{\textstyle \prod _{x\in X}S_{x}}
9770:topology of pointwise convergence
9494:{\displaystyle x_{\bullet }\to x}
8619:
8551:
8452:
8323:
8204:
8091:{\displaystyle s(x)\in B_{r_{x}}}
6163:{\displaystyle X\to \mathbb {K} }
5927:) is compact with respect to the
4657:{\displaystyle U^{\circ \circ };}
4631:the same must be true of the set
3806:{\displaystyle U^{\#}=U^{\circ }}
2169:{\displaystyle f\in X^{\prime },}
925:in the domain, the net of values
768:topology of pointwise convergence
223:is a vector space over the field
32212:
32211:
31497:
31496:
31423:Topological quantum field theory
27431:a neighborhood of the origin in
27256:Said differently, the polar set
27140:{\displaystyle U^{\circ ,\tau }}
26146:definition of convex compactness
25333:Then there exists a subsequence
24971:compact in the norm topology by
23837:property in the weak* topology.
23724:which approaches the infimum of
23151:be a separable normed space and
16783:{\displaystyle \mathbb {K} ^{X}}
16434:{\displaystyle f(U)\subseteq B.}
13262:{\displaystyle f\in X^{\prime }}
13027:the conclusion is equivalent to
9014:{\displaystyle x_{\bullet }(i).}
7360:{\displaystyle \mathbb {K} ^{X}}
7094:is identified with the function
6852:{\displaystyle \mathbb {K} ^{X}}
6731:{\displaystyle \mathbb {K} ^{X}}
6135:{\displaystyle \mathbb {K} ^{X}}
4291:{\displaystyle f\in U^{\circ },}
4255:{\displaystyle f\in X^{\prime }}
3547:{\displaystyle f_{i}(u)\to f(u)}
32243:Theorems in functional analysis
32199:With the approximation property
30705:A Course in Functional Analysis
29392:{\displaystyle \,\subseteq .\,}
28717:
28362:{\displaystyle R_{x}\leq S_{x}}
28164:
27489:{\displaystyle U^{\circ ,\nu }}
27289:'s defining "requirement" that
25829:Weak version of Alaoglu theorem
25599:Consequences for Hilbert spaces
25475:
24844:
24716:
24628:
23512:denotes the duality pairing of
22220:which shows that these numbers
19755:be the addition map defined by
17951:
17945:
17895:
17857:
17253:is a linear functional. So let
15973:{\displaystyle Y:=\mathbb {K} }
15545:
15539:
13189:is bounded on the neighborhood
11703:to endowed with any topology.
11366:is replaced with the open ball
11189:
11183:
7479:
7473:
6099:linear functional is continuous
5935:When the continuous dual space
4584:where because the neighborhood
4185:is bounded on the neighborhood
31662:Open mapping (Banach–Schauder)
30409:
30397:
30345:
30276:
30272:
30266:
30259:
30157:
30151:
30133:
30127:
30035:
30031:
30025:
30018:
29961:
29957:
29951:
29944:
28748:
28736:
28476:
28464:
28461:
27703:
27598:
27535:
27523:
27450:
27438:
27348:
27335:
27087:
27083:
27077:
27070:
27035:
27022:
26947:
26935:
26889:
26876:
26856:
26844:
26821:
26809:
26624:
26620:
26614:
26607:
26452:
26234:
26199: – theorem in mathematics
25966:{\displaystyle {\mathcal {C}}}
25707:
25701:
25671:
25665:
25634:
25628:
25581:
25575:
25503:
25497:
25456:
25153:
25147:
25107:.) For example, suppose that
24850:Consequences for normed spaces
24793:
24787:
24700:
24694:
24666:
24615:
24595:
24589:
24480:
24474:
24310:
24306:
24300:
24294:
24286:
24264:
24114:guarantees that their product
24001:
23993:
23898:
23890:
23624:
23595:partial differential equations
23340:
23328:
22951:
22938:
22857:
22845:
22842:
21332:
21326:
21154:
21142:
20778:is equipped with the topology
20576:
20570:
20561:
20555:
20546:
20534:
20508:
20496:
20490:
20487:
20475:
20442:
20436:
20427:
20421:
20415:
20412:
20400:
20370:
20358:
20325:
20313:
20274:
20268:
20252:
20246:
20202:
20196:
20180:
20174:
20017:
20011:
20002:
19996:
19987:
19984:
19978:
19969:
19963:
19957:
19912:
19909:
19903:
19894:
19888:
19882:
19876:
19777:
19765:
19737:
19665:
19662:
19656:
19647:
19641:
19635:
19632:
19599:
19593:
19587:
19567:
19561:
19539:
19533:
19503:
19497:
19491:
19471:
19465:
19443:
19437:
19373:
19367:
19351:
19345:
19261:
19188:
19182:
19173:
19167:
19158:
19146:
19116:
19107:
19098:
19092:
19044:
19035:
19029:
19026:
19017:
18984:
18978:
18969:
18966:
18957:
18915:
18906:
18846:
18837:
18806:
18800:
18753:
18747:
18706:
18700:
18661:
18635:
18629:
18607:
18601:
18549:
18543:
18496:
18490:
18441:
18435:
18393:
18387:
18375:
18372:
18366:
18360:
18337:
18334:
18328:
18322:
18316:
18308:
18302:
18239:
18233:
18227:
18224:
18218:
18133:
18127:
18079:
18045:
18039:
18027:
18018:
17989:
17980:
17974:
17971:
17962:
17939:
17927:
17921:
17918:
17906:
17889:
17883:
17877:
17874:
17868:
17851:
17845:
17839:
17836:
17830:
17647:
17641:
17635:
17632:
17626:
17561:
17511:
17505:
17460:
17454:
17371:
17362:
17356:
16687:
16681:
16609:
16603:
16587:
16581:
16531:
16525:
16519:
16499:
16493:
16419:
16413:
16177:
16171:
16078:subset of a topological space
15921:
15915:
15333:
15327:
15228:
15224:
15218:
15211:
14937:
14931:
14891:
14887:
14881:
14874:
14844:
14840:
14834:
14827:
14719:
14713:
14681:
14677:
14671:
14664:
14616:
14612:
14606:
14599:
14301:
14295:
14269:
14263:
14252:
14246:
13961:
13640:
13634:
13575:
13571:
13565:
13558:
13429:
13421:
13145:
13141:
13135:
13128:
12862:
12858:
12852:
12845:
12097:
12093:
12087:
12080:
11915:
11907:
11733:be a subset of a vector space
11527:
11523:
11517:
11510:
11464:
11460:
11454:
11447:
11399:
11391:
11308:
11302:
11267:
11261:
11199:
11193:
11173:
11169:
11163:
11156:
11087:
11083:
11077:
11070:
10503:
10497:
10491:
10471:
10465:
10376:
10370:
10200:
10194:
10185:
10179:
10118:
10112:
10097:
9987:
9688:
9682:
9676:
9640:
9634:
9628:
9587:
9581:
9575:
9485:
9452:
9133:
9041:
9005:
8999:
8856:
8801:
8789:
8763:
8572:
8520:
8377:
8269:
8210:
8198:
8065:
8059:
7673:
7667:
7635:
7629:
7592:
7524:
7465:
7300:and why the Cartesian product
7144:
7138:
7110:
6980:
6942:
6938:
6932:
6926:
6812:
6781:
6748:
6415:
6407:
6152:
4897:-bounded if and only if it is
4141:
4137:
4131:
4124:
3751:
3747:
3741:
3734:
3713:
3707:
3669:
3661:
3599:
3593:
3541:
3535:
3529:
3526:
3520:
3454:
3450:
3444:
3437:
2875:
2871:
2865:
2858:
2698:
2694:
2688:
2681:
2521:
2517:
2511:
2504:
1494:Proof involving duality theory
1181:
1177:
1171:
1164:
1000:
994:
954:
948:
426:
420:
347:
118:
115:
103:
100:
86:most important fact about the
16:Theorem in functional analysis
1:
31219:Uniform boundedness principle
30709:Graduate Texts in Mathematics
30594:See Theorem 3.15, p. 68.
30469:
30392:Narici & Beckenstein 2011
30369:Narici & Beckenstein 2011
30325:{\displaystyle \blacksquare }
30048:{\displaystyle |f(u)|\leq 1.}
27696:and strictly weaker than the
26140:of closed subsets having the
25763:. However, the theorem does
25532:The corresponding result for
24834:{\displaystyle \blacksquare }
23813:The last step often requires
22747:{\displaystyle \,\subseteq ;}
21129:is any subset that satisfies
21088:{\displaystyle \blacksquare }
20925:Because the underlying field
20605:{\displaystyle \blacksquare }
19947:where the right hand side is
18399:{\displaystyle M(f(x))=sf(x)}
18350:where the right hand side is
18270:{\displaystyle \mathbb {K} ,}
17807:{\displaystyle \mathbb {K} :}
16722:{\displaystyle \blacksquare }
15849:{\displaystyle \blacksquare }
15737:{\displaystyle \mathbb {K} .}
15012:is always a closed subset of
13979:denote the projection to the
13282:{\displaystyle \blacksquare }
10534:{\displaystyle \mathbb {K} ,}
10409:this happens if and only if
10149:{\displaystyle \mathbb {K} ,}
7689:Stated in words, for a point
6573:{\displaystyle U\subseteq X,}
6214:and the relationship between
6142:of all functions of the form
6107:continuous linear functionals
5719:{\displaystyle \blacksquare }
3764:{\displaystyle |f(u)|\leq 1.}
3697:so too must this net's limit
1593:{\displaystyle \mathbb {C} .}
1540:{\displaystyle \mathbb {K} ,}
30601:; Wolff, Manfred P. (1999).
30338:
30010:and it remains to show that
28871:{\displaystyle \mathbb {K} }
26830:{\displaystyle (X,\sigma ).}
26142:finite intersection property
26077:finite intersection property
25902:{\displaystyle X^{\prime }.}
24160:{\displaystyle X^{\prime },}
23913:Riesz representation theorem
23194:{\displaystyle X^{\prime }.}
23055:{\displaystyle m_{\bullet }}
22963:{\displaystyle U^{\#}=^{\#}}
22240:{\displaystyle m_{\bullet }}
21676:{\displaystyle m_{\bullet }}
21122:{\displaystyle U\subseteq X}
20940:{\displaystyle \mathbb {K} }
19940:{\displaystyle \mathbb {K} }
19125:{\displaystyle sf(x)=f(sx),}
19079:are unique, it follows that
19072:{\displaystyle \mathbb {K} }
18054:{\displaystyle f(sx)=sf(x):}
17675:{\displaystyle \mathbb {K} }
17410:{\displaystyle f_{\bullet }}
17226:{\displaystyle f\in X^{\#},}
16926:{\displaystyle \mathbb {C} }
16904:{\displaystyle \mathbb {R} }
16882:{\displaystyle \mathbb {K} }
16860:{\displaystyle \mathbb {K} }
16344:that converges pointwise to
16067:{\displaystyle B\subseteq Y}
16041:{\displaystyle U\subseteq X}
15859:The conclusion that the set
14220:{\displaystyle f\in U^{\#}.}
13234:continuous linear functional
12679:is therefore compact in the
12645:{\displaystyle x\in r_{x}U.}
12222:{\displaystyle x\in r_{x}U,}
11977:{\displaystyle \mathbb {K} }
11823:) and for every real number
11768:{\displaystyle \mathbb {K} }
10923:{\displaystyle \mathbb {K} }
10743:{\displaystyle \mathbb {K} }
9299:{\displaystyle x_{\bullet }}
9077:{\displaystyle x_{\bullet }}
8895:{\displaystyle x_{\bullet }}
8740:is by definition a function
8148:{\displaystyle U\subseteq X}
8014:is the set of all functions
7824:In particular, suppose that
6013:to be a compact subset when
5986:for the closed unit ball in
5499:{\displaystyle X^{\prime }:}
4817:{\displaystyle X^{\prime },}
4764:{\displaystyle X^{\prime }.}
4537:{\displaystyle X^{\prime }:}
4400:{\displaystyle X^{\prime },}
4227:continuous linear functional
3627:{\displaystyle \mathbb {K} }
3569:{\displaystyle \mathbb {K} }
3467:{\displaystyle |f(u)|\leq 1}
3422:{\displaystyle f\in U^{\#},}
1565:{\displaystyle \mathbb {R} }
1321:{\displaystyle X^{\prime }.}
1087:{\displaystyle X^{\prime },}
878:{\displaystyle f_{\bullet }}
566:{\displaystyle X^{\prime },}
238:{\displaystyle \mathbb {K} }
184:
7:
31883:Radially convex/Star-shaped
31868:Pre-compact/Totally bounded
30482:Topological Vector Spaces I
29650:{\displaystyle m_{u}\leq 1}
28668:{\displaystyle m_{u}\leq 1}
28602:{\displaystyle m_{x}\leq r}
27548:is not guaranteed to equal
26953:{\displaystyle (X,\sigma )}
26862:{\displaystyle (X,\sigma )}
26434:{\displaystyle B^{\prime }}
26331:{\displaystyle B^{\prime }}
26300:{\displaystyle X^{\prime }}
26159:
26020:{\displaystyle X^{\prime }}
25745:Zermelo–Fraenkel set theory
25618:Closed and bounded sets in
25587:{\displaystyle L^{1}(\mu )}
25159:{\displaystyle L^{p}(\mu )}
25032:-compact; this is known as
24960:{\displaystyle X^{\prime }}
24933:{\displaystyle X^{\prime }}
24898:{\displaystyle X^{\prime }}
23860:{\displaystyle X^{\prime }}
23532:{\displaystyle X^{\prime }}
20918:Proof of corollary to lemma
18871: by linearity of
16709:which completes the proof.
16396:{\displaystyle f\in U_{B},}
14399:{\displaystyle x\in r_{x}U}
13096:{\displaystyle f\in U^{\#}}
12580:there exists a real number
12454:{\displaystyle X^{\prime }}
12434:with continuous dual space
12324:, which is also called the
12190:is a real number such that
11485:with the strict inequality
7772:" is the same as "plugging
6033:{\displaystyle X^{\prime }}
6006:{\displaystyle X^{\prime }}
5955:{\displaystyle X^{\prime }}
5916:{\displaystyle X^{\prime }}
5835:{\displaystyle X^{\prime }}
5371:-totally bounded subset of
5219:{\displaystyle X^{\prime }}
5192:{\displaystyle X^{\prime }}
5096:-totally bounded subset of
4844:{\displaystyle X^{\prime }}
4092:{\displaystyle f\in U^{\#}}
3840:is a closed subset of both
3206:{\displaystyle f\in X^{\#}}
2200:{\displaystyle X^{\prime }}
2015:{\displaystyle X^{\prime }}
710:{\displaystyle X^{\prime }}
10:
32269:
31569:Continuous linear operator
31362:Invariant subspace problem
27167:{\displaystyle U^{\circ }}
26478:denotes the topology that
25856:be a normed space and let
25732:
24940:is weak-* compact. So if
23562:Sequential compactness of
20030:and the left hand side is
18406:and the left hand side is
18097:be the "multiplication by
16702:{\displaystyle f(u)\in B,}
16628:belongs to the closed (in
16403:which by definition means
16262:{\displaystyle f\in Y^{X}}
14363:{\displaystyle r_{x}>0}
13821:is a closed subset of the
12746:{\displaystyle U^{\circ }}
12606:{\displaystyle r_{x}>0}
12183:{\displaystyle r_{x}>0}
11186: if and only if
9600:may instead be written as
8810:{\displaystyle (I,\leq ).}
8223:then the linear bijection
7814:{\displaystyle \Pr {}_{z}}
7569:{\displaystyle \Pr {}_{z}}
5698:{\displaystyle U^{\circ }}
5599:{\displaystyle U^{\circ }}
5423:{\displaystyle U^{\circ }}
5318:{\displaystyle U^{\circ }}
5043:{\displaystyle U^{\circ }}
4967:{\displaystyle U^{\circ }}
4684:{\displaystyle U^{\circ }}
4457:{\displaystyle U^{\circ }}
4427:{\displaystyle U^{\circ }}
3833:{\displaystyle U^{\circ }}
3612:belongs to the closed (in
1610:continuous linear operator
1348:{\displaystyle U^{\circ }}
613:always holds. Denote the
188:
175:weak-* compactness theorem
77:
32248:Topological vector spaces
32207:
31952:
31914:Algebraic interior (core)
31896:
31794:
31682:
31656:Vector-valued Hahn–Banach
31617:
31551:
31544:Topological vector spaces
31492:
31451:
31375:
31354:
31313:
31252:
31194:
31140:
31082:
31075:
31010:
30989:
30981:Transpose of a linear map
30973:
30952:
30873:
30822:
30603:Topological Vector Spaces
30536:Topological Vector Spaces
28723:For any non-empty subset
28398:Thus, every neighborhood
27631:{\displaystyle B_{r_{x}}}
26176:Delta-compactness theorem
22247:are unique; indeed, this
21656:where these real numbers
17301:{\displaystyle x,y\in X.}
16009:{\displaystyle B:=B_{1}.}
11683:require the vector space
11240:{\displaystyle \,\sup \,}
9772:. This is because given
6103:bounded on a neighborhood
3726:belong to this set. Thus
1355:is equal to the polar of
62:of compact sets with the
31744:Topological homomorphism
31604:Topological vector space
31331:Spectrum of a C*-algebra
27541:{\displaystyle (X,\nu )}
27456:{\displaystyle (X,\nu )}
26908:and denote the polar of
26227:
26221:Topological vector space
26186:Eberlein–Šmulian theorem
25105:Eberlein–Šmulian theorem
24913:The closed unit ball in
24137:The closed unit ball in
23770:and then establish that
23584:diagonalization argument
23312:{\displaystyle x,y\in B}
23171:the closed unit ball in
21765:{\displaystyle P=U^{\#}}
20907:topological vector space
16933:) is a closed subset of
16367:It remains to show that
14574:function that satisfies
14432:{\displaystyle r_{u}:=1}
14285:it remains to show that
13353:(since each closed ball
12432:topological vector space
12381:topological vector space
9461:{\displaystyle F:X\to Y}
9050:{\displaystyle F:X\to Y}
6859:is identified with its (
5923:(endowed with its usual
3605:{\displaystyle f_{i}(u)}
1027:topological vector space
537:topological vector space
155:Bourbaki–Alaoglu theorem
23:and related branches of
31428:Noncommutative geometry
30452:10.4064/fm-77-2-167-170
30440:Fundamenta Mathematicae
30406:, Theorem (4) in §20.9.
29679:{\displaystyle u\in U.}
29617:{\displaystyle u\in U,}
29588:{\displaystyle r_{u}=1}
28900:{\displaystyle x\in X,}
28697:{\displaystyle u\in U.}
28635:{\displaystyle m_{0}=0}
28609:so that in particular,
28569:{\displaystyle x\in rU}
28514:{\displaystyle x\in X,}
28391:{\displaystyle x\in X.}
28154:{\displaystyle X^{\#}.}
27249:{\displaystyle U^{\#}.}
26762:is any TVS topology on
26755:{\displaystyle \sigma }
25818:(also sometimes called
23948:{\displaystyle x\in X,}
23917:Helly selection theorem
23833:to obey a (sequential)
23120:normed vector space is
22970:). Similarly, because
22874:and it is unchanged if
22132:{\displaystyle x\in X,}
21390:{\displaystyle X^{\#}.}
19413:{\displaystyle i\in I.}
17704:{\displaystyle z\in X.}
17332:{\displaystyle z\in X,}
16463:{\displaystyle u\in U,}
14158:{\displaystyle x\in X.}
13911:{\displaystyle z\in X,}
12573:{\displaystyle x\in X,}
12151:{\displaystyle x\in X,}
10952:{\displaystyle x\in X,}
10683:{\displaystyle x\in X,}
10435:{\displaystyle z\in X,}
10032:{\displaystyle z\in X,}
9104:" is just the function
8120:{\displaystyle x\in X.}
5394:{\displaystyle X^{\#}.}
5119:{\displaystyle X^{\#}:}
3813:and then conclude that
3496:{\displaystyle u\in U.}
3173:{\displaystyle X^{\#}:}
984:converges to the value
66:. As a consequence of
31802:Absolutely convex/disk
31484:Tomita–Takesaki theory
31459:Approximation property
31403:Calculus of variations
30326:
30306:
30245:
30111:
30049:
30004:
30003:{\displaystyle u\in U}
29978:
29914:
29852:
29790:
29680:
29651:
29618:
29589:
29556:
29508:
29481:
29454:
29423:
29393:
29368:
29337:
29307:
29261:
29221:
29185:
28984:
28901:
28872:
28850:
28830:
28758:
28698:
28669:
28636:
28603:
28570:
28541:
28540:{\displaystyle r>0}
28515:
28486:
28432:
28412:
28392:
28363:
28323:
28279:
28248:
28155:
28123:
28085:
28084:{\displaystyle U^{\#}}
28058:
27909:
27908:{\displaystyle U^{\#}}
27882:
27766:
27686:
27642:, the conclusion that
27632:
27589:
27569:
27568:{\displaystyle U^{\#}}
27542:
27510:
27490:
27457:
27421:
27401:
27381:
27361:
27316:
27283:
27250:
27220:
27168:
27141:
27108:
26954:
26922:
26902:
26863:
26831:
26796:
26776:
26756:
26736:
26716:
26715:{\displaystyle X^{\#}}
26689:
26665:
26645:
26532:
26492:
26472:
26435:
26408:
26332:
26301:
26274:
26119:
26069:
26021:
25987:
25967:
25923:
25903:
25870:
25850:
25714:
25681:
25649:weak operator topology
25641:
25588:
25552:
25526:
25413:
25412:{\displaystyle f\in X}
25387:
25327:
25304:
25258:
25212:
25192:
25160:
25121:
25097:
25077:
25061:reflexive Banach space
25053:
25026:
24961:
24934:
24899:
24868:
24835:
24816:
24771:
24738:
24678:
24644:
24550:
24549:{\displaystyle i\in I}
24524:
24504:
24445:
24371:
24336:
24217:
24197:
24161:
24128:
24104:
24073:
24024:
23949:
23905:
23861:
23827:
23807:
23784:
23764:
23741:
23718:
23659:
23636:
23576:
23556:
23533:
23506:
23474:
23366:
23313:
23281:
23215:
23195:
23165:
23145:
23103:
23076:
23056:
23030:
22964:
22912:
22888:
22864:
22762:) set. The function
22748:
22724:
22693:
22618:
22588:
22558:
22492:
22298:
22275:
22241:
22214:
22133:
22104:
22064:
21941:
21862:
21766:
21733:
21677:
21650:
21474:
21413:
21391:
21357:
21250:
21226:
21123:
21089:
21069:
21000:
20941:
20890:
20818:
20772:
20752:
20751:{\displaystyle X^{\#}}
20717:
20716:{\displaystyle X^{\#}}
20685:
20684:{\displaystyle X^{\#}}
20662:complete uniform space
20654:
20606:
20586:
20518:
20452:
20377:
20024:
19941:
19919:
19842:
19822:
19749:
19705:
19672:
19609:
19510:
19414:
19385:
19281:
19198:
19126:
19073:
19051:
18994:
18934:
18400:
18344:
18271:
18246:
18195:
18175:
18111:
18091:
18055:
17999:
17808:
17783:
17760:
17705:
17676:
17654:
17603:
17571:
17538:
17431:
17411:
17383:
17333:
17302:
17267:
17247:
17233:it must be shown that
17227:
17191:
17159:
17139:
17138:{\displaystyle X^{\#}}
17112:
17051:
17000:
16980:
16927:
16905:
16883:
16861:
16839:
16819:
16818:{\displaystyle X^{\#}}
16784:
16755:
16754:{\displaystyle X^{\#}}
16723:
16703:
16665:
16642:
16622:
16558:
16538:
16464:
16435:
16397:
16361:
16338:
16311:
16263:
16222:
16201:is a closed subset of
16195:
16095:
16068:
16042:
16010:
15974:
15946:
15850:
15830:
15798:
15738:
15713:
15656:
15059:
15006:
15005:{\displaystyle X^{\#}}
14967:
14915:
14858:
14634:
14564:
14544:
14459:
14458:{\displaystyle u\in U}
14433:
14400:
14364:
14331:
14279:
14221:
14185:
14184:{\displaystyle x\in X}
14159:
14130:
14063:
13993:
13973:
13912:
13874:
13815:
13814:{\displaystyle U^{\#}}
13787:
13718:
13671:is a closed subset of
13665:
13453:
13343:
13283:
13263:
13226:
13206:
13183:
13163:
13097:
13064:
13021:
12964:
12918:
12747:
12720:
12673:
12672:{\displaystyle U^{\#}}
12646:
12607:
12574:
12545:
12518:
12495:
12475:
12455:
12424:
12397:
12373:
12349:
12348:{\displaystyle U^{\#}}
12310:
12250:
12249:{\displaystyle U^{\#}}
12223:
12184:
12152:
12121:
11978:
11956:
11932:
11846:endow the closed ball
11840:
11817:
11769:
11747:
11727:
11697:
11665:
11645:
11593:
11542:
11479:
11416:
11360:
11333:
11241:
11219:
11127:
11101:
11037:
11002:is a closed subset of
10996:
10953:
10924:
10902:
10875:
10834:
10787:
10744:
10726:(topological) subspace
10718:
10684:
10649:
10626:
10606:
10586:
10535:
10510:
10436:
10403:
10208:
10150:
10125:
10033:
9997:
9943:
9887:
9860:
9840:
9786:
9758:
9698:
9647:
9594:
9518:
9495:
9462:
9430:
9300:
9273:
9207:
9146:
9098:
9078:
9051:
9015:
8976:
8949:
8922:
8921:{\displaystyle i\in I}
8896:
8869:
8838:
8811:
8773:
8734:
8714:
8643:
8473:
8217:
8171:
8149:
8121:
8092:
8043:
8008:
7958:
7866:
7815:
7786:
7766:
7746:
7726:
7703:
7683:
7604:
7570:
7541:
7476: defined by
7416:
7415:{\displaystyle z\in X}
7361:
7332:
7294:
7238:
7187:
7122:
7088:
7050:
6987:
6964:
6873:
6853:
6824:
6788:
6769:as is now described:
6763:
6732:
6703:
6639:
6619:
6587:The Cartesian product
6574:
6545:
6472:
6452:
6432:
6341:
6303:
6250:
6249:{\displaystyle X^{\#}}
6208:
6164:
6136:
6034:
6007:
5976:
5956:
5917:
5890:
5871:Banach–Alaoglu theorem
5856:
5836:
5809:
5789:
5769:
5745:
5720:
5699:
5672:
5600:
5569:
5500:
5470:
5424:
5395:
5365:
5319:
5292:
5220:
5193:
5166:
5120:
5090:
5044:
5014:
4968:
4937:
4891:
4845:
4818:
4785:
4765:
4731:
4685:
4658:
4625:
4598:
4578:
4538:
4504:
4458:
4428:
4401:
4371:
4292:
4256:
4219:
4199:
4179:
4159:
4093:
4060:
4015:
3975:
3903:
3834:
3807:
3765:
3720:
3691:
3628:
3606:
3570:
3548:
3497:
3468:
3423:
3387:
3315:
3295:
3294:{\displaystyle U^{\#}}
3268:
3207:
3174:
3144:
3098:
3097:{\displaystyle U^{\#}}
3068:
3016:
2960:
2914:
2894:
2783:
2739:
2717:
2603:
2560:
2540:
2428:
2408:
2386:
2337:
2287:
2241:
2221:
2201:
2170:
2134:
2085:
2016:
1981:
1909:
1836:
1767:
1688:
1636:
1635:{\displaystyle X^{\#}}
1594:
1566:
1541:
1516:
1484:
1412:
1369:
1349:
1322:
1292:
1243:
1223:
1200:
1088:
1043:
1010:
978:
919:
899:
879:
852:
784:
760:
711:
684:
638:
637:{\displaystyle X^{\#}}
607:
567:
529:
498:
433:
359:
290:
266:
265:{\displaystyle X^{\#}}
239:
217:
171:Banach–Alaoglu theorem
144:sequential compactness
125:
29:Banach–Alaoglu theorem
31837:Complemented subspace
31651:hyperplane separation
31479:Banach–Mazur distance
31442:Generalized functions
30416:Meise & Vogt 1997
30327:
30307:
30246:
30112:
30050:
30005:
29979:
29915:
29853:
29791:
29681:
29652:
29619:
29590:
29557:
29509:
29507:{\displaystyle T_{P}}
29482:
29455:
29453:{\displaystyle T_{P}}
29429:is closed under (non-
29424:
29394:
29369:
29338:
29308:
29262:
29222:
29186:
28985:
28902:
28873:
28851:
28831:
28759:
28699:
28670:
28637:
28604:
28571:
28542:
28516:
28487:
28433:
28413:
28393:
28364:
28324:
28280:
28278:{\displaystyle T_{P}}
28249:
28156:
28124:
28086:
28059:
27910:
27883:
27767:
27687:
27633:
27590:
27570:
27543:
27511:
27491:
27458:
27422:
27402:
27387:is a TVS topology on
27382:
27362:
27317:
27284:
27251:
27221:
27169:
27142:
27109:
26955:
26923:
26903:
26864:
26832:
26797:
26777:
26757:
26737:
26735:{\displaystyle \tau }
26717:
26690:
26666:
26646:
26538:shows that the polar
26533:
26493:
26473:
26471:{\displaystyle \tau }
26436:
26409:
26333:
26302:
26275:
26240:Explicitly, a subset
26166:Bishop–Phelps theorem
26120:
26070:
26022:
25988:
25968:
25924:
25904:
25878:continuous dual space
25871:
25851:
25715:
25682:
25680:{\displaystyle B(H),}
25642:
25589:
25553:
25527:
25414:
25388:
25328:
25305:
25259:
25213:
25193:
25161:
25122:
25098:
25078:
25054:
25027:
24962:
24935:
24900:
24869:
24836:
24817:
24772:
24739:
24719: for every
24679:
24645:
24551:
24525:
24505:
24446:
24372:
24337:
24218:
24198:
24162:
24129:
24105:
24103:{\displaystyle D_{x}}
24074:
24025:
23950:
23906:
23862:
23835:lower semi-continuity
23828:
23808:
23785:
23765:
23742:
23719:
23660:
23637:
23588:Arzelà–Ascoli theorem
23577:
23557:
23534:
23507:
23475:
23346:
23314:
23282:
23216:
23196:
23166:
23146:
23104:
23077:
23062:is also unchanged if
23057:
23031:
22965:
22913:
22889:
22865:
22749:
22725:
22694:
22619:
22589:
22559:
22493:
22299:
22276:
22242:
22215:
22134:
22105:
22065:
21942:
21863:
21767:
21734:
21678:
21651:
21475:
21414:
21392:
21358:
21251:
21227:
21124:
21090:
21070:
21001:
20942:
20891:
20819:
20773:
20753:
20718:
20686:
20655:
20607:
20587:
20519:
20453:
20378:
20025:
19942:
19920:
19843:
19823:
19750:
19706:
19673:
19610:
19511:
19415:
19386:
19282:
19199:
19127:
19074:
19052:
18995:
18935:
18401:
18345:
18272:
18247:
18196:
18176:
18112:
18092:
18056:
18000:
17809:
17784:
17761:
17706:
17677:
17655:
17604:
17572:
17539:
17432:
17412:
17384:
17334:
17303:
17268:
17248:
17228:
17192:
17160:
17140:
17113:
17052:
17001:
16981:
16928:
16906:
16884:
16862:
16840:
16820:
16785:
16756:
16724:
16704:
16666:
16643:
16623:
16559:
16539:
16465:
16436:
16398:
16362:
16339:
16337:{\displaystyle U_{B}}
16312:
16264:
16223:
16221:{\displaystyle Y^{X}}
16196:
16096:
16069:
16043:
16011:
15975:
15947:
15851:
15831:
15799:
15739:
15714:
15657:
15060:
15007:
14968:
14916:
14859:
14635:
14565:
14545:
14460:
14434:
14401:
14365:
14332:
14280:
14222:
14186:
14160:
14131:
14064:
13994:
13974:
13913:
13875:
13816:
13788:
13719:
13666:
13454:
13344:
13284:
13264:
13227:
13207:
13184:
13164:
13098:
13065:
13022:
12965:
12919:
12748:
12721:
12674:
12647:
12608:
12575:
12546:
12519:
12496:
12476:
12456:
12425:
12398:
12374:
12350:
12311:
12251:
12224:
12185:
12153:
12122:
11979:
11957:
11933:
11841:
11818:
11770:
11748:
11728:
11698:
11666:
11646:
11594:
11543:
11480:
11417:
11361:
11359:{\displaystyle B_{r}}
11334:
11242:
11220:
11128:
11102:
11038:
10997:
10954:
10925:
10903:
10901:{\displaystyle S_{x}}
10876:
10835:
10788:
10745:
10719:
10685:
10659:topological subspaces
10650:
10627:
10607:
10587:
10536:
10511:
10437:
10404:
10209:
10151:
10126:
10034:
9998:
9944:
9888:
9886:{\displaystyle f_{i}}
9861:
9841:
9787:
9759:
9699:
9648:
9595:
9519:
9496:
9463:
9431:
9301:
9274:
9208:
9147:
9099:
9079:
9052:
9016:
8977:
8975:{\displaystyle x_{i}}
8950:
8948:{\displaystyle x_{i}}
8923:
8897:
8870:
8839:
8812:
8774:
8735:
8715:
8644:
8474:
8218:
8172:
8150:
8122:
8093:
8044:
8009:
7959:
7867:
7816:
7787:
7767:
7747:
7727:
7704:
7684:
7605:
7571:
7542:
7417:
7369:(categorical) product
7362:
7333:
7295:
7239:
7188:
7123:
7089:
7051:
6988:
6965:
6874:
6854:
6825:
6789:
6764:
6733:
6704:
6640:
6620:
6575:
6546:
6473:
6453:
6433:
6342:
6304:
6251:
6216:pointwise convergence
6209:
6170:is identified as the
6165:
6137:
6111:sublinear functionals
6101:if and only if it is
6035:
6008:
5977:
5957:
5918:
5891:
5857:
5837:
5810:
5790:
5770:
5746:
5721:
5700:
5673:
5601:
5570:
5501:
5471:
5425:
5403:Finally, deduce that
5396:
5366:
5320:
5293:
5221:
5194:
5167:
5121:
5091:
5045:
5015:
4969:
4938:
4892:
4846:
4819:
4786:
4766:
4732:
4686:
4659:
4626:
4599:
4579:
4539:
4505:
4459:
4434:being closed follows.
4429:
4402:
4372:
4293:
4257:
4220:
4200:
4180:
4160:
4094:
4061:
4016:
3976:
3904:
3835:
3808:
3766:
3721:
3692:
3629:
3607:
3571:
3549:
3498:
3469:
3424:
3388:
3316:
3296:
3269:
3208:
3175:
3145:
3099:
3069:
3017:
2961:
2915:
2895:
2784:
2740:
2718:
2604:
2561:
2541:
2429:
2409:
2387:
2338:
2288:
2242:
2222:
2202:
2171:
2135:
2086:
2017:
1982:
1910:
1837:
1768:
1689:
1637:
1595:
1567:
1542:
1517:
1485:
1413:
1370:
1350:
1323:
1293:
1244:
1224:
1201:
1089:
1063:continuous dual space
1044:
1011:
1009:{\displaystyle f(x).}
979:
920:
900:
880:
853:
785:
761:
712:
685:
639:
608:
568:
541:continuous dual space
530:
510:canonical dual system
499:
434:
360:
291:
267:
240:
218:
167:locally convex spaces
126:
32238:Compactness theorems
32087:Locally convex space
31637:Closed graph theorem
31589:Locally convex space
31224:Kakutani fixed-point
31209:Riesz representation
30316:
30255:
30121:
30059:
30014:
29988:
29924:
29862:
29800:
29690:
29661:
29628:
29599:
29566:
29518:
29491:
29464:
29437:
29406:
29378:
29351:
29317:
29271:
29231:
29195:
28994:
28911:
28882:
28860:
28840:
28768:
28727:
28679:
28646:
28613:
28580:
28551:
28525:
28496:
28442:
28422:
28402:
28373:
28333:
28293:
28262:
28174:
28135:
28095:
28068:
27919:
27892:
27776:
27713:
27646:
27608:
27588:{\displaystyle \nu }
27579:
27575:and so the topology
27552:
27520:
27500:
27467:
27435:
27411:
27391:
27380:{\displaystyle \nu }
27371:
27332:
27293:
27260:
27230:
27178:
27151:
27118:
26964:
26932:
26912:
26873:
26841:
26806:
26786:
26766:
26746:
26726:
26699:
26679:
26655:
26542:
26502:
26482:
26462:
26418:
26346:
26315:
26284:
26244:
26206:Krein-Milman theorem
26171:Banach–Mazur theorem
26083:
26031:
26004:
25977:
25953:
25913:
25883:
25860:
25840:
25790:, which implies the
25722:Heine–Borel property
25713:{\displaystyle B(H)}
25695:
25659:
25640:{\displaystyle B(H)}
25622:
25562:
25536:
25423:
25397:
25337:
25314:
25268:
25222:
25202:
25170:
25134:
25111:
25087:
25067:
25043:
24990:
24944:
24917:
24882:
24858:
24825:
24781:
24748:
24688:
24654:
24560:
24534:
24514:
24455:
24388:
24348:
24227:
24207:
24171:
24141:
24118:
24087:
24079:be endowed with the
24034:
23959:
23930:
23871:
23844:
23817:
23794:
23774:
23751:
23728:
23669:
23646:
23605:
23599:variational problems
23566:
23543:
23516:
23484:
23322:
23291:
23225:
23205:
23175:
23155:
23135:
23122:sequentially compact
23090:
23066:
23039:
22974:
22922:
22902:
22896:convex balanced hull
22878:
22766:
22734:
22707:
22630:
22598:
22568:
22502:
22307:
22285:
22258:
22224:
22142:
22114:
22074:
21951:
21872:
21776:
21743:
21687:
21660:
21484:
21423:
21403:
21371:
21260:
21240:
21133:
21107:
21079:
21010:
20954:
20929:
20831:
20782:
20762:
20735:
20730:algebraic dual space
20700:
20696:Corollary to lemma (
20668:
20626:
20596:
20528:
20462:
20387:
20034:
19951:
19929:
19852:
19832:
19759:
19715:
19682:
19619:
19520:
19424:
19395:
19291:
19208:
19140:
19083:
19061:
19004:
18944:
18923: notation
18410:
18354:
18281:
18256:
18205:
18185:
18121:
18101:
18065:
18012:
17817:
17793:
17770:
17715:
17686:
17664:
17613:
17581:
17548:
17441:
17421:
17417:'s net of values at
17394:
17343:
17314:
17277:
17273:be a scalar and let
17257:
17237:
17201:
17169:
17149:
17122:
17061:
17026:
16990:
16937:
16915:
16893:
16871:
16849:
16829:
16825:of any vector space
16802:
16797:algebraic dual space
16765:
16738:
16713:
16675:
16652:
16632:
16568:
16548:
16474:
16445:
16407:
16371:
16348:
16321:
16273:
16240:
16234:Proof of observation
16205:
16105:
16082:
16052:
16026:
15984:
15956:
15863:
15840:
15808:
15748:
15723:
15670:
15073:
15016:
14989:
14984:algebraic dual space
14925:
14870:
14643:
14578:
14572:positive homogeneous
14554:
14469:
14443:
14410:
14374:
14341:
14289:
14231:
14195:
14169:
14140:
14073:
14007:
13983:
13922:
13893:
13828:
13798:
13731:
13675:
13470:
13357:
13303:
13293:Proof of Proposition
13273:
13240:
13216:
13193:
13173:
13107:
13074:
13031:
12975:
12931:
12757:
12730:
12687:
12656:
12617:
12584:
12555:
12532:
12505:
12485:
12465:
12438:
12414:
12387:
12363:
12332:
12320:is identical to the
12316:(where because this
12270:
12233:
12194:
12161:
12133:
11992:
11966:
11946:
11850:
11827:
11779:
11757:
11737:
11717:
11687:
11655:
11606:
11552:
11489:
11426:
11370:
11343:
11255:
11229:
11136:
11114:
11050:
11046:Characterization of
11006:
10963:
10934:
10912:
10885:
10844:
10801:
10754:
10732:
10693:
10665:
10636:
10616:
10596:
10548:
10520:
10446:
10417:
10218:
10163:
10135:
10043:
10014:
9953:
9897:
9870:
9850:
9799:
9776:
9715:
9657:
9604:
9528:
9505:
9472:
9440:
9310:
9283:
9217:
9156:
9108:
9088:
9061:
9029:
9023:function composition
8986:
8959:
8932:
8906:
8879:
8848:
8825:
8786:
8744:
8724:
8663:
8487:
8227:
8181:
8161:
8133:
8102:
8053:
8018:
7968:
7876:
7828:
7796:
7776:
7756:
7736:
7713:
7693:
7614:
7580:
7551:
7426:
7400:
7379:(which is a type of
7342:
7304:
7251:
7197:
7132:
7098:
7060:
6999:
6986:{\displaystyle \to }
6977:
6887:
6863:
6834:
6800:
6787:{\displaystyle \to }
6778:
6742:
6713:
6652:
6629:
6591:
6555:
6482:
6462:
6442:
6350:
6328:
6260:
6233:
6228:algebraic dual space
6177:
6146:
6117:
6017:
5990:
5966:
5939:
5900:
5880:
5846:
5819:
5799:
5779:
5759:
5735:
5710:
5682:
5610:
5583:
5510:
5480:
5434:
5407:
5375:
5329:
5302:
5230:
5203:
5176:
5130:
5100:
5054:
5027:
4978:
4951:
4947:. So in particular,
4901:
4855:
4828:
4798:
4792:distinguishes points
4775:
4745:
4695:
4668:
4635:
4612:
4588:
4552:
4518:
4468:
4441:
4411:
4381:
4302:
4266:
4233:
4209:
4189:
4169:
4103:
4070:
4025:
3985:
3913:
3844:
3817:
3777:
3730:
3719:{\displaystyle f(u)}
3701:
3638:
3616:
3580:
3558:
3554:in the scalar field
3507:
3478:
3433:
3397:
3325:
3305:
3278:
3217:
3184:
3154:
3108:
3081:
3029:
2970:
2924:
2904:
2795:
2750:
2729:
2615:
2570:
2550:
2441:
2418:
2398:
2350:
2301:
2254:
2231:
2211:
2184:
2144:
2095:
2026:
1999:
1991:. Importantly, the
1919:
1850:
1777:
1705:
1649:
1642:is endowed with the
1619:
1579:
1554:
1547:which is either the
1526:
1506:
1422:
1379:
1359:
1332:
1302:
1256:
1233:
1213:
1101:
1068:
1033:
988:
929:
909:
889:
862:
798:
774:
770:because given a map
721:
694:
648:
621:
577:
547:
519:
443:
369:
306:
280:
274:algebraic dual space
249:
227:
207:
94:
32067:Interpolation space
31599:Operator topologies
31408:Functional calculus
31367:Mahler's conjecture
31346:Von Neumann algebra
31060:Functional analysis
31018:Biorthogonal system
30850:Operator topologies
30744:Functional Analysis
30599:Schaefer, Helmut H.
30567:Functional Analysis
30394:, pp. 225–273.
30371:, pp. 235–240.
30251:which implies that
29624:which implies that
27595:can not be ignored.
27322:be a subset of the
25834: —
25808:Hahn–Banach theorem
25792:Hahn–Banach theorem
25741:Tychonoff's theorem
25613:Hahn–Banach theorem
25551:{\displaystyle p=1}
25478: for all
25382:
24766:
24366:
24259:
24189:
24112:Tychonoff's theorem
23276:
23082:is replaced by its
22894:is replaced by the
20726: —
20723:is weak-* complete)
18588: because
16793: —
15484: for all
15351: for all
15065:(this is proved in
13351:Tychonoff's theorem
11711: —
9268:
6224:subspace topologies
5874: —
5753:normed vector space
5476:-compact subset of
1023: —
543:will be denoted by
124:{\displaystyle C()}
68:Tychonoff's theorem
48:normed vector space
21:functional analysis
32253:Linear functionals
32097:(Pseudo)Metrizable
31929:Minkowski addition
31781:Sublinear function
31433:Riemann hypothesis
31132:Topological vector
30322:
30302:
30241:
30195:
30107:
30045:
30000:
29974:
29942:
29910:
29892:
29848:
29786:
29676:
29647:
29614:
29585:
29552:
29504:
29477:
29450:
29419:
29389:
29364:
29333:
29303:
29257:
29217:
29181:
29160:
29124:
29094:
29058:
29042:
28980:
28933:
28897:
28868:
28846:
28826:
28820:
28754:
28694:
28665:
28632:
28599:
28566:
28537:
28511:
28482:
28428:
28408:
28388:
28359:
28319:
28275:
28244:
28205:
28151:
28119:
28113:
28081:
28054:
28040:
27998:
27950:
27905:
27878:
27869:
27827:
27772:can be written as
27762:
27751:
27682:
27664:
27628:
27585:
27565:
27538:
27506:
27486:
27453:
27417:
27397:
27377:
27357:
27312:
27279:
27246:
27216:
27174:from above. Then
27164:
27137:
27104:
27068:
27001:
26950:
26918:
26898:
26859:
26827:
26792:
26782:such that the set
26772:
26752:
26732:
26712:
26685:
26661:
26641:
26605:
26528:
26488:
26468:
26431:
26404:
26328:
26297:
26270:
26181:Dixmier–Ng theorem
26134:convex compactness
26115:
26111:
26065:
26017:
25983:
25963:
25945:convex compactness
25919:
25899:
25866:
25846:
25832:
25820:convex compactness
25796:real vector spaces
25743:, which under the
25710:
25677:
25653:ultraweak topology
25637:
25584:
25548:
25522:
25409:
25383:
25340:
25326:{\displaystyle X.}
25323:
25300:
25254:
25208:
25188:
25156:
25117:
25093:
25073:
25049:
25022:
24973:F. Riesz's theorem
24957:
24930:
24895:
24864:
24831:
24812:
24767:
24751:
24734:
24674:
24640:
24572:
24546:
24520:
24500:
24441:
24367:
24351:
24332:
24330:
24244:
24223:in a natural way:
24213:
24193:
24174:
24157:
24124:
24100:
24069:
24058:
24020:
23945:
23924:
23901:
23857:
23823:
23806:{\displaystyle F.}
23803:
23790:is a minimizer of
23780:
23763:{\displaystyle x,}
23760:
23740:{\displaystyle F,}
23737:
23714:
23658:{\displaystyle X,}
23655:
23632:
23572:
23555:{\displaystyle X.}
23552:
23529:
23502:
23470:
23309:
23277:
23241:
23221:is separable, let
23211:
23191:
23161:
23141:
23131:Specifically, let
23102:{\displaystyle X.}
23099:
23072:
23052:
23026:
22960:
22908:
22884:
22860:
22797:
22744:
22720:
22689:
22671:
22614:
22584:
22554:
22488:
22341:
22297:{\displaystyle P,}
22294:
22271:
22237:
22210:
22129:
22100:
22060:
21982:
21937:
21858:
21840:
21817:
21762:
21729:
21711:
21673:
21646:
21644:
21624:
21563:
21470:
21409:
21387:
21353:
21291:
21246:
21222:
21178:
21119:
21085:
21065:
20996:
20987:
20937:
20886:
20814:
20768:
20758:of a vector space
20748:
20724:
20713:
20681:
20650:
20644:
20602:
20582:
20514:
20448:
20383:which proves that
20373:
20076:
20020:
19937:
19915:
19838:
19828:The continuity of
19818:
19798:
19745:
19701:
19668:
19605:
19506:
19410:
19381:
19322:
19277:
19194:
19122:
19069:
19047:
18990:
18940:which proves that
18930:
18928:
18779:
18464:
18396:
18340:
18267:
18242:
18201:is continuous and
18191:
18171:
18154:
18107:
18087:
18051:
17995:
17804:
17782:{\displaystyle z,}
17779:
17756:
17701:
17672:
17650:
17599:
17567:
17534:
17481:
17427:
17407:
17379:
17329:
17298:
17263:
17243:
17223:
17187:
17155:
17135:
17108:
17047:
16996:
16976:
16970:
16923:
16901:
16879:
16857:
16835:
16815:
16791:
16780:
16751:
16719:
16699:
16664:{\displaystyle B,}
16661:
16638:
16618:
16554:
16534:
16460:
16431:
16393:
16360:{\displaystyle f.}
16357:
16334:
16307:
16259:
16218:
16191:
16136:
16094:{\displaystyle Y,}
16091:
16064:
16048:is any set and if
16038:
16006:
15970:
15942:
15846:
15836:which proves (2).
15826:
15794:
15734:
15709:
15688:
15652:
15650:
15643:
15574:
15528:
15436:
15209:
15117:
15055:
15049:
15002:
14963:
14911:
14854:
14825:
14630:
14597:
14560:
14540:
14501:
14455:
14429:
14396:
14360:
14327:
14275:
14217:
14181:
14155:
14126:
14059:
14038:
14003:). To prove that
13989:
13969:
13955:
13908:
13870:
13846:
13811:
13783:
13762:
13714:
13693:
13661:
13556:
13501:
13449:
13395:
13339:
13321:
13294:
13279:
13259:
13222:
13205:{\displaystyle U;}
13202:
13179:
13159:
13126:
13093:
13060:
13017:
12960:
12914:
12843:
12788:
12743:
12726:where recall that
12716:
12669:
12642:
12603:
12570:
12544:{\displaystyle X,}
12541:
12517:{\displaystyle X,}
12514:
12491:
12471:
12451:
12420:
12408:
12393:
12369:
12345:
12306:
12288:
12246:
12219:
12180:
12148:
12117:
12078:
12023:
11986:Euclidean topology
11974:
11952:
11928:
11881:
11839:{\displaystyle r,}
11836:
11813:
11765:
11743:
11723:
11709:
11693:
11661:
11641:
11630:
11589:
11576:
11538:
11508:
11475:
11445:
11412:
11356:
11329:
11288:
11237:
11215:
11154:
11126:{\displaystyle r,}
11123:
11097:
11068:
11033:
11024:
10992:
10981:
10949:
10920:
10898:
10871:
10862:
10830:
10819:
10783:
10772:
10740:
10714:
10680:
10648:{\displaystyle X.}
10645:
10622:
10602:
10582:
10531:
10506:
10432:
10399:
10290:
10204:
10146:
10121:
10029:
9993:
9939:
9930:
9883:
9856:
9836:
9782:
9754:
9748:
9694:
9643:
9590:
9517:{\displaystyle X,}
9514:
9491:
9468:is continuous and
9458:
9426:
9399:
9296:
9269:
9220:
9203:
9142:
9094:
9074:
9047:
9011:
8972:
8945:
8918:
8892:
8865:
8837:{\displaystyle X,}
8834:
8807:
8769:
8730:
8710:
8639:
8634:
8469:
8467:
8330:
8295:
8259:
8213:
8167:
8145:
8117:
8088:
8039:
8004:
7986:
7954:
7930:
7894:
7862:
7811:
7782:
7762:
7742:
7725:{\displaystyle s,}
7722:
7699:
7679:
7656:
7600:
7566:
7537:
7459:
7412:
7357:
7328:
7322:
7290:
7284:
7234:
7183:
7165:
7118:
7084:
7078:
7046:
6983:
6960:
6918:
6869:
6849:
6820:
6784:
6759:
6728:
6699:
6635:
6615:
6609:
6570:
6541:
6509:
6468:
6448:
6428:
6381:
6340:{\displaystyle r,}
6337:
6299:
6278:
6246:
6204:
6195:
6160:
6132:
6095:topological spaces
6030:
6003:
5972:
5952:
5913:
5886:
5872:
5852:
5832:
5805:
5795:then the polar of
5785:
5765:
5741:
5716:
5695:
5668:
5596:
5565:
5496:
5466:
5420:
5391:
5361:
5315:
5288:
5216:
5189:
5162:
5116:
5086:
5040:
5010:
4964:
4933:
4887:
4841:
4814:
4781:
4761:
4727:
4681:
4654:
4624:{\displaystyle X,}
4621:
4594:
4574:
4534:
4500:
4454:
4424:
4397:
4367:
4288:
4252:
4215:
4195:
4175:
4155:
4122:
4089:
4056:
4011:
3971:
3899:
3830:
3803:
3761:
3716:
3687:
3624:
3602:
3566:
3544:
3493:
3464:
3419:
3383:
3311:
3301:that converges to
3291:
3264:
3203:
3170:
3150:-closed subset of
3140:
3094:
3064:
3012:
2956:
2910:
2890:
2856:
2779:
2735:
2713:
2679:
2599:
2556:
2536:
2502:
2424:
2404:
2382:
2333:
2283:
2237:
2217:
2197:
2166:
2130:
2081:
2012:
1977:
1905:
1832:
1763:
1684:
1632:
1590:
1562:
1537:
1512:
1500:
1480:
1408:
1365:
1345:
1318:
1288:
1249:is compact in the
1239:
1219:
1196:
1162:
1084:
1039:
1021:
1006:
974:
915:
895:
875:
848:
780:
756:
707:
680:
634:
603:
563:
525:
494:
429:
409:
355:
286:
262:
235:
213:
121:
35:) states that the
32225:
32224:
31944:Relative interior
31690:Bilinear operator
31574:Linear functional
31510:
31509:
31413:Integral operator
31190:
31189:
31026:
31025:
30915:in Hilbert spaces
30757:978-0-471-55604-6
30722:978-0-387-97245-9
30676:978-0-486-45352-1
30646:978-0-12-622760-4
30616:978-1-4612-7155-0
30581:978-0-07-054236-5
30491:978-3-642-64988-2
30180:
29927:
29899:
29894:
29890:
29876:
29749:
29743:
29704:
29698:
29145:
29095:
29079:
29043:
29013:
28927:
28849:{\displaystyle 0}
28814:
28431:{\displaystyle X}
28418:of the origin in
28411:{\displaystyle U}
28212:
28207:
28203:
28189:
28098:
28019:
27983:
27957:
27952:
27948:
27934:
27848:
27812:
27804:
27798:
27736:
27694:ultrafilter lemma
27649:
27509:{\displaystyle U}
27420:{\displaystyle U}
27400:{\displaystyle X}
27053:
27052:
27046:
27008:
27003:
26999:
26985:
26921:{\displaystyle U}
26795:{\displaystyle U}
26775:{\displaystyle X}
26688:{\displaystyle U}
26664:{\displaystyle U}
26590:
26589:
26583:
26491:{\displaystyle X}
26340:subspace topology
26192:Goldstine theorem
26092:
25986:{\displaystyle B}
25922:{\displaystyle B}
25869:{\displaystyle B}
25849:{\displaystyle X}
25826:
25806:). However, the
25788:ultrafilter lemma
25777:ultrafilter Lemma
25594:is not reflexive.
25479:
25246:
25233:
25211:{\displaystyle q}
25120:{\displaystyle X}
25096:{\displaystyle X}
25076:{\displaystyle X}
25052:{\displaystyle X}
24867:{\displaystyle X}
24720:
24632:
24563:
24523:{\displaystyle F}
24216:{\displaystyle D}
24127:{\displaystyle D}
24083:. Because every
24043:
23922:
23826:{\displaystyle F}
23783:{\displaystyle x}
23575:{\displaystyle B}
23468:
23214:{\displaystyle X}
23164:{\displaystyle B}
23144:{\displaystyle X}
23075:{\displaystyle U}
22911:{\displaystyle U}
22887:{\displaystyle U}
22804:
22799:
22795:
22781:
22656:
22453:
22447:
22419:
22413:
22382:
22376:
22348:
22343:
22339:
22325:
22028:
22022:
21989:
21984:
21980:
21966:
21825:
21824:
21819:
21815:
21801:
21718:
21713:
21709:
21695:
21609:
21548:
21412:{\displaystyle X}
21298:
21293:
21289:
21275:
21249:{\displaystyle X}
21185:
21180:
21176:
21162:
21099:
21098:
20972:
20826:topological space
20771:{\displaystyle X}
20695:
20629:
20616:
20615:
20083:
20078:
20074:
20060:
19841:{\displaystyle A}
19805:
19800:
19796:
19782:
19329:
19324:
19320:
19306:
18924:
18895:
18872:
18820:
18786:
18781:
18777:
18763:
18680:
18589:
18582:
18579:
18576:
18518:
18505:
18473:
18466:
18462:
18194:{\displaystyle M}
18161:
18156:
18152:
18138:
18117:" map defined by
18110:{\displaystyle s}
17949:
17742:
17488:
17483:
17479:
17465:
17430:{\displaystyle z}
17266:{\displaystyle s}
17246:{\displaystyle f}
17197:To conclude that
17158:{\displaystyle f}
17145:the converges to
17057:and suppose that
16999:{\displaystyle X}
16955:
16838:{\displaystyle X}
16733:
16641:{\displaystyle Y}
16557:{\displaystyle Y}
16269:and suppose that
16143:
16138:
16134:
16120:
15673:
15662:is closed in the
15630:
15605:
15581:
15576:
15572:
15558:
15543:
15542: where
15513:
15485:
15468:
15455:
15451:
15445:
15421:
15352:
15320:
15317:
15313:
15310:
15288:
15285:
15281:
15278:
15274:
15271:
15267:
15264:
15194:
15190:
15187:
15183:
15180:
15162:
15156:
15153:
15149:
15146:
15142:
15139:
15135:
15132:
15119:
15115:
15034:
14921:which shows that
14810:
14758:
14708:
14661:
14582:
14563:{\displaystyle f}
14524:
14508:
14503:
14499:
14485:
14465:), which implies
14406:(so for example,
14023:
13992:{\displaystyle z}
13940:
13831:
13747:
13678:
13630:
13624:
13600:
13594:
13541:
13540:
13534:
13508:
13503:
13499:
13485:
13402:
13397:
13393:
13379:
13306:
13292:
13225:{\displaystyle f}
13182:{\displaystyle f}
13111:
12887:
12881:
12828:
12827:
12821:
12795:
12790:
12786:
12772:
12494:{\displaystyle U}
12474:{\displaystyle U}
12423:{\displaystyle X}
12406:
12396:{\displaystyle U}
12372:{\displaystyle X}
12273:
12063:
12062:
12056:
12030:
12025:
12021:
12007:
11955:{\displaystyle X}
11888:
11883:
11879:
11865:
11798:
11746:{\displaystyle X}
11726:{\displaystyle U}
11707:
11696:{\displaystyle X}
11676:
11675:
11664:{\displaystyle X}
11637:
11632:
11628:
11614:
11583:
11578:
11574:
11560:
11493:
11430:
11295:
11290:
11286:
11272:
11187:
11139:
11053:
11009:
10966:
10847:
10804:
10795:subspace topology
10757:
10625:{\displaystyle f}
10605:{\displaystyle f}
10297:
10292:
10288:
10274:
9915:
9893:is an element of
9859:{\displaystyle f}
9785:{\displaystyle f}
9733:
9406:
9401:
9397:
9383:
9097:{\displaystyle F}
8779:from a non-empty
8733:{\displaystyle X}
8309:
8280:
8244:
8170:{\displaystyle X}
7971:
7915:
7879:
7785:{\displaystyle s}
7765:{\displaystyle s}
7745:{\displaystyle z}
7702:{\displaystyle z}
7663:
7658:
7654:
7640:
7576:sends a function
7477:
7444:
7396:at a given point
7307:
7269:
7172:
7167:
7163:
7149:
7063:
6925:
6920:
6916:
6902:
6872:{\displaystyle X}
6638:{\displaystyle X}
6594:
6516:
6511:
6507:
6493:
6471:{\displaystyle 0}
6451:{\displaystyle r}
6388:
6383:
6379:
6365:
6263:
6180:
6172:Cartesian product
6082:does not utilize
5975:{\displaystyle X}
5889:{\displaystyle X}
5870:
5855:{\displaystyle X}
5808:{\displaystyle U}
5788:{\displaystyle X}
5768:{\displaystyle U}
5744:{\displaystyle X}
5020:-totally bounded.
4784:{\displaystyle X}
4597:{\displaystyle U}
4218:{\displaystyle f}
4198:{\displaystyle U}
4178:{\displaystyle f}
4107:
3393:To conclude that
3314:{\displaystyle f}
3213:and suppose that
2913:{\displaystyle U}
2841:
2840:
2834:
2738:{\displaystyle U}
2657:
2656:
2650:
2559:{\displaystyle U}
2487:
2486:
2480:
2427:{\displaystyle X}
2407:{\displaystyle U}
2240:{\displaystyle f}
2220:{\displaystyle f}
1993:subspace topology
1515:{\displaystyle X}
1498:
1368:{\displaystyle U}
1242:{\displaystyle X}
1222:{\displaystyle U}
1147:
1146:
1140:
1042:{\displaystyle X}
1019:
918:{\displaystyle x}
898:{\displaystyle f}
783:{\displaystyle f}
528:{\displaystyle X}
439:where the triple
416:
411:
407:
393:
289:{\displaystyle X}
216:{\displaystyle X}
33:Alaoglu's theorem
32260:
32215:
32214:
32189:Uniformly smooth
31858:
31850:
31817:Balanced/Circled
31807:Absorbing/Radial
31537:
31530:
31523:
31514:
31513:
31500:
31499:
31418:Jones polynomial
31336:Operator algebra
31080:
31079:
31053:
31046:
31039:
31030:
31029:
30997:Saturated family
30895:Ultraweak/Weak-*
30804:
30797:
30790:
30781:
30780:
30776:
30774:
30772:
30749:
30734:
30688:
30663:Trèves, François
30658:
30628:
30593:
30557:
30530:
30511:
30478:Köthe, Gottfried
30463:
30462:
30460:
30458:
30437:
30428:
30419:
30413:
30407:
30401:
30395:
30389:
30372:
30366:
30355:
30349:
30332:
30331:
30329:
30328:
30323:
30311:
30309:
30308:
30303:
30292:
30291:
30279:
30262:
30250:
30248:
30247:
30242:
30237:
30236:
30235:
30234:
30217:
30213:
30212:
30211:
30210:
30209:
30194:
30174:
30173:
30168:
30150:
30149:
30144:
30117:it follows that
30116:
30114:
30113:
30108:
30103:
30102:
30101:
30100:
30080:
30079:
30054:
30052:
30051:
30046:
30038:
30021:
30009:
30007:
30006:
30001:
29983:
29981:
29980:
29975:
29964:
29947:
29941:
29919:
29917:
29916:
29911:
29909:
29908:
29897:
29896:
29895:
29893:
29891:
29888:
29884:
29879:
29874:
29857:
29855:
29854:
29849:
29844:
29843:
29831:
29827:
29826:
29825:
29795:
29793:
29792:
29787:
29785:
29784:
29772:
29768:
29767:
29766:
29747:
29741:
29740:
29739:
29727:
29723:
29722:
29721:
29702:
29696:
29685:
29683:
29682:
29677:
29656:
29654:
29653:
29648:
29640:
29639:
29623:
29621:
29620:
29615:
29594:
29592:
29591:
29586:
29578:
29577:
29561:
29559:
29558:
29553:
29551:
29550:
29539:
29535:
29534:
29513:
29511:
29510:
29505:
29503:
29502:
29486:
29484:
29483:
29478:
29476:
29475:
29459:
29457:
29456:
29451:
29449:
29448:
29428:
29426:
29425:
29420:
29418:
29417:
29398:
29396:
29395:
29390:
29374:with respect to
29373:
29371:
29370:
29365:
29363:
29362:
29342:
29340:
29339:
29334:
29332:
29331:
29312:
29310:
29309:
29304:
29299:
29298:
29286:
29285:
29266:
29264:
29263:
29258:
29256:
29255:
29243:
29242:
29227:it follows that
29226:
29224:
29223:
29218:
29216:
29215:
29190:
29188:
29187:
29182:
29177:
29176:
29175:
29174:
29159:
29141:
29140:
29139:
29138:
29123:
29122:
29121:
29109:
29108:
29093:
29075:
29074:
29073:
29072:
29057:
29041:
29040:
29039:
29027:
29026:
29009:
29008:
28989:
28987:
28986:
28981:
28979:
28975:
28974:
28973:
28961:
28960:
28948:
28947:
28932:
28923:
28922:
28906:
28904:
28903:
28898:
28877:
28875:
28874:
28869:
28867:
28855:
28853:
28852:
28847:
28835:
28833:
28832:
28827:
28825:
28824:
28819:
28805:
28801:
28788:
28787:
28763:
28761:
28760:
28755:
28721:
28704:
28703:
28701:
28700:
28695:
28674:
28672:
28671:
28666:
28658:
28657:
28641:
28639:
28638:
28633:
28625:
28624:
28608:
28606:
28605:
28600:
28592:
28591:
28575:
28573:
28572:
28567:
28546:
28544:
28543:
28538:
28520:
28518:
28517:
28512:
28491:
28489:
28488:
28483:
28454:
28453:
28437:
28435:
28434:
28429:
28417:
28415:
28414:
28409:
28397:
28395:
28394:
28389:
28368:
28366:
28365:
28360:
28358:
28357:
28345:
28344:
28328:
28326:
28325:
28320:
28318:
28317:
28305:
28304:
28284:
28282:
28281:
28276:
28274:
28273:
28253:
28251:
28250:
28245:
28243:
28242:
28231:
28227:
28226:
28210:
28209:
28208:
28206:
28204:
28201:
28197:
28192:
28187:
28186:
28185:
28168:
28162:
28160:
28158:
28157:
28152:
28147:
28146:
28131:this is true of
28128:
28126:
28125:
28120:
28118:
28112:
28090:
28088:
28087:
28082:
28080:
28079:
28063:
28061:
28060:
28055:
28050:
28046:
28045:
28039:
28015:
28014:
28008:
28007:
27997:
27982:
27981:
27967:
27966:
27955:
27954:
27953:
27951:
27949:
27946:
27942:
27937:
27932:
27931:
27930:
27914:
27912:
27911:
27906:
27904:
27903:
27887:
27885:
27884:
27879:
27874:
27868:
27844:
27843:
27837:
27836:
27826:
27811:
27810:
27802:
27796:
27795:
27794:
27793:
27792:
27771:
27769:
27768:
27763:
27761:
27760:
27750:
27732:
27731:
27730:
27729:
27707:
27701:
27691:
27689:
27688:
27683:
27681:
27680:
27679:
27678:
27663:
27637:
27635:
27634:
27629:
27627:
27626:
27625:
27624:
27602:
27596:
27594:
27592:
27591:
27586:
27574:
27572:
27571:
27566:
27564:
27563:
27547:
27545:
27544:
27539:
27516:with respect to
27515:
27513:
27512:
27507:
27495:
27493:
27492:
27487:
27485:
27484:
27462:
27460:
27459:
27454:
27426:
27424:
27423:
27418:
27406:
27404:
27403:
27398:
27386:
27384:
27383:
27378:
27366:
27364:
27363:
27358:
27356:
27355:
27321:
27319:
27318:
27313:
27311:
27310:
27288:
27286:
27285:
27280:
27278:
27277:
27255:
27253:
27252:
27247:
27242:
27241:
27225:
27223:
27222:
27217:
27215:
27214:
27196:
27195:
27173:
27171:
27170:
27165:
27163:
27162:
27147:is just the set
27146:
27144:
27143:
27138:
27136:
27135:
27113:
27111:
27110:
27105:
27103:
27102:
27090:
27073:
27067:
27050:
27044:
27043:
27042:
27015:
27014:
27006:
27005:
27004:
27002:
27000:
26997:
26993:
26988:
26983:
26982:
26981:
26959:
26957:
26956:
26951:
26928:with respect to
26927:
26925:
26924:
26919:
26907:
26905:
26904:
26899:
26897:
26896:
26868:
26866:
26865:
26860:
26836:
26834:
26833:
26828:
26801:
26799:
26798:
26793:
26781:
26779:
26778:
26773:
26761:
26759:
26758:
26753:
26741:
26739:
26738:
26733:
26721:
26719:
26718:
26713:
26711:
26710:
26694:
26692:
26691:
26686:
26670:
26668:
26667:
26662:
26650:
26648:
26647:
26642:
26640:
26639:
26627:
26610:
26604:
26587:
26581:
26580:
26579:
26564:
26563:
26554:
26553:
26537:
26535:
26534:
26529:
26527:
26526:
26514:
26513:
26497:
26495:
26494:
26489:
26477:
26475:
26474:
26469:
26456:
26450:
26440:
26438:
26437:
26432:
26430:
26429:
26413:
26411:
26410:
26405:
26400:
26396:
26395:
26391:
26384:
26383:
26363:
26362:
26337:
26335:
26334:
26329:
26327:
26326:
26306:
26304:
26303:
26298:
26296:
26295:
26279:
26277:
26276:
26271:
26269:
26268:
26256:
26255:
26238:
26211:
26202:
26124:
26122:
26121:
26116:
26110:
26109:
26108:
26074:
26072:
26071:
26066:
26061:
26060:
26026:
26024:
26023:
26018:
26016:
26015:
25992:
25990:
25989:
25984:
25972:
25970:
25969:
25964:
25962:
25961:
25947:
25946:
25939:
25938:
25937:quasicompactness
25928:
25926:
25925:
25920:
25908:
25906:
25905:
25900:
25895:
25894:
25875:
25873:
25872:
25867:
25855:
25853:
25852:
25847:
25835:
25831:
25830:
25816:quasicompactness
25719:
25717:
25716:
25711:
25686:
25684:
25683:
25678:
25646:
25644:
25643:
25638:
25593:
25591:
25590:
25585:
25574:
25573:
25558:is not true, as
25557:
25555:
25554:
25549:
25531:
25529:
25528:
25523:
25518:
25517:
25496:
25495:
25480:
25477:
25445:
25444:
25443:
25442:
25418:
25416:
25415:
25410:
25392:
25390:
25389:
25384:
25381:
25376:
25365:
25361:
25360:
25359:
25358:
25332:
25330:
25329:
25324:
25309:
25307:
25306:
25301:
25293:
25292:
25280:
25279:
25263:
25261:
25260:
25255:
25247:
25239:
25234:
25226:
25217:
25215:
25214:
25209:
25197:
25195:
25194:
25189:
25165:
25163:
25162:
25157:
25146:
25145:
25126:
25124:
25123:
25118:
25102:
25100:
25099:
25094:
25082:
25080:
25079:
25074:
25058:
25056:
25055:
25050:
25031:
25029:
25028:
25023:
25021:
25017:
25016:
25015:
24966:
24964:
24963:
24958:
24956:
24955:
24939:
24937:
24936:
24931:
24929:
24928:
24904:
24902:
24901:
24896:
24894:
24893:
24873:
24871:
24870:
24865:
24840:
24838:
24837:
24832:
24821:
24819:
24818:
24813:
24808:
24807:
24776:
24774:
24773:
24768:
24765:
24759:
24743:
24741:
24740:
24735:
24721:
24718:
24715:
24714:
24683:
24681:
24680:
24675:
24673:
24649:
24647:
24646:
24641:
24633:
24630:
24627:
24626:
24614:
24613:
24602:
24598:
24588:
24587:
24571:
24555:
24553:
24552:
24547:
24529:
24527:
24526:
24521:
24510:in the image of
24509:
24507:
24506:
24501:
24499:
24498:
24487:
24483:
24473:
24472:
24450:
24448:
24447:
24442:
24434:
24433:
24422:
24418:
24417:
24400:
24399:
24376:
24374:
24373:
24368:
24365:
24359:
24341:
24339:
24338:
24333:
24331:
24324:
24323:
24283:
24277:
24276:
24261:
24258:
24252:
24242:
24222:
24220:
24219:
24214:
24202:
24200:
24199:
24194:
24188:
24182:
24166:
24164:
24163:
24158:
24153:
24152:
24133:
24131:
24130:
24125:
24109:
24107:
24106:
24101:
24099:
24098:
24081:product topology
24078:
24076:
24075:
24070:
24068:
24067:
24057:
24029:
24027:
24026:
24021:
24004:
23996:
23988:
23971:
23970:
23954:
23952:
23951:
23946:
23910:
23908:
23907:
23902:
23897:
23889:
23888:
23866:
23864:
23863:
23858:
23856:
23855:
23832:
23830:
23829:
23824:
23812:
23810:
23809:
23804:
23789:
23787:
23786:
23781:
23769:
23767:
23766:
23761:
23746:
23744:
23743:
23738:
23723:
23721:
23720:
23715:
23713:
23712:
23694:
23693:
23681:
23680:
23664:
23662:
23661:
23656:
23641:
23639:
23638:
23633:
23631:
23623:
23622:
23581:
23579:
23578:
23573:
23561:
23559:
23558:
23553:
23538:
23536:
23535:
23530:
23528:
23527:
23511:
23509:
23508:
23503:
23479:
23477:
23476:
23471:
23469:
23467:
23466:
23462:
23458:
23457:
23421:
23417:
23413:
23412:
23383:
23380:
23379:
23365:
23360:
23318:
23316:
23315:
23310:
23286:
23284:
23283:
23278:
23275:
23270:
23259:
23255:
23254:
23237:
23236:
23220:
23218:
23217:
23212:
23200:
23198:
23197:
23192:
23187:
23186:
23170:
23168:
23167:
23162:
23150:
23148:
23147:
23142:
23108:
23106:
23105:
23100:
23081:
23079:
23078:
23073:
23061:
23059:
23058:
23053:
23051:
23050:
23035:
23033:
23032:
23027:
23022:
23021:
23016:
23012:
23005:
23004:
22986:
22985:
22969:
22967:
22966:
22961:
22959:
22958:
22934:
22933:
22917:
22915:
22914:
22909:
22893:
22891:
22890:
22885:
22869:
22867:
22866:
22861:
22835:
22834:
22823:
22819:
22818:
22802:
22801:
22800:
22798:
22796:
22793:
22789:
22784:
22779:
22778:
22777:
22753:
22751:
22750:
22745:
22730:with respect to
22729:
22727:
22726:
22721:
22719:
22718:
22698:
22696:
22695:
22690:
22688:
22687:
22686:
22685:
22670:
22652:
22651:
22650:
22649:
22623:
22621:
22620:
22615:
22610:
22609:
22593:
22591:
22590:
22585:
22583:
22582:
22563:
22561:
22560:
22555:
22553:
22552:
22540:
22539:
22524:
22523:
22522:
22521:
22497:
22495:
22494:
22489:
22484:
22480:
22479:
22478:
22477:
22476:
22451:
22445:
22444:
22443:
22442:
22441:
22417:
22411:
22410:
22406:
22405:
22404:
22392:
22391:
22380:
22374:
22373:
22372:
22371:
22370:
22346:
22345:
22344:
22342:
22340:
22337:
22333:
22328:
22323:
22319:
22318:
22303:
22301:
22300:
22295:
22280:
22278:
22277:
22272:
22270:
22269:
22246:
22244:
22243:
22238:
22236:
22235:
22219:
22217:
22216:
22211:
22206:
22202:
22201:
22200:
22188:
22187:
22175:
22174:
22154:
22153:
22138:
22136:
22135:
22130:
22109:
22107:
22106:
22101:
22099:
22098:
22086:
22085:
22069:
22067:
22066:
22061:
22059:
22055:
22054:
22053:
22052:
22051:
22026:
22020:
22019:
22018:
22013:
22004:
22003:
21987:
21986:
21985:
21983:
21981:
21978:
21974:
21969:
21964:
21963:
21962:
21946:
21944:
21943:
21938:
21933:
21932:
21927:
21918:
21917:
21906:
21902:
21901:
21884:
21883:
21867:
21865:
21864:
21859:
21857:
21856:
21855:
21854:
21839:
21822:
21821:
21820:
21818:
21816:
21813:
21809:
21804:
21799:
21798:
21797:
21796:
21795:
21771:
21769:
21768:
21763:
21761:
21760:
21738:
21736:
21735:
21730:
21728:
21727:
21716:
21715:
21714:
21712:
21710:
21707:
21703:
21698:
21693:
21682:
21680:
21679:
21674:
21672:
21671:
21655:
21653:
21652:
21647:
21645:
21641:
21640:
21639:
21638:
21623:
21602:
21600:
21599:
21584:
21580:
21579:
21578:
21577:
21562:
21541:
21539:
21538:
21523:
21520:
21519:
21517:
21516:
21500:
21499:
21479:
21477:
21476:
21471:
21469:
21468:
21457:
21453:
21452:
21435:
21434:
21418:
21416:
21415:
21410:
21396:
21394:
21393:
21388:
21383:
21382:
21362:
21360:
21359:
21354:
21352:
21348:
21347:
21346:
21319:
21318:
21296:
21295:
21294:
21292:
21290:
21287:
21283:
21278:
21273:
21272:
21271:
21255:
21253:
21252:
21247:
21234:absorbing subset
21231:
21229:
21228:
21223:
21183:
21182:
21181:
21179:
21177:
21174:
21170:
21165:
21160:
21128:
21126:
21125:
21120:
21094:
21092:
21091:
21086:
21074:
21072:
21071:
21066:
21064:
21060:
21059:
21055:
21048:
21047:
21027:
21026:
21005:
21003:
21002:
20997:
20992:
20986:
20968:
20967:
20962:
20946:
20944:
20943:
20938:
20936:
20914:
20913:
20895:
20893:
20892:
20887:
20885:
20881:
20880:
20876:
20869:
20868:
20848:
20847:
20823:
20821:
20820:
20815:
20813:
20809:
20802:
20801:
20777:
20775:
20774:
20769:
20757:
20755:
20754:
20749:
20747:
20746:
20727:
20722:
20720:
20719:
20714:
20712:
20711:
20690:
20688:
20687:
20682:
20680:
20679:
20659:
20657:
20656:
20651:
20649:
20643:
20611:
20609:
20608:
20603:
20591:
20589:
20588:
20583:
20524:it follows that
20523:
20521:
20520:
20515:
20474:
20473:
20457:
20455:
20454:
20449:
20399:
20398:
20382:
20380:
20379:
20374:
20357:
20356:
20344:
20343:
20332:
20328:
20312:
20311:
20293:
20292:
20281:
20277:
20267:
20266:
20245:
20244:
20226:
20225:
20214:
20210:
20209:
20205:
20195:
20194:
20173:
20172:
20146:
20145:
20134:
20130:
20129:
20125:
20124:
20099:
20098:
20081:
20080:
20079:
20077:
20075:
20072:
20068:
20063:
20058:
20057:
20053:
20052:
20029:
20027:
20026:
20021:
19946:
19944:
19943:
19938:
19936:
19924:
19922:
19921:
19916:
19875:
19871:
19870:
19847:
19845:
19844:
19839:
19827:
19825:
19824:
19819:
19803:
19802:
19801:
19799:
19797:
19794:
19790:
19785:
19780:
19754:
19752:
19751:
19746:
19744:
19736:
19728:
19710:
19708:
19707:
19702:
19697:
19689:
19677:
19675:
19674:
19669:
19631:
19630:
19615:it follows that
19614:
19612:
19611:
19606:
19586:
19585:
19574:
19570:
19560:
19559:
19532:
19531:
19515:
19513:
19512:
19507:
19490:
19489:
19478:
19474:
19464:
19463:
19436:
19435:
19419:
19417:
19416:
19411:
19390:
19388:
19387:
19382:
19380:
19376:
19366:
19365:
19344:
19343:
19327:
19326:
19325:
19323:
19321:
19318:
19314:
19309:
19304:
19303:
19302:
19286:
19284:
19283:
19278:
19276:
19268:
19254:
19253:
19242:
19238:
19237:
19220:
19219:
19203:
19201:
19200:
19195:
19131:
19129:
19128:
19123:
19078:
19076:
19075:
19070:
19068:
19056:
19054:
19053:
19048:
19016:
19015:
18999:
18997:
18996:
18991:
18956:
18955:
18939:
18937:
18936:
18931:
18929:
18925:
18922:
18919:
18905:
18904:
18893:
18883:
18882:
18873:
18870:
18867:
18865:
18864:
18853:
18849:
18836:
18835:
18818:
18799:
18798:
18784:
18783:
18782:
18780:
18778:
18775:
18771:
18766:
18761:
18760:
18756:
18746:
18745:
18727:
18725:
18724:
18713:
18709:
18699:
18698:
18678:
18668:
18654:
18653:
18642:
18638:
18628:
18627:
18600:
18599:
18590:
18587:
18584:
18580:
18577:
18574:
18573:
18572:
18561:
18557:
18556:
18552:
18542:
18541:
18516:
18506:
18503:
18500:
18489:
18488:
18471:
18468:
18467:
18465:
18463:
18460:
18456:
18451:
18448:
18444:
18434:
18433:
18405:
18403:
18402:
18397:
18349:
18347:
18346:
18341:
18315:
18311:
18301:
18300:
18277:it follows that
18276:
18274:
18273:
18268:
18263:
18251:
18249:
18248:
18243:
18217:
18216:
18200:
18198:
18197:
18192:
18180:
18178:
18177:
18172:
18159:
18158:
18157:
18155:
18153:
18150:
18146:
18141:
18136:
18116:
18114:
18113:
18108:
18096:
18094:
18093:
18088:
18086:
18078:
18060:
18058:
18057:
18052:
18004:
18002:
18001:
17996:
17961:
17960:
17950:
17947:
17905:
17904:
17867:
17866:
17829:
17828:
17813:
17811:
17810:
17805:
17800:
17788:
17786:
17785:
17780:
17765:
17763:
17762:
17757:
17743:
17740:
17710:
17708:
17707:
17702:
17681:
17679:
17678:
17673:
17671:
17659:
17657:
17656:
17651:
17625:
17624:
17608:
17606:
17605:
17600:
17595:
17594:
17589:
17576:
17574:
17573:
17568:
17560:
17559:
17543:
17541:
17540:
17535:
17530:
17529:
17518:
17514:
17504:
17503:
17486:
17485:
17484:
17482:
17480:
17477:
17473:
17468:
17463:
17453:
17452:
17436:
17434:
17433:
17428:
17416:
17414:
17413:
17408:
17406:
17405:
17388:
17386:
17385:
17380:
17378:
17355:
17354:
17338:
17336:
17335:
17330:
17307:
17305:
17304:
17299:
17272:
17270:
17269:
17264:
17252:
17250:
17249:
17244:
17232:
17230:
17229:
17224:
17219:
17218:
17196:
17194:
17193:
17188:
17183:
17182:
17177:
17164:
17162:
17161:
17156:
17144:
17142:
17141:
17136:
17134:
17133:
17117:
17115:
17114:
17109:
17107:
17106:
17095:
17091:
17090:
17073:
17072:
17056:
17054:
17053:
17048:
17046:
17045:
17040:
17011:
17010:
17005:
17003:
17002:
16997:
16985:
16983:
16982:
16977:
16975:
16969:
16951:
16950:
16945:
16932:
16930:
16929:
16924:
16922:
16910:
16908:
16907:
16902:
16900:
16888:
16886:
16885:
16880:
16878:
16866:
16864:
16863:
16858:
16856:
16844:
16842:
16841:
16836:
16824:
16822:
16821:
16816:
16814:
16813:
16794:
16789:
16787:
16786:
16781:
16779:
16778:
16773:
16760:
16758:
16757:
16752:
16750:
16749:
16728:
16726:
16725:
16720:
16708:
16706:
16705:
16700:
16670:
16668:
16667:
16662:
16647:
16645:
16644:
16639:
16627:
16625:
16624:
16619:
16602:
16601:
16580:
16579:
16564:and every value
16563:
16561:
16560:
16555:
16543:
16541:
16540:
16535:
16518:
16517:
16506:
16502:
16492:
16491:
16469:
16467:
16466:
16461:
16440:
16438:
16437:
16432:
16402:
16400:
16399:
16394:
16389:
16388:
16366:
16364:
16363:
16358:
16343:
16341:
16340:
16335:
16333:
16332:
16316:
16314:
16313:
16308:
16306:
16305:
16294:
16290:
16289:
16268:
16266:
16265:
16260:
16258:
16257:
16227:
16225:
16224:
16219:
16217:
16216:
16200:
16198:
16197:
16192:
16190:
16186:
16164:
16163:
16141:
16140:
16139:
16137:
16135:
16132:
16128:
16123:
16118:
16117:
16116:
16100:
16098:
16097:
16092:
16073:
16071:
16070:
16065:
16047:
16045:
16044:
16039:
16015:
16013:
16012:
16007:
16002:
16001:
15979:
15977:
15976:
15971:
15969:
15951:
15949:
15948:
15943:
15941:
15937:
15936:
15935:
15908:
15907:
15902:
15882:
15881:
15880:
15879:
15855:
15853:
15852:
15847:
15835:
15833:
15832:
15827:
15822:
15821:
15816:
15803:
15801:
15800:
15795:
15793:
15792:
15780:
15779:
15767:
15766:
15765:
15764:
15743:
15741:
15740:
15735:
15730:
15718:
15716:
15715:
15710:
15708:
15707:
15702:
15693:
15687:
15664:product topology
15661:
15659:
15658:
15653:
15651:
15647:
15646:
15631:
15628:
15624:
15606:
15603:
15599:
15598:
15579:
15578:
15577:
15575:
15573:
15570:
15566:
15561:
15556:
15555:
15554:
15544:
15541:
15538:
15537:
15527:
15506:
15502:
15501:
15486:
15483:
15481:
15480:
15466:
15465:
15464:
15453:
15449:
15443:
15441:
15435:
15417:
15416:
15405:
15401:
15400:
15386:
15385:
15373:
15369:
15368:
15353:
15350:
15348:
15347:
15318:
15315:
15311:
15308:
15307:
15306:
15301:
15286:
15283:
15279:
15276:
15272:
15269:
15265:
15262:
15261:
15260:
15248:
15244:
15243:
15231:
15214:
15208:
15188:
15185:
15181:
15178:
15177:
15176:
15171:
15160:
15154:
15151:
15147:
15144:
15140:
15137:
15133:
15130:
15129:
15128:
15121:
15120:
15118:
15116:
15113:
15109:
15104:
15096:
15095:
15094:
15093:
15064:
15062:
15061:
15056:
15054:
15048:
15030:
15029:
15024:
15011:
15009:
15008:
15003:
15001:
15000:
14972:
14970:
14969:
14964:
14959:
14958:
14957:
14956:
14920:
14918:
14917:
14912:
14907:
14906:
14894:
14877:
14863:
14861:
14860:
14855:
14847:
14830:
14824:
14806:
14802:
14801:
14797:
14796:
14772:
14768:
14767:
14763:
14759:
14757:
14756:
14744:
14726:
14722:
14709:
14707:
14706:
14694:
14684:
14667:
14662:
14660:
14659:
14647:
14639:
14637:
14636:
14631:
14619:
14602:
14596:
14569:
14567:
14566:
14561:
14549:
14547:
14546:
14541:
14525:
14523:
14522:
14510:
14506:
14505:
14504:
14502:
14500:
14497:
14493:
14488:
14483:
14482:
14481:
14464:
14462:
14461:
14456:
14438:
14436:
14435:
14430:
14422:
14421:
14405:
14403:
14402:
14397:
14392:
14391:
14369:
14367:
14366:
14361:
14353:
14352:
14336:
14334:
14333:
14328:
14323:
14322:
14321:
14320:
14284:
14282:
14281:
14276:
14245:
14244:
14239:
14226:
14224:
14223:
14218:
14213:
14212:
14190:
14188:
14187:
14182:
14164:
14162:
14161:
14156:
14135:
14133:
14132:
14127:
14125:
14124:
14123:
14122:
14105:
14101:
14100:
14087:
14086:
14081:
14068:
14066:
14065:
14060:
14055:
14054:
14053:
14052:
14037:
14019:
14018:
14001:as defined above
13998:
13996:
13995:
13990:
13978:
13976:
13975:
13970:
13968:
13960:
13954:
13936:
13935:
13930:
13917:
13915:
13914:
13909:
13879:
13877:
13876:
13871:
13866:
13865:
13860:
13851:
13845:
13820:
13818:
13817:
13812:
13810:
13809:
13792:
13790:
13789:
13784:
13779:
13778:
13777:
13776:
13761:
13743:
13742:
13723:
13721:
13720:
13715:
13710:
13709:
13708:
13707:
13692:
13670:
13668:
13667:
13662:
13660:
13656:
13655:
13654:
13628:
13622:
13621:
13620:
13598:
13592:
13591:
13590:
13578:
13561:
13555:
13538:
13532:
13531:
13530:
13515:
13514:
13506:
13505:
13504:
13502:
13500:
13497:
13493:
13488:
13483:
13482:
13481:
13458:
13456:
13455:
13450:
13445:
13444:
13432:
13424:
13416:
13400:
13399:
13398:
13396:
13394:
13391:
13387:
13382:
13377:
13376:
13375:
13374:
13373:
13348:
13346:
13345:
13340:
13338:
13337:
13336:
13335:
13320:
13288:
13286:
13285:
13280:
13269:), as desired.
13268:
13266:
13265:
13260:
13258:
13257:
13231:
13229:
13228:
13223:
13211:
13209:
13208:
13203:
13188:
13186:
13185:
13180:
13168:
13166:
13165:
13160:
13148:
13131:
13125:
13102:
13100:
13099:
13094:
13092:
13091:
13069:
13067:
13066:
13061:
13056:
13055:
13043:
13042:
13026:
13024:
13023:
13018:
13013:
13012:
13000:
12999:
12987:
12986:
12969:
12967:
12966:
12961:
12956:
12955:
12943:
12942:
12923:
12921:
12920:
12915:
12910:
12909:
12897:
12896:
12885:
12879:
12878:
12877:
12865:
12848:
12842:
12825:
12819:
12818:
12817:
12802:
12801:
12793:
12792:
12791:
12789:
12787:
12784:
12780:
12775:
12770:
12769:
12768:
12752:
12750:
12749:
12744:
12742:
12741:
12725:
12723:
12722:
12717:
12712:
12711:
12699:
12698:
12678:
12676:
12675:
12670:
12668:
12667:
12651:
12649:
12648:
12643:
12635:
12634:
12612:
12610:
12609:
12604:
12596:
12595:
12579:
12577:
12576:
12571:
12550:
12548:
12547:
12542:
12526:absorbing subset
12523:
12521:
12520:
12515:
12500:
12498:
12497:
12492:
12480:
12478:
12477:
12472:
12460:
12458:
12457:
12452:
12450:
12449:
12429:
12427:
12426:
12421:
12402:
12400:
12399:
12394:
12378:
12376:
12375:
12370:
12354:
12352:
12351:
12346:
12344:
12343:
12318:product topology
12315:
12313:
12312:
12307:
12305:
12304:
12303:
12302:
12287:
12256:is a closed and
12255:
12253:
12252:
12247:
12245:
12244:
12228:
12226:
12225:
12220:
12212:
12211:
12189:
12187:
12186:
12181:
12173:
12172:
12157:
12155:
12154:
12149:
12126:
12124:
12123:
12118:
12113:
12112:
12100:
12083:
12077:
12060:
12054:
12053:
12052:
12037:
12036:
12028:
12027:
12026:
12024:
12022:
12019:
12015:
12010:
12005:
12004:
12003:
11983:
11981:
11980:
11975:
11973:
11961:
11959:
11958:
11953:
11937:
11935:
11934:
11929:
11918:
11910:
11902:
11886:
11885:
11884:
11882:
11880:
11877:
11873:
11868:
11863:
11862:
11861:
11845:
11843:
11842:
11837:
11822:
11820:
11819:
11814:
11812:
11804:
11799:
11796:
11794:
11786:
11774:
11772:
11771:
11766:
11764:
11752:
11750:
11749:
11744:
11732:
11730:
11729:
11724:
11712:
11702:
11700:
11699:
11694:
11670:
11668:
11667:
11662:
11650:
11648:
11647:
11642:
11635:
11634:
11633:
11631:
11629:
11626:
11622:
11617:
11612:
11598:
11596:
11595:
11590:
11588:
11581:
11580:
11579:
11577:
11575:
11572:
11568:
11563:
11558:
11547:
11545:
11544:
11539:
11530:
11513:
11507:
11484:
11482:
11481:
11476:
11467:
11450:
11444:
11421:
11419:
11418:
11413:
11402:
11394:
11386:
11365:
11363:
11362:
11357:
11355:
11354:
11338:
11336:
11335:
11330:
11293:
11292:
11291:
11289:
11287:
11284:
11280:
11275:
11270:
11246:
11244:
11243:
11238:
11224:
11222:
11221:
11216:
11214:
11213:
11188:
11185:
11176:
11159:
11153:
11132:
11130:
11129:
11124:
11106:
11104:
11103:
11098:
11090:
11073:
11067:
11042:
11040:
11039:
11034:
11029:
11023:
11001:
10999:
10998:
10993:
10991:
10990:
10980:
10958:
10956:
10955:
10950:
10929:
10927:
10926:
10921:
10919:
10907:
10905:
10904:
10899:
10897:
10896:
10880:
10878:
10877:
10872:
10867:
10861:
10839:
10837:
10836:
10831:
10829:
10828:
10818:
10793:is equal to the
10792:
10790:
10789:
10784:
10782:
10781:
10771:
10749:
10747:
10746:
10741:
10739:
10723:
10721:
10720:
10715:
10713:
10705:
10704:
10689:
10687:
10686:
10681:
10654:
10652:
10651:
10646:
10631:
10629:
10628:
10623:
10611:
10609:
10608:
10603:
10591:
10589:
10588:
10583:
10581:
10580:
10569:
10565:
10564:
10540:
10538:
10537:
10532:
10527:
10515:
10513:
10512:
10507:
10490:
10489:
10478:
10474:
10464:
10463:
10441:
10439:
10438:
10433:
10408:
10406:
10405:
10400:
10395:
10394:
10383:
10379:
10369:
10368:
10350:
10349:
10338:
10334:
10333:
10329:
10328:
10315:
10314:
10309:
10295:
10294:
10293:
10291:
10289:
10286:
10282:
10277:
10272:
10271:
10267:
10266:
10255:
10251:
10250:
10232:
10231:
10226:
10213:
10211:
10210:
10205:
10178:
10177:
10172:
10155:
10153:
10152:
10147:
10142:
10130:
10128:
10127:
10122:
10111:
10110:
10105:
10096:
10092:
10091:
10080:
10076:
10075:
10057:
10056:
10051:
10038:
10036:
10035:
10030:
10002:
10000:
9999:
9994:
9986:
9985:
9974:
9970:
9969:
9948:
9946:
9945:
9940:
9935:
9929:
9911:
9910:
9905:
9892:
9890:
9889:
9884:
9882:
9881:
9865:
9863:
9862:
9857:
9845:
9843:
9842:
9837:
9832:
9831:
9820:
9816:
9815:
9791:
9789:
9788:
9783:
9766:product topology
9763:
9761:
9760:
9755:
9753:
9747:
9729:
9728:
9723:
9703:
9701:
9700:
9695:
9675:
9674:
9652:
9650:
9649:
9644:
9627:
9623:
9622:
9599:
9597:
9596:
9591:
9574:
9573:
9562:
9558:
9557:
9553:
9552:
9523:
9521:
9520:
9515:
9500:
9498:
9497:
9492:
9484:
9483:
9467:
9465:
9464:
9459:
9435:
9433:
9432:
9427:
9422:
9421:
9404:
9403:
9402:
9400:
9398:
9395:
9391:
9386:
9381:
9380:
9379:
9368:
9364:
9363:
9359:
9358:
9333:
9329:
9328:
9305:
9303:
9302:
9297:
9295:
9294:
9278:
9276:
9275:
9270:
9267:
9262:
9251:
9247:
9246:
9242:
9241:
9212:
9210:
9209:
9204:
9202:
9201:
9190:
9186:
9185:
9181:
9180:
9151:
9149:
9148:
9143:
9126:
9125:
9103:
9101:
9100:
9095:
9083:
9081:
9080:
9075:
9073:
9072:
9056:
9054:
9053:
9048:
9020:
9018:
9017:
9012:
8998:
8997:
8981:
8979:
8978:
8973:
8971:
8970:
8954:
8952:
8951:
8946:
8944:
8943:
8927:
8925:
8924:
8919:
8901:
8899:
8898:
8893:
8891:
8890:
8874:
8872:
8871:
8866:
8855:
8843:
8841:
8840:
8835:
8816:
8814:
8813:
8808:
8778:
8776:
8775:
8770:
8756:
8755:
8739:
8737:
8736:
8731:
8719:
8717:
8716:
8711:
8709:
8708:
8697:
8693:
8692:
8675:
8674:
8648:
8646:
8645:
8640:
8635:
8631:
8627:
8626:
8625:
8614:
8613:
8599:
8598:
8593:
8592:
8569:
8563:
8562:
8558:
8557:
8546:
8537:
8536:
8531:
8517:
8515:
8514:
8509:
8502:
8478:
8476:
8475:
8470:
8468:
8464:
8460:
8459:
8458:
8441:
8437:
8436:
8418:
8417:
8406:
8402:
8401:
8374:
8372:
8371:
8360:
8356:
8355:
8340:
8339:
8335:
8329:
8305:
8301:
8300:
8294:
8266:
8264:
8258:
8242:
8222:
8220:
8219:
8214:
8176:
8174:
8173:
8168:
8154:
8152:
8151:
8146:
8126:
8124:
8123:
8118:
8097:
8095:
8094:
8089:
8087:
8086:
8085:
8084:
8048:
8046:
8045:
8040:
8038:
8037:
8032:
8013:
8011:
8010:
8005:
8003:
8002:
8001:
8000:
7985:
7963:
7961:
7960:
7955:
7950:
7949:
7944:
7935:
7929:
7911:
7910:
7909:
7908:
7893:
7871:
7869:
7868:
7863:
7861:
7860:
7849:
7845:
7844:
7820:
7818:
7817:
7812:
7810:
7809:
7804:
7791:
7789:
7788:
7783:
7771:
7769:
7768:
7763:
7751:
7749:
7748:
7743:
7731:
7729:
7728:
7723:
7708:
7706:
7705:
7700:
7688:
7686:
7685:
7680:
7661:
7660:
7659:
7657:
7655:
7652:
7648:
7643:
7638:
7628:
7627:
7622:
7609:
7607:
7606:
7601:
7599:
7575:
7573:
7572:
7567:
7565:
7564:
7559:
7546:
7544:
7543:
7538:
7536:
7535:
7523:
7522:
7511:
7507:
7506:
7489:
7488:
7478:
7475:
7472:
7464:
7458:
7440:
7439:
7434:
7422:is the function
7421:
7419:
7418:
7413:
7395:
7394:
7366:
7364:
7363:
7358:
7356:
7355:
7350:
7337:
7335:
7334:
7329:
7327:
7321:
7299:
7297:
7296:
7291:
7289:
7283:
7265:
7264:
7259:
7243:
7241:
7240:
7235:
7230:
7229:
7218:
7214:
7213:
7192:
7190:
7189:
7184:
7182:
7181:
7170:
7169:
7168:
7166:
7164:
7161:
7157:
7152:
7147:
7127:
7125:
7124:
7119:
7117:
7093:
7091:
7090:
7085:
7083:
7077:
7055:
7053:
7052:
7047:
7045:
7044:
7033:
7029:
7028:
7011:
7010:
6992:
6990:
6989:
6984:
6969:
6967:
6966:
6961:
6956:
6955:
6923:
6922:
6921:
6919:
6917:
6914:
6910:
6905:
6900:
6899:
6898:
6878:
6876:
6875:
6870:
6858:
6856:
6855:
6850:
6848:
6847:
6842:
6829:
6827:
6826:
6821:
6819:
6793:
6791:
6790:
6785:
6768:
6766:
6765:
6760:
6755:
6737:
6735:
6734:
6729:
6727:
6726:
6721:
6708:
6706:
6705:
6700:
6698:
6697:
6686:
6682:
6681:
6664:
6663:
6644:
6642:
6641:
6636:
6624:
6622:
6621:
6616:
6614:
6608:
6579:
6577:
6576:
6571:
6550:
6548:
6547:
6542:
6514:
6513:
6512:
6510:
6508:
6505:
6501:
6496:
6491:
6477:
6475:
6474:
6469:
6457:
6455:
6454:
6449:
6437:
6435:
6434:
6429:
6418:
6410:
6402:
6386:
6385:
6384:
6382:
6380:
6377:
6373:
6368:
6363:
6362:
6361:
6346:
6344:
6343:
6338:
6313:
6312:
6308:
6306:
6305:
6300:
6295:
6294:
6293:
6292:
6277:
6255:
6253:
6252:
6247:
6245:
6244:
6220:product topology
6213:
6211:
6210:
6205:
6200:
6194:
6169:
6167:
6166:
6161:
6159:
6141:
6139:
6138:
6133:
6131:
6130:
6125:
6080:elementary proof
6074:Elementary proof
6042:F. Riesz theorem
6039:
6037:
6036:
6031:
6029:
6028:
6012:
6010:
6009:
6004:
6002:
6001:
5981:
5979:
5978:
5973:
5961:
5959:
5958:
5953:
5951:
5950:
5922:
5920:
5919:
5914:
5912:
5911:
5895:
5893:
5892:
5887:
5875:
5861:
5859:
5858:
5853:
5841:
5839:
5838:
5833:
5831:
5830:
5814:
5812:
5811:
5806:
5794:
5792:
5791:
5786:
5774:
5772:
5771:
5766:
5750:
5748:
5747:
5742:
5725:
5723:
5722:
5717:
5704:
5702:
5701:
5696:
5694:
5693:
5678:it follows that
5677:
5675:
5674:
5669:
5664:
5660:
5659:
5655:
5648:
5647:
5627:
5626:
5605:
5603:
5602:
5597:
5595:
5594:
5574:
5572:
5571:
5566:
5564:
5560:
5559:
5555:
5548:
5547:
5527:
5526:
5505:
5503:
5502:
5497:
5492:
5491:
5475:
5473:
5472:
5467:
5465:
5461:
5454:
5453:
5429:
5427:
5426:
5421:
5419:
5418:
5400:
5398:
5397:
5392:
5387:
5386:
5370:
5368:
5367:
5362:
5360:
5356:
5349:
5348:
5324:
5322:
5321:
5316:
5314:
5313:
5297:
5295:
5294:
5289:
5284:
5280:
5279:
5275:
5268:
5267:
5247:
5246:
5225:
5223:
5222:
5217:
5215:
5214:
5198:
5196:
5195:
5190:
5188:
5187:
5171:
5169:
5168:
5163:
5161:
5157:
5150:
5149:
5126:Recall that the
5125:
5123:
5122:
5117:
5112:
5111:
5095:
5093:
5092:
5087:
5085:
5081:
5074:
5073:
5049:
5047:
5046:
5041:
5039:
5038:
5019:
5017:
5016:
5011:
5009:
5005:
4998:
4997:
4973:
4971:
4970:
4965:
4963:
4962:
4942:
4940:
4939:
4934:
4932:
4928:
4921:
4920:
4896:
4894:
4893:
4888:
4886:
4882:
4875:
4874:
4850:
4848:
4847:
4842:
4840:
4839:
4823:
4821:
4820:
4815:
4810:
4809:
4790:
4788:
4787:
4782:
4770:
4768:
4767:
4762:
4757:
4756:
4736:
4734:
4733:
4728:
4726:
4722:
4715:
4714:
4690:
4688:
4687:
4682:
4680:
4679:
4663:
4661:
4660:
4655:
4650:
4649:
4630:
4628:
4627:
4622:
4606:absorbing subset
4603:
4601:
4600:
4595:
4583:
4581:
4580:
4575:
4573:
4572:
4543:
4541:
4540:
4535:
4530:
4529:
4509:
4507:
4506:
4501:
4499:
4495:
4488:
4487:
4463:
4461:
4460:
4455:
4453:
4452:
4433:
4431:
4430:
4425:
4423:
4422:
4407:the claim about
4406:
4404:
4403:
4398:
4393:
4392:
4376:
4374:
4373:
4368:
4366:
4365:
4353:
4352:
4340:
4339:
4327:
4326:
4314:
4313:
4297:
4295:
4294:
4289:
4284:
4283:
4261:
4259:
4258:
4253:
4251:
4250:
4224:
4222:
4221:
4216:
4204:
4202:
4201:
4196:
4184:
4182:
4181:
4176:
4164:
4162:
4161:
4156:
4144:
4127:
4121:
4098:
4096:
4095:
4090:
4088:
4087:
4065:
4063:
4062:
4057:
4051:
4050:
4038:
4037:
4020:
4018:
4017:
4012:
4010:
4009:
3997:
3996:
3980:
3978:
3977:
3972:
3967:
3963:
3962:
3958:
3951:
3950:
3930:
3929:
3908:
3906:
3905:
3900:
3898:
3894:
3893:
3889:
3882:
3881:
3861:
3860:
3839:
3837:
3836:
3831:
3829:
3828:
3812:
3810:
3809:
3804:
3802:
3801:
3789:
3788:
3770:
3768:
3767:
3762:
3754:
3737:
3725:
3723:
3722:
3717:
3696:
3694:
3693:
3688:
3683:
3679:
3672:
3664:
3656:
3633:
3631:
3630:
3625:
3623:
3611:
3609:
3608:
3603:
3592:
3591:
3576:and every value
3575:
3573:
3572:
3567:
3565:
3553:
3551:
3550:
3545:
3519:
3518:
3502:
3500:
3499:
3494:
3473:
3471:
3470:
3465:
3457:
3440:
3428:
3426:
3425:
3420:
3415:
3414:
3392:
3390:
3389:
3384:
3379:
3375:
3374:
3370:
3363:
3362:
3342:
3341:
3320:
3318:
3317:
3312:
3300:
3298:
3297:
3292:
3290:
3289:
3273:
3271:
3270:
3265:
3263:
3262:
3251:
3247:
3246:
3229:
3228:
3212:
3210:
3209:
3204:
3202:
3201:
3179:
3177:
3176:
3171:
3166:
3165:
3149:
3147:
3146:
3141:
3139:
3135:
3128:
3127:
3103:
3101:
3100:
3095:
3093:
3092:
3073:
3071:
3070:
3065:
3060:
3059:
3047:
3046:
3021:
3019:
3018:
3013:
3008:
3007:
2995:
2994:
2982:
2981:
2965:
2963:
2962:
2957:
2952:
2948:
2947:
2946:
2919:
2917:
2916:
2911:
2900:be the polar of
2899:
2897:
2896:
2891:
2889:
2885:
2878:
2861:
2855:
2838:
2832:
2831:
2830:
2807:
2806:
2788:
2786:
2785:
2780:
2778:
2774:
2773:
2772:
2746:with respect to
2744:
2742:
2741:
2736:
2722:
2720:
2719:
2714:
2712:
2708:
2701:
2684:
2678:
2677:
2676:
2654:
2648:
2630:
2629:
2608:
2606:
2605:
2600:
2598:
2594:
2593:
2592:
2565:
2563:
2562:
2557:
2546:be the polar of
2545:
2543:
2542:
2537:
2535:
2531:
2524:
2507:
2501:
2484:
2478:
2477:
2476:
2453:
2452:
2433:
2431:
2430:
2425:
2413:
2411:
2410:
2405:
2391:
2389:
2388:
2383:
2378:
2374:
2373:
2372:
2342:
2340:
2339:
2334:
2329:
2325:
2324:
2323:
2297:although unlike
2292:
2290:
2289:
2284:
2282:
2278:
2277:
2276:
2246:
2244:
2243:
2238:
2226:
2224:
2223:
2218:
2206:
2204:
2203:
2198:
2196:
2195:
2175:
2173:
2172:
2167:
2162:
2161:
2139:
2137:
2136:
2131:
2126:
2122:
2115:
2114:
2090:
2088:
2087:
2082:
2080:
2076:
2075:
2071:
2064:
2063:
2043:
2042:
2021:
2019:
2018:
2013:
2011:
2010:
1986:
1984:
1983:
1978:
1973:
1969:
1968:
1964:
1957:
1956:
1936:
1935:
1914:
1912:
1911:
1906:
1904:
1900:
1899:
1895:
1888:
1887:
1867:
1866:
1841:
1839:
1838:
1833:
1831:
1827:
1826:
1822:
1815:
1814:
1794:
1793:
1772:
1770:
1769:
1764:
1759:
1755:
1754:
1750:
1743:
1742:
1722:
1721:
1693:
1691:
1690:
1685:
1680:
1676:
1669:
1668:
1641:
1639:
1638:
1633:
1631:
1630:
1599:
1597:
1596:
1591:
1586:
1571:
1569:
1568:
1563:
1561:
1546:
1544:
1543:
1538:
1533:
1521:
1519:
1518:
1513:
1489:
1487:
1486:
1481:
1476:
1472:
1471:
1467:
1460:
1459:
1439:
1438:
1417:
1415:
1414:
1409:
1407:
1403:
1402:
1401:
1374:
1372:
1371:
1366:
1354:
1352:
1351:
1346:
1344:
1343:
1327:
1325:
1324:
1319:
1314:
1313:
1297:
1295:
1294:
1289:
1287:
1283:
1276:
1275:
1248:
1246:
1245:
1240:
1228:
1226:
1225:
1220:
1205:
1203:
1202:
1197:
1195:
1191:
1184:
1167:
1161:
1144:
1138:
1137:
1136:
1113:
1112:
1093:
1091:
1090:
1085:
1080:
1079:
1048:
1046:
1045:
1040:
1024:
1015:
1013:
1012:
1007:
983:
981:
980:
975:
973:
972:
961:
957:
947:
946:
924:
922:
921:
916:
904:
902:
901:
896:
884:
882:
881:
876:
874:
873:
857:
855:
854:
849:
844:
843:
832:
828:
827:
810:
809:
789:
787:
786:
781:
765:
763:
762:
757:
752:
748:
741:
740:
716:
714:
713:
708:
706:
705:
689:
687:
686:
681:
679:
675:
668:
667:
643:
641:
640:
635:
633:
632:
612:
610:
609:
604:
602:
601:
589:
588:
572:
570:
569:
564:
559:
558:
534:
532:
531:
526:
503:
501:
500:
495:
493:
489:
488:
484:
466:
465:
438:
436:
435:
430:
414:
413:
412:
410:
408:
405:
401:
396:
391:
390:
386:
364:
362:
361:
356:
354:
346:
345:
327:
323:
295:
293:
292:
287:
272:will denote the
271:
269:
268:
263:
261:
260:
244:
242:
241:
236:
234:
222:
220:
219:
214:
148:Leonidas Alaoglu
130:
128:
127:
122:
64:product topology
32268:
32267:
32263:
32262:
32261:
32259:
32258:
32257:
32228:
32227:
32226:
32221:
32203:
31965:B-complete/Ptak
31948:
31892:
31856:
31848:
31827:Bounding points
31790:
31732:Densely defined
31678:
31667:Bounded inverse
31613:
31547:
31541:
31511:
31506:
31488:
31452:Advanced topics
31447:
31371:
31350:
31309:
31275:Hilbert–Schmidt
31248:
31239:Gelfand–Naimark
31186:
31136:
31071:
31057:
31027:
31022:
31006:
30985:
30969:
30948:
30869:
30818:
30808:
30770:
30768:
30758:
30747:
30723:
30713:Springer-Verlag
30701:Conway, John B.
30696:
30694:Further reading
30691:
30677:
30647:
30633:Schechter, Eric
30617:
30582:
30546:
30527:
30492:
30472:
30467:
30466:
30456:
30454:
30435:
30429:
30422:
30418:, Theorem 23.5.
30414:
30410:
30402:
30398:
30390:
30375:
30367:
30358:
30354:, Theorem 3.15.
30350:
30346:
30341:
30336:
30335:
30317:
30314:
30313:
30287:
30283:
30275:
30258:
30256:
30253:
30252:
30230:
30226:
30225:
30221:
30205:
30201:
30200:
30196:
30184:
30179:
30175:
30169:
30167:
30166:
30145:
30143:
30142:
30122:
30119:
30118:
30096:
30092:
30091:
30087:
30075:
30071:
30060:
30057:
30056:
30034:
30017:
30015:
30012:
30011:
29989:
29986:
29985:
29960:
29943:
29931:
29925:
29922:
29921:
29920:if and only if
29904:
29900:
29887:
29885:
29880:
29878:
29877:
29863:
29860:
29859:
29858:By definition,
29839:
29835:
29821:
29817:
29813:
29809:
29801:
29798:
29797:
29780:
29776:
29762:
29758:
29754:
29750:
29735:
29731:
29717:
29713:
29709:
29705:
29691:
29688:
29687:
29662:
29659:
29658:
29635:
29631:
29629:
29626:
29625:
29600:
29597:
29596:
29573:
29569:
29567:
29564:
29563:
29562:that satisfies
29540:
29530:
29526:
29522:
29521:
29519:
29516:
29515:
29514:has an element
29498:
29494:
29492:
29489:
29488:
29471:
29467:
29465:
29462:
29461:
29444:
29440:
29438:
29435:
29434:
29413:
29409:
29407:
29404:
29403:
29379:
29376:
29375:
29358:
29354:
29352:
29349:
29348:
29327:
29323:
29318:
29315:
29314:
29313:thereby making
29294:
29290:
29281:
29277:
29272:
29269:
29268:
29251:
29247:
29238:
29234:
29232:
29229:
29228:
29211:
29207:
29196:
29193:
29192:
29170:
29166:
29165:
29161:
29149:
29134:
29130:
29129:
29125:
29117:
29113:
29104:
29100:
29099:
29083:
29068:
29064:
29063:
29059:
29047:
29035:
29031:
29022:
29018:
29017:
29004:
29000:
28995:
28992:
28991:
28969:
28965:
28956:
28952:
28943:
28939:
28938:
28934:
28931:
28918:
28914:
28912:
28909:
28908:
28883:
28880:
28879:
28863:
28861:
28858:
28857:
28841:
28838:
28837:
28818:
28813:
28809:
28783:
28779:
28778:
28774:
28769:
28766:
28765:
28728:
28725:
28724:
28722:
28718:
28708:
28707:
28680:
28677:
28676:
28653:
28649:
28647:
28644:
28643:
28620:
28616:
28614:
28611:
28610:
28587:
28583:
28581:
28578:
28577:
28552:
28549:
28548:
28526:
28523:
28522:
28497:
28494:
28493:
28449:
28445:
28443:
28440:
28439:
28423:
28420:
28419:
28403:
28400:
28399:
28374:
28371:
28370:
28353:
28349:
28340:
28336:
28334:
28331:
28330:
28329:if and only if
28313:
28309:
28300:
28296:
28294:
28291:
28290:
28269:
28265:
28263:
28260:
28259:
28232:
28222:
28218:
28214:
28213:
28200:
28198:
28193:
28191:
28190:
28181:
28177:
28175:
28172:
28171:
28169:
28165:
28142:
28138:
28136:
28133:
28132:
28114:
28102:
28096:
28093:
28092:
28075:
28071:
28069:
28066:
28065:
28041:
28023:
28010:
28009:
28003:
27999:
27987:
27977:
27976:
27975:
27971:
27962:
27958:
27945:
27943:
27938:
27936:
27935:
27926:
27922:
27920:
27917:
27916:
27899:
27895:
27893:
27890:
27889:
27870:
27852:
27839:
27838:
27832:
27828:
27816:
27806:
27805:
27788:
27784:
27783:
27779:
27777:
27774:
27773:
27756:
27752:
27740:
27725:
27721:
27720:
27716:
27714:
27711:
27710:
27709:The conclusion
27708:
27704:
27698:axiom of choice
27674:
27670:
27669:
27665:
27653:
27647:
27644:
27643:
27640:Hausdorff space
27620:
27616:
27615:
27611:
27609:
27606:
27605:
27603:
27599:
27580:
27577:
27576:
27559:
27555:
27553:
27550:
27549:
27521:
27518:
27517:
27501:
27498:
27497:
27474:
27470:
27468:
27465:
27464:
27463:then the polar
27436:
27433:
27432:
27412:
27409:
27408:
27392:
27389:
27388:
27372:
27369:
27368:
27351:
27347:
27333:
27330:
27329:
27300:
27296:
27294:
27291:
27290:
27267:
27263:
27261:
27258:
27257:
27237:
27233:
27231:
27228:
27227:
27204:
27200:
27185:
27181:
27179:
27176:
27175:
27158:
27154:
27152:
27149:
27148:
27125:
27121:
27119:
27116:
27115:
27098:
27097:
27086:
27069:
27057:
27038:
27034:
27010:
27009:
26996:
26994:
26989:
26987:
26986:
26971:
26967:
26965:
26962:
26961:
26933:
26930:
26929:
26913:
26910:
26909:
26892:
26888:
26874:
26871:
26870:
26842:
26839:
26838:
26807:
26804:
26803:
26787:
26784:
26783:
26767:
26764:
26763:
26747:
26744:
26743:
26727:
26724:
26723:
26706:
26702:
26700:
26697:
26696:
26680:
26677:
26676:
26656:
26653:
26652:
26635:
26634:
26623:
26606:
26594:
26575:
26571:
26559:
26558:
26549:
26545:
26543:
26540:
26539:
26522:
26518:
26509:
26505:
26503:
26500:
26499:
26483:
26480:
26479:
26463:
26460:
26459:
26457:
26453:
26447:totally bounded
26425:
26421:
26419:
26416:
26415:
26379:
26375:
26374:
26370:
26358:
26354:
26353:
26349:
26347:
26344:
26343:
26342:inherited from
26322:
26318:
26316:
26313:
26312:
26311:and the subset
26309:weak-* topology
26291:
26287:
26285:
26282:
26281:
26264:
26260:
26251:
26247:
26245:
26242:
26241:
26239:
26235:
26230:
26209:
26200:
26162:
26127:
26104:
26103:
26096:
26084:
26081:
26080:
26056:
26055:
26032:
26029:
26028:
26011:
26007:
26005:
26002:
26001:
25978:
25975:
25974:
25957:
25956:
25954:
25951:
25950:
25944:
25943:
25936:
25935:
25914:
25911:
25910:
25890:
25886:
25884:
25881:
25880:
25861:
25858:
25857:
25841:
25838:
25837:
25833:
25828:
25827:
25812:weak-* topology
25753:axiom of choice
25737:
25731:
25696:
25693:
25692:
25660:
25657:
25656:
25623:
25620:
25619:
25601:
25569:
25565:
25563:
25560:
25559:
25537:
25534:
25533:
25513:
25509:
25491:
25487:
25476:
25438:
25434:
25433:
25429:
25424:
25421:
25420:
25398:
25395:
25394:
25377:
25366:
25354:
25350:
25349:
25345:
25341:
25338:
25335:
25334:
25315:
25312:
25311:
25288:
25284:
25275:
25271:
25269:
25266:
25265:
25238:
25225:
25223:
25220:
25219:
25203:
25200:
25199:
25171:
25168:
25167:
25141:
25137:
25135:
25132:
25131:
25112:
25109:
25108:
25088:
25085:
25084:
25068:
25065:
25064:
25044:
25041:
25040:
25011:
25007:
25000:
24996:
24991:
24988:
24987:
24951:
24947:
24945:
24942:
24941:
24924:
24920:
24918:
24915:
24914:
24905:with the usual
24889:
24885:
24883:
24880:
24879:
24859:
24856:
24855:
24852:
24847:
24842:
24826:
24823:
24822:
24803:
24799:
24782:
24779:
24778:
24760:
24755:
24749:
24746:
24745:
24717:
24710:
24706:
24689:
24686:
24685:
24669:
24655:
24652:
24651:
24650:the functional
24629:
24622:
24618:
24603:
24583:
24579:
24578:
24574:
24573:
24567:
24561:
24558:
24557:
24535:
24532:
24531:
24515:
24512:
24511:
24488:
24468:
24464:
24463:
24459:
24458:
24456:
24453:
24452:
24423:
24413:
24409:
24405:
24404:
24395:
24391:
24389:
24386:
24385:
24379:weak-* topology
24360:
24355:
24349:
24346:
24345:
24329:
24328:
24313:
24309:
24290:
24282:
24274:
24273:
24268:
24260:
24253:
24248:
24241:
24230:
24228:
24225:
24224:
24208:
24205:
24204:
24183:
24178:
24172:
24169:
24168:
24148:
24144:
24142:
24139:
24138:
24119:
24116:
24115:
24094:
24090:
24088:
24085:
24084:
24063:
24059:
24047:
24035:
24032:
24031:
24000:
23992:
23984:
23966:
23962:
23960:
23957:
23956:
23931:
23928:
23927:
23893:
23884:
23880:
23872:
23869:
23868:
23851:
23847:
23845:
23842:
23841:
23818:
23815:
23814:
23795:
23792:
23791:
23775:
23772:
23771:
23752:
23749:
23748:
23729:
23726:
23725:
23708:
23704:
23689:
23685:
23676:
23672:
23670:
23667:
23666:
23647:
23644:
23643:
23627:
23618:
23614:
23606:
23603:
23602:
23567:
23564:
23563:
23544:
23541:
23540:
23523:
23519:
23517:
23514:
23513:
23485:
23482:
23481:
23453:
23449:
23433:
23429:
23422:
23408:
23404:
23388:
23384:
23382:
23372:
23368:
23361:
23350:
23323:
23320:
23319:
23292:
23289:
23288:
23271:
23260:
23250:
23246:
23242:
23232:
23228:
23226:
23223:
23222:
23206:
23203:
23202:
23182:
23178:
23176:
23173:
23172:
23156:
23153:
23152:
23136:
23133:
23132:
23114:
23091:
23088:
23087:
23067:
23064:
23063:
23046:
23042:
23040:
23037:
23036:
23017:
23000:
22996:
22995:
22991:
22990:
22981:
22977:
22975:
22972:
22971:
22954:
22950:
22929:
22925:
22923:
22920:
22919:
22903:
22900:
22899:
22879:
22876:
22875:
22824:
22814:
22810:
22806:
22805:
22792:
22790:
22785:
22783:
22782:
22773:
22769:
22767:
22764:
22763:
22735:
22732:
22731:
22714:
22710:
22708:
22705:
22704:
22681:
22677:
22676:
22672:
22660:
22645:
22641:
22640:
22636:
22631:
22628:
22627:
22605:
22601:
22599:
22596:
22595:
22578:
22574:
22569:
22566:
22565:
22548:
22544:
22535:
22531:
22517:
22513:
22512:
22508:
22503:
22500:
22499:
22472:
22468:
22467:
22463:
22437:
22433:
22432:
22428:
22424:
22420:
22400:
22396:
22387:
22383:
22366:
22362:
22361:
22357:
22353:
22349:
22336:
22334:
22329:
22327:
22326:
22314:
22310:
22308:
22305:
22304:
22286:
22283:
22282:
22265:
22261:
22259:
22256:
22255:
22231:
22227:
22225:
22222:
22221:
22196:
22192:
22183:
22179:
22170:
22166:
22165:
22161:
22149:
22145:
22143:
22140:
22139:
22115:
22112:
22111:
22094:
22090:
22081:
22077:
22075:
22072:
22071:
22047:
22043:
22042:
22038:
22014:
22009:
22008:
21999:
21995:
21994:
21990:
21977:
21975:
21970:
21968:
21967:
21958:
21954:
21952:
21949:
21948:
21928:
21923:
21922:
21907:
21897:
21893:
21889:
21888:
21879:
21875:
21873:
21870:
21869:
21850:
21846:
21845:
21841:
21829:
21812:
21810:
21805:
21803:
21802:
21791:
21787:
21786:
21782:
21777:
21774:
21773:
21756:
21752:
21744:
21741:
21740:
21723:
21719:
21706:
21704:
21699:
21697:
21696:
21688:
21685:
21684:
21667:
21663:
21661:
21658:
21657:
21643:
21642:
21634:
21630:
21629:
21625:
21613:
21601:
21595:
21591:
21582:
21581:
21573:
21569:
21568:
21564:
21552:
21540:
21534:
21530:
21521:
21518:
21512:
21508:
21501:
21495:
21491:
21487:
21485:
21482:
21481:
21458:
21448:
21444:
21440:
21439:
21430:
21426:
21424:
21421:
21420:
21404:
21401:
21400:
21378:
21374:
21372:
21369:
21368:
21342:
21338:
21314:
21310:
21303:
21299:
21286:
21284:
21279:
21277:
21276:
21267:
21263:
21261:
21258:
21257:
21241:
21238:
21237:
21173:
21171:
21166:
21164:
21163:
21134:
21131:
21130:
21108:
21105:
21104:
21102:
21100:
21080:
21077:
21076:
21043:
21039:
21038:
21034:
21022:
21018:
21017:
21013:
21011:
21008:
21007:
20988:
20976:
20963:
20958:
20957:
20955:
20952:
20951:
20932:
20930:
20927:
20926:
20919:
20911:
20864:
20860:
20859:
20855:
20843:
20839:
20838:
20834:
20832:
20829:
20828:
20797:
20793:
20792:
20788:
20783:
20780:
20779:
20763:
20760:
20759:
20742:
20738:
20736:
20733:
20732:
20725:
20707:
20703:
20701:
20698:
20697:
20675:
20671:
20669:
20666:
20665:
20660:is a Hausdorff
20645:
20633:
20627:
20624:
20623:
20620:The lemma above
20617:
20597:
20594:
20593:
20529:
20526:
20525:
20469:
20465:
20463:
20460:
20459:
20394:
20390:
20388:
20385:
20384:
20352:
20348:
20333:
20307:
20303:
20302:
20298:
20297:
20282:
20262:
20258:
20240:
20236:
20235:
20231:
20230:
20215:
20190:
20186:
20168:
20164:
20163:
20159:
20155:
20151:
20150:
20135:
20120:
20116:
20112:
20108:
20104:
20103:
20094:
20090:
20071:
20069:
20064:
20062:
20061:
20048:
20044:
20040:
20035:
20032:
20031:
19952:
19949:
19948:
19932:
19930:
19927:
19926:
19866:
19862:
19858:
19853:
19850:
19849:
19833:
19830:
19829:
19793:
19791:
19786:
19784:
19783:
19760:
19757:
19756:
19740:
19732:
19724:
19716:
19713:
19712:
19693:
19685:
19683:
19680:
19679:
19626:
19622:
19620:
19617:
19616:
19575:
19555:
19551:
19550:
19546:
19545:
19527:
19523:
19521:
19518:
19517:
19479:
19459:
19455:
19454:
19450:
19449:
19431:
19427:
19425:
19422:
19421:
19396:
19393:
19392:
19361:
19357:
19339:
19335:
19334:
19330:
19317:
19315:
19310:
19308:
19307:
19298:
19294:
19292:
19289:
19288:
19272:
19264:
19243:
19233:
19229:
19225:
19224:
19215:
19211:
19209:
19206:
19205:
19141:
19138:
19137:
19135:
19084:
19081:
19080:
19064:
19062:
19059:
19058:
19011:
19007:
19005:
19002:
19001:
18951:
18947:
18945:
18942:
18941:
18927:
18926:
18921:
18918:
18900:
18896:
18891:
18885:
18884:
18878:
18874:
18869:
18866:
18854:
18831:
18827:
18826:
18822:
18821:
18816:
18810:
18809:
18794:
18790:
18774:
18772:
18767:
18765:
18764:
18741:
18737:
18736:
18732:
18726:
18714:
18694:
18690:
18686:
18682:
18681:
18676:
18670:
18669:
18664:
18643:
18623:
18619:
18618:
18614:
18613:
18595:
18591:
18586:
18583:
18562:
18537:
18533:
18532:
18528:
18524:
18520:
18519:
18514:
18508:
18507:
18502:
18499:
18484:
18480:
18469:
18459:
18457:
18452:
18450:
18449:
18429:
18425:
18424:
18420:
18413:
18411:
18408:
18407:
18355:
18352:
18351:
18296:
18292:
18291:
18287:
18282:
18279:
18278:
18259:
18257:
18254:
18253:
18212:
18208:
18206:
18203:
18202:
18186:
18183:
18182:
18149:
18147:
18142:
18140:
18139:
18122:
18119:
18118:
18102:
18099:
18098:
18082:
18074:
18066:
18063:
18062:
18013:
18010:
18009:
18007:
17956:
17952:
17948: and
17946:
17900:
17896:
17862:
17858:
17824:
17820:
17818:
17815:
17814:
17796:
17794:
17791:
17790:
17771:
17768:
17767:
17741: and
17739:
17716:
17713:
17712:
17687:
17684:
17683:
17667:
17665:
17662:
17661:
17620:
17616:
17614:
17611:
17610:
17590:
17585:
17584:
17582:
17579:
17578:
17555:
17551:
17549:
17546:
17545:
17519:
17499:
17495:
17494:
17490:
17489:
17476:
17474:
17469:
17467:
17466:
17448:
17444:
17442:
17439:
17438:
17422:
17419:
17418:
17401:
17397:
17395:
17392:
17391:
17374:
17350:
17346:
17344:
17341:
17340:
17315:
17312:
17311:
17278:
17275:
17274:
17258:
17255:
17254:
17238:
17235:
17234:
17214:
17210:
17202:
17199:
17198:
17178:
17173:
17172:
17170:
17167:
17166:
17150:
17147:
17146:
17129:
17125:
17123:
17120:
17119:
17096:
17086:
17082:
17078:
17077:
17068:
17064:
17062:
17059:
17058:
17041:
17036:
17035:
17027:
17024:
17023:
17016:
17008:
16991:
16988:
16987:
16971:
16959:
16946:
16941:
16940:
16938:
16935:
16934:
16918:
16916:
16913:
16912:
16896:
16894:
16891:
16890:
16874:
16872:
16869:
16868:
16852:
16850:
16847:
16846:
16830:
16827:
16826:
16809:
16805:
16803:
16800:
16799:
16792:
16774:
16769:
16768:
16766:
16763:
16762:
16745:
16741:
16739:
16736:
16735:
16714:
16711:
16710:
16676:
16673:
16672:
16653:
16650:
16649:
16633:
16630:
16629:
16597:
16593:
16575:
16571:
16569:
16566:
16565:
16549:
16546:
16545:
16507:
16487:
16483:
16482:
16478:
16477:
16475:
16472:
16471:
16446:
16443:
16442:
16408:
16405:
16404:
16384:
16380:
16372:
16369:
16368:
16349:
16346:
16345:
16328:
16324:
16322:
16319:
16318:
16295:
16285:
16281:
16277:
16276:
16274:
16271:
16270:
16253:
16249:
16241:
16238:
16237:
16212:
16208:
16206:
16203:
16202:
16159:
16155:
16148:
16144:
16131:
16129:
16124:
16122:
16121:
16112:
16108:
16106:
16103:
16102:
16083:
16080:
16079:
16053:
16050:
16049:
16027:
16024:
16023:
15997:
15993:
15985:
15982:
15981:
15965:
15957:
15954:
15953:
15931:
15927:
15903:
15898:
15897:
15890:
15886:
15875:
15871:
15870:
15866:
15864:
15861:
15860:
15857:
15841:
15838:
15837:
15817:
15812:
15811:
15809:
15806:
15805:
15788:
15784:
15775:
15771:
15760:
15756:
15755:
15751:
15749:
15746:
15745:
15726:
15724:
15721:
15720:
15703:
15698:
15697:
15689:
15677:
15671:
15668:
15667:
15649:
15648:
15642:
15641:
15627:
15625:
15620:
15617:
15616:
15602:
15600:
15594:
15590:
15583:
15582:
15569:
15567:
15562:
15560:
15559:
15550:
15546:
15540:
15533:
15529:
15517:
15504:
15503:
15497:
15496:
15482:
15476:
15472:
15460:
15456:
15437:
15425:
15406:
15396:
15392:
15388:
15387:
15381:
15380:
15371:
15370:
15364:
15363:
15349:
15343:
15339:
15302:
15297:
15296:
15256:
15255:
15246:
15245:
15239:
15238:
15227:
15210:
15198:
15172:
15167:
15166:
15124:
15123:
15112:
15110:
15105:
15103:
15102:
15097:
15089:
15085:
15084:
15080:
15076:
15074:
15071:
15070:
15067:the lemma below
15050:
15038:
15025:
15020:
15019:
15017:
15014:
15013:
14996:
14992:
14990:
14987:
14986:
14952:
14948:
14947:
14943:
14926:
14923:
14922:
14902:
14898:
14890:
14873:
14871:
14868:
14867:
14843:
14826:
14814:
14792:
14788:
14784:
14780:
14776:
14752:
14748:
14743:
14742:
14738:
14734:
14730:
14702:
14698:
14693:
14692:
14688:
14680:
14663:
14655:
14651:
14646:
14644:
14641:
14640:
14615:
14598:
14586:
14579:
14576:
14575:
14555:
14552:
14551:
14518:
14514:
14509:
14496:
14494:
14489:
14487:
14486:
14477:
14473:
14470:
14467:
14466:
14444:
14441:
14440:
14417:
14413:
14411:
14408:
14407:
14387:
14383:
14375:
14372:
14371:
14348:
14344:
14342:
14339:
14338:
14316:
14312:
14311:
14307:
14290:
14287:
14286:
14240:
14238:
14237:
14232:
14229:
14228:
14208:
14204:
14196:
14193:
14192:
14170:
14167:
14166:
14141:
14138:
14137:
14118:
14114:
14113:
14109:
14096:
14092:
14088:
14082:
14080:
14079:
14074:
14071:
14070:
14048:
14044:
14043:
14039:
14027:
14014:
14010:
14008:
14005:
14004:
13984:
13981:
13980:
13964:
13956:
13944:
13931:
13929:
13928:
13923:
13920:
13919:
13894:
13891:
13890:
13861:
13856:
13855:
13847:
13835:
13829:
13826:
13825:
13805:
13801:
13799:
13796:
13795:
13772:
13768:
13767:
13763:
13751:
13738:
13734:
13732:
13729:
13728:
13703:
13699:
13698:
13694:
13682:
13676:
13673:
13672:
13650:
13646:
13616:
13612:
13605:
13601:
13586:
13585:
13574:
13557:
13545:
13526:
13522:
13510:
13509:
13496:
13494:
13489:
13487:
13486:
13477:
13473:
13471:
13468:
13467:
13440:
13436:
13428:
13420:
13412:
13390:
13388:
13383:
13381:
13380:
13369:
13365:
13364:
13360:
13358:
13355:
13354:
13331:
13327:
13326:
13322:
13310:
13304:
13301:
13300:
13290:
13274:
13271:
13270:
13253:
13249:
13241:
13238:
13237:
13217:
13214:
13213:
13194:
13191:
13190:
13174:
13171:
13170:
13144:
13127:
13115:
13108:
13105:
13104:
13087:
13083:
13075:
13072:
13071:
13051:
13047:
13038:
13034:
13032:
13029:
13028:
13008:
13004:
12995:
12991:
12982:
12978:
12976:
12973:
12972:
12951:
12947:
12938:
12934:
12932:
12929:
12928:
12905:
12901:
12892:
12888:
12873:
12872:
12861:
12844:
12832:
12813:
12809:
12797:
12796:
12783:
12781:
12776:
12774:
12773:
12764:
12760:
12758:
12755:
12754:
12753:was defined as
12737:
12733:
12731:
12728:
12727:
12707:
12703:
12694:
12690:
12688:
12685:
12684:
12681:weak-* topology
12663:
12659:
12657:
12654:
12653:
12630:
12626:
12618:
12615:
12614:
12591:
12587:
12585:
12582:
12581:
12556:
12553:
12552:
12533:
12530:
12529:
12506:
12503:
12502:
12486:
12483:
12482:
12466:
12463:
12462:
12445:
12441:
12439:
12436:
12435:
12415:
12412:
12411:
12388:
12385:
12384:
12383:(TVS) and that
12364:
12361:
12360:
12357:
12339:
12335:
12333:
12330:
12329:
12326:weak-* topology
12298:
12294:
12293:
12289:
12277:
12271:
12268:
12267:
12240:
12236:
12234:
12231:
12230:
12207:
12203:
12195:
12192:
12191:
12168:
12164:
12162:
12159:
12158:
12134:
12131:
12130:
12108:
12107:
12096:
12079:
12067:
12048:
12044:
12032:
12031:
12018:
12016:
12011:
12009:
12008:
11999:
11995:
11993:
11990:
11989:
11969:
11967:
11964:
11963:
11947:
11944:
11943:
11914:
11906:
11898:
11876:
11874:
11869:
11867:
11866:
11857:
11853:
11851:
11848:
11847:
11828:
11825:
11824:
11808:
11800:
11795:
11790:
11782:
11780:
11777:
11776:
11760:
11758:
11755:
11754:
11753:over the field
11738:
11735:
11734:
11718:
11715:
11714:
11710:
11688:
11685:
11684:
11677:
11656:
11653:
11652:
11625:
11623:
11618:
11616:
11615:
11607:
11604:
11603:
11584:
11571:
11569:
11564:
11562:
11561:
11553:
11550:
11549:
11526:
11509:
11497:
11490:
11487:
11486:
11463:
11446:
11434:
11427:
11424:
11423:
11422:(and replacing
11398:
11390:
11382:
11371:
11368:
11367:
11350:
11346:
11344:
11341:
11340:
11283:
11281:
11276:
11274:
11273:
11256:
11253:
11252:
11230:
11227:
11226:
11209:
11205:
11184:
11172:
11155:
11143:
11137:
11134:
11133:
11115:
11112:
11111:
11086:
11069:
11057:
11051:
11048:
11047:
11025:
11013:
11007:
11004:
11003:
10986:
10982:
10970:
10964:
10961:
10960:
10935:
10932:
10931:
10915:
10913:
10910:
10909:
10892:
10888:
10886:
10883:
10882:
10863:
10851:
10845:
10842:
10841:
10824:
10820:
10808:
10802:
10799:
10798:
10777:
10773:
10761:
10755:
10752:
10751:
10735:
10733:
10730:
10729:
10709:
10700:
10696:
10694:
10691:
10690:
10666:
10663:
10662:
10637:
10634:
10633:
10617:
10614:
10613:
10597:
10594:
10593:
10570:
10560:
10556:
10552:
10551:
10549:
10546:
10545:
10523:
10521:
10518:
10517:
10479:
10459:
10455:
10454:
10450:
10449:
10447:
10444:
10443:
10418:
10415:
10414:
10384:
10364:
10360:
10359:
10355:
10354:
10339:
10324:
10320:
10316:
10310:
10308:
10307:
10303:
10299:
10298:
10285:
10283:
10278:
10276:
10275:
10256:
10246:
10242:
10238:
10237:
10233:
10227:
10225:
10224:
10219:
10216:
10215:
10173:
10171:
10170:
10164:
10161:
10160:
10138:
10136:
10133:
10132:
10106:
10104:
10103:
10081:
10071:
10067:
10063:
10062:
10058:
10052:
10050:
10049:
10044:
10041:
10040:
10015:
10012:
10011:
9975:
9965:
9961:
9957:
9956:
9954:
9951:
9950:
9931:
9919:
9906:
9901:
9900:
9898:
9895:
9894:
9877:
9873:
9871:
9868:
9867:
9851:
9848:
9847:
9821:
9811:
9807:
9803:
9802:
9800:
9797:
9796:
9777:
9774:
9773:
9749:
9737:
9724:
9719:
9718:
9716:
9713:
9712:
9670:
9666:
9658:
9655:
9654:
9618:
9614:
9610:
9605:
9602:
9601:
9563:
9548:
9544:
9540:
9536:
9532:
9531:
9529:
9526:
9525:
9506:
9503:
9502:
9479:
9475:
9473:
9470:
9469:
9441:
9438:
9437:
9417:
9413:
9394:
9392:
9387:
9385:
9384:
9369:
9354:
9350:
9346:
9342:
9338:
9337:
9324:
9320:
9316:
9311:
9308:
9307:
9290:
9286:
9284:
9281:
9280:
9263:
9252:
9237:
9233:
9229:
9225:
9221:
9218:
9215:
9214:
9191:
9176:
9172:
9168:
9164:
9160:
9159:
9157:
9154:
9153:
9121:
9117:
9109:
9106:
9105:
9089:
9086:
9085:
9068:
9064:
9062:
9059:
9058:
9030:
9027:
9026:
8993:
8989:
8987:
8984:
8983:
8966:
8962:
8960:
8957:
8956:
8939:
8935:
8933:
8930:
8929:
8907:
8904:
8903:
8886:
8882:
8880:
8877:
8876:
8851:
8849:
8846:
8845:
8826:
8823:
8822:
8787:
8784:
8783:
8751:
8747:
8745:
8742:
8741:
8725:
8722:
8721:
8698:
8688:
8684:
8680:
8679:
8670:
8666:
8664:
8661:
8660:
8633:
8632:
8615:
8609:
8608:
8607:
8594:
8588:
8587:
8586:
8582:
8578:
8576:
8568:
8560:
8559:
8547:
8542:
8541:
8532:
8527:
8526:
8524:
8516:
8510:
8505:
8504:
8501:
8490:
8488:
8485:
8484:
8466:
8465:
8442:
8432:
8428:
8424:
8423:
8407:
8397:
8393:
8389:
8388:
8387:
8383:
8381:
8373:
8361:
8351:
8347:
8343:
8342:
8337:
8336:
8331:
8313:
8296:
8284:
8279:
8275:
8273:
8265:
8260:
8248:
8241:
8230:
8228:
8225:
8224:
8182:
8179:
8178:
8162:
8159:
8158:
8134:
8131:
8130:
8103:
8100:
8099:
8080:
8076:
8075:
8071:
8054:
8051:
8050:
8033:
8028:
8027:
8019:
8016:
8015:
7996:
7992:
7991:
7987:
7975:
7969:
7966:
7965:
7945:
7940:
7939:
7931:
7919:
7904:
7900:
7899:
7895:
7883:
7877:
7874:
7873:
7850:
7840:
7836:
7832:
7831:
7829:
7826:
7825:
7805:
7803:
7802:
7797:
7794:
7793:
7777:
7774:
7773:
7757:
7754:
7753:
7737:
7734:
7733:
7714:
7711:
7710:
7694:
7691:
7690:
7651:
7649:
7644:
7642:
7641:
7623:
7621:
7620:
7615:
7612:
7611:
7595:
7581:
7578:
7577:
7560:
7558:
7557:
7552:
7549:
7548:
7531:
7527:
7512:
7502:
7498:
7494:
7493:
7484:
7480:
7474:
7468:
7460:
7448:
7435:
7433:
7432:
7427:
7424:
7423:
7401:
7398:
7397:
7392:
7391:
7351:
7346:
7345:
7343:
7340:
7339:
7323:
7311:
7305:
7302:
7301:
7285:
7273:
7260:
7255:
7254:
7252:
7249:
7248:
7219:
7209:
7205:
7201:
7200:
7198:
7195:
7194:
7177:
7173:
7160:
7158:
7153:
7151:
7150:
7133:
7130:
7129:
7113:
7099:
7096:
7095:
7079:
7067:
7061:
7058:
7057:
7034:
7024:
7020:
7016:
7015:
7006:
7002:
7000:
6997:
6996:
6978:
6975:
6974:
6945:
6941:
6913:
6911:
6906:
6904:
6903:
6894:
6890:
6888:
6885:
6884:
6881:tuple of values
6864:
6861:
6860:
6843:
6838:
6837:
6835:
6832:
6831:
6815:
6801:
6798:
6797:
6779:
6776:
6775:
6751:
6743:
6740:
6739:
6722:
6717:
6716:
6714:
6711:
6710:
6687:
6677:
6673:
6669:
6668:
6659:
6655:
6653:
6650:
6649:
6630:
6627:
6626:
6610:
6598:
6592:
6589:
6588:
6556:
6553:
6552:
6504:
6502:
6497:
6495:
6494:
6483:
6480:
6479:
6463:
6460:
6459:
6443:
6440:
6439:
6414:
6406:
6398:
6376:
6374:
6369:
6367:
6366:
6357:
6353:
6351:
6348:
6347:
6329:
6326:
6325:
6324:For every real
6318:
6288:
6284:
6283:
6279:
6267:
6261:
6258:
6257:
6240:
6236:
6234:
6231:
6230:
6196:
6184:
6178:
6175:
6174:
6155:
6147:
6144:
6143:
6126:
6121:
6120:
6118:
6115:
6114:
6076:
6065:locally compact
6057:strong topology
6053:locally compact
6024:
6020:
6018:
6015:
6014:
5997:
5993:
5991:
5988:
5987:
5967:
5964:
5963:
5946:
5942:
5940:
5937:
5936:
5933:
5929:weak-* topology
5907:
5903:
5901:
5898:
5897:
5881:
5878:
5877:
5873:
5864:usual dual norm
5847:
5844:
5843:
5826:
5822:
5820:
5817:
5816:
5800:
5797:
5796:
5780:
5777:
5776:
5760:
5757:
5756:
5736:
5733:
5732:
5729:
5711:
5708:
5707:
5689:
5685:
5683:
5680:
5679:
5643:
5639:
5638:
5634:
5622:
5618:
5617:
5613:
5611:
5608:
5607:
5590:
5586:
5584:
5581:
5580:
5543:
5539:
5538:
5534:
5522:
5518:
5517:
5513:
5511:
5508:
5507:
5487:
5483:
5481:
5478:
5477:
5449:
5445:
5444:
5440:
5435:
5432:
5431:
5414:
5410:
5408:
5405:
5404:
5382:
5378:
5376:
5373:
5372:
5344:
5340:
5339:
5335:
5330:
5327:
5326:
5309:
5305:
5303:
5300:
5299:
5263:
5259:
5258:
5254:
5242:
5238:
5237:
5233:
5231:
5228:
5227:
5210:
5206:
5204:
5201:
5200:
5183:
5179:
5177:
5174:
5173:
5145:
5141:
5140:
5136:
5131:
5128:
5127:
5107:
5103:
5101:
5098:
5097:
5069:
5065:
5064:
5060:
5055:
5052:
5051:
5034:
5030:
5028:
5025:
5024:
4993:
4989:
4988:
4984:
4979:
4976:
4975:
4958:
4954:
4952:
4949:
4948:
4945:totally bounded
4916:
4912:
4911:
4907:
4902:
4899:
4898:
4870:
4866:
4865:
4861:
4856:
4853:
4852:
4835:
4831:
4829:
4826:
4825:
4805:
4801:
4799:
4796:
4795:
4776:
4773:
4772:
4752:
4748:
4746:
4743:
4742:
4710:
4706:
4705:
4701:
4696:
4693:
4692:
4675:
4671:
4669:
4666:
4665:
4642:
4638:
4636:
4633:
4632:
4613:
4610:
4609:
4589:
4586:
4585:
4565:
4561:
4553:
4550:
4549:
4546:bipolar theorem
4525:
4521:
4519:
4516:
4515:
4512:totally bounded
4483:
4479:
4478:
4474:
4469:
4466:
4465:
4448:
4444:
4442:
4439:
4438:
4418:
4414:
4412:
4409:
4408:
4388:
4384:
4382:
4379:
4378:
4361:
4357:
4348:
4344:
4335:
4331:
4322:
4318:
4309:
4305:
4303:
4300:
4299:
4279:
4275:
4267:
4264:
4263:
4246:
4242:
4234:
4231:
4230:
4210:
4207:
4206:
4190:
4187:
4186:
4170:
4167:
4166:
4140:
4123:
4111:
4104:
4101:
4100:
4083:
4079:
4071:
4068:
4067:
4046:
4042:
4033:
4029:
4026:
4023:
4022:
4005:
4001:
3992:
3988:
3986:
3983:
3982:
3946:
3942:
3941:
3937:
3925:
3921:
3920:
3916:
3914:
3911:
3910:
3877:
3873:
3872:
3868:
3856:
3852:
3851:
3847:
3845:
3842:
3841:
3824:
3820:
3818:
3815:
3814:
3797:
3793:
3784:
3780:
3778:
3775:
3774:
3750:
3733:
3731:
3728:
3727:
3702:
3699:
3698:
3668:
3660:
3652:
3645:
3641:
3639:
3636:
3635:
3619:
3617:
3614:
3613:
3587:
3583:
3581:
3578:
3577:
3561:
3559:
3556:
3555:
3514:
3510:
3508:
3505:
3504:
3479:
3476:
3475:
3453:
3436:
3434:
3431:
3430:
3410:
3406:
3398:
3395:
3394:
3358:
3354:
3353:
3349:
3337:
3333:
3332:
3328:
3326:
3323:
3322:
3306:
3303:
3302:
3285:
3281:
3279:
3276:
3275:
3252:
3242:
3238:
3234:
3233:
3224:
3220:
3218:
3215:
3214:
3197:
3193:
3185:
3182:
3181:
3161:
3157:
3155:
3152:
3151:
3123:
3119:
3118:
3114:
3109:
3106:
3105:
3088:
3084:
3082:
3079:
3078:
3055:
3051:
3036:
3032:
3030:
3027:
3026:
3003:
2999:
2990:
2986:
2977:
2973:
2971:
2968:
2967:
2942:
2938:
2931:
2927:
2925:
2922:
2921:
2905:
2902:
2901:
2874:
2857:
2845:
2826:
2822:
2815:
2811:
2802:
2798:
2796:
2793:
2792:
2768:
2764:
2757:
2753:
2751:
2748:
2747:
2730:
2727:
2726:
2697:
2680:
2672:
2668:
2661:
2638:
2634:
2622:
2618:
2616:
2613:
2612:
2588:
2584:
2577:
2573:
2571:
2568:
2567:
2551:
2548:
2547:
2520:
2503:
2491:
2472:
2468:
2461:
2457:
2448:
2444:
2442:
2439:
2438:
2419:
2416:
2415:
2399:
2396:
2395:
2368:
2364:
2357:
2353:
2351:
2348:
2347:
2319:
2315:
2308:
2304:
2302:
2299:
2298:
2272:
2268:
2261:
2257:
2255:
2252:
2251:
2232:
2229:
2228:
2212:
2209:
2208:
2191:
2187:
2185:
2182:
2181:
2157:
2153:
2145:
2142:
2141:
2110:
2106:
2105:
2101:
2096:
2093:
2092:
2059:
2055:
2054:
2050:
2038:
2034:
2033:
2029:
2027:
2024:
2023:
2006:
2002:
2000:
1997:
1996:
1989:totally bounded
1952:
1948:
1947:
1943:
1931:
1927:
1926:
1922:
1920:
1917:
1916:
1883:
1879:
1878:
1874:
1862:
1858:
1857:
1853:
1851:
1848:
1847:
1810:
1806:
1805:
1801:
1789:
1785:
1784:
1780:
1778:
1775:
1774:
1738:
1734:
1733:
1729:
1717:
1713:
1712:
1708:
1706:
1703:
1702:
1664:
1660:
1659:
1655:
1650:
1647:
1646:
1644:weak-* topology
1626:
1622:
1620:
1617:
1616:
1582:
1580:
1577:
1576:
1574:complex numbers
1557:
1555:
1552:
1551:
1529:
1527:
1524:
1523:
1507:
1504:
1503:
1496:
1491:
1455:
1451:
1450:
1446:
1434:
1430:
1429:
1425:
1423:
1420:
1419:
1397:
1393:
1386:
1382:
1380:
1377:
1376:
1360:
1357:
1356:
1339:
1335:
1333:
1330:
1329:
1309:
1305:
1303:
1300:
1299:
1271:
1267:
1266:
1262:
1257:
1254:
1253:
1251:weak-* topology
1234:
1231:
1230:
1214:
1211:
1210:
1180:
1163:
1151:
1132:
1128:
1121:
1117:
1108:
1104:
1102:
1099:
1098:
1075:
1071:
1069:
1066:
1065:
1034:
1031:
1030:
1022:
1020:Alaoglu theorem
989:
986:
985:
962:
942:
938:
937:
933:
932:
930:
927:
926:
910:
907:
906:
890:
887:
886:
869:
865:
863:
860:
859:
833:
823:
819:
815:
814:
805:
801:
799:
796:
795:
775:
772:
771:
736:
732:
731:
727:
722:
719:
718:
701:
697:
695:
692:
691:
663:
659:
658:
654:
649:
646:
645:
628:
624:
622:
619:
618:
615:weak-* topology
597:
593:
584:
580:
578:
575:
574:
554:
550:
548:
545:
544:
539:(TVS) then its
520:
517:
516:
474:
470:
461:
457:
450:
446:
444:
441:
440:
404:
402:
397:
395:
394:
376:
372:
370:
367:
366:
350:
341:
337:
313:
309:
307:
304:
303:
281:
278:
277:
256:
252:
250:
247:
246:
230:
228:
225:
224:
208:
205:
204:
201:
187:
179:Alaoglu theorem
163:dual topologies
95:
92:
91:
88:weak-* topology
80:
31:(also known as
17:
12:
11:
5:
32266:
32256:
32255:
32250:
32245:
32240:
32223:
32222:
32220:
32219:
32208:
32205:
32204:
32202:
32201:
32196:
32191:
32186:
32184:Ultrabarrelled
32176:
32170:
32165:
32159:
32154:
32149:
32144:
32139:
32134:
32125:
32119:
32114:
32112:Quasi-complete
32109:
32107:Quasibarrelled
32104:
32099:
32094:
32089:
32084:
32079:
32074:
32069:
32064:
32059:
32054:
32049:
32048:
32047:
32037:
32032:
32027:
32022:
32017:
32012:
32007:
32002:
31997:
31987:
31982:
31972:
31967:
31962:
31956:
31954:
31950:
31949:
31947:
31946:
31936:
31931:
31926:
31921:
31916:
31906:
31900:
31898:
31897:Set operations
31894:
31893:
31891:
31890:
31885:
31880:
31875:
31870:
31865:
31860:
31852:
31844:
31839:
31834:
31829:
31824:
31819:
31814:
31809:
31804:
31798:
31796:
31792:
31791:
31789:
31788:
31783:
31778:
31773:
31768:
31767:
31766:
31761:
31756:
31746:
31741:
31740:
31739:
31734:
31729:
31724:
31719:
31714:
31709:
31699:
31698:
31697:
31686:
31684:
31680:
31679:
31677:
31676:
31671:
31670:
31669:
31659:
31653:
31644:
31639:
31634:
31632:Banach–Alaoglu
31629:
31627:Anderson–Kadec
31623:
31621:
31615:
31614:
31612:
31611:
31606:
31601:
31596:
31591:
31586:
31581:
31576:
31571:
31566:
31561:
31555:
31553:
31552:Basic concepts
31549:
31548:
31540:
31539:
31532:
31525:
31517:
31508:
31507:
31505:
31504:
31493:
31490:
31489:
31487:
31486:
31481:
31476:
31471:
31469:Choquet theory
31466:
31461:
31455:
31453:
31449:
31448:
31446:
31445:
31435:
31430:
31425:
31420:
31415:
31410:
31405:
31400:
31395:
31390:
31385:
31379:
31377:
31373:
31372:
31370:
31369:
31364:
31358:
31356:
31352:
31351:
31349:
31348:
31343:
31338:
31333:
31328:
31323:
31321:Banach algebra
31317:
31315:
31311:
31310:
31308:
31307:
31302:
31297:
31292:
31287:
31282:
31277:
31272:
31267:
31262:
31256:
31254:
31250:
31249:
31247:
31246:
31244:Banach–Alaoglu
31241:
31236:
31231:
31226:
31221:
31216:
31211:
31206:
31200:
31198:
31192:
31191:
31188:
31187:
31185:
31184:
31179:
31174:
31172:Locally convex
31169:
31155:
31150:
31144:
31142:
31138:
31137:
31135:
31134:
31129:
31124:
31119:
31114:
31109:
31104:
31099:
31094:
31089:
31083:
31077:
31073:
31072:
31056:
31055:
31048:
31041:
31033:
31024:
31023:
31021:
31020:
31014:
31012:
31011:Other concepts
31008:
31007:
31005:
31004:
30999:
30993:
30991:
30987:
30986:
30984:
30983:
30977:
30975:
30971:
30970:
30968:
30967:
30962:
30960:Banach–Alaoglu
30956:
30954:
30950:
30949:
30947:
30946:
30941:
30940:
30939:
30934:
30932:polar topology
30924:
30919:
30918:
30917:
30912:
30907:
30897:
30892:
30891:
30890:
30879:
30877:
30871:
30870:
30868:
30867:
30862:
30860:Polar topology
30857:
30852:
30847:
30842:
30837:
30832:
30826:
30824:
30823:Basic concepts
30820:
30819:
30813:and spaces of
30807:
30806:
30799:
30792:
30784:
30778:
30777:
30756:
30735:
30721:
30695:
30692:
30690:
30689:
30675:
30659:
30645:
30629:
30615:
30595:
30580:
30558:
30545:978-1584888666
30544:
30531:
30525:
30512:
30490:
30473:
30471:
30468:
30465:
30464:
30446:(2): 167–170.
30420:
30408:
30396:
30373:
30356:
30343:
30342:
30340:
30337:
30334:
30333:
30321:
30301:
30298:
30295:
30290:
30286:
30282:
30278:
30274:
30271:
30268:
30265:
30261:
30240:
30233:
30229:
30224:
30220:
30216:
30208:
30204:
30199:
30193:
30190:
30187:
30183:
30178:
30172:
30165:
30162:
30159:
30156:
30153:
30148:
30141:
30138:
30135:
30132:
30129:
30126:
30106:
30099:
30095:
30090:
30086:
30083:
30078:
30074:
30070:
30067:
30064:
30044:
30041:
30037:
30033:
30030:
30027:
30024:
30020:
29999:
29996:
29993:
29973:
29970:
29967:
29963:
29959:
29956:
29953:
29950:
29946:
29940:
29937:
29934:
29930:
29907:
29903:
29883:
29873:
29870:
29867:
29847:
29842:
29838:
29834:
29830:
29824:
29820:
29816:
29812:
29808:
29805:
29783:
29779:
29775:
29771:
29765:
29761:
29757:
29753:
29746:
29738:
29734:
29730:
29726:
29720:
29716:
29712:
29708:
29701:
29695:
29686:The inclusion
29675:
29672:
29669:
29666:
29646:
29643:
29638:
29634:
29613:
29610:
29607:
29604:
29584:
29581:
29576:
29572:
29549:
29546:
29543:
29538:
29533:
29529:
29525:
29501:
29497:
29474:
29470:
29447:
29443:
29416:
29412:
29399:(In fact, the
29387:
29384:
29361:
29357:
29330:
29326:
29322:
29302:
29297:
29293:
29289:
29284:
29280:
29276:
29254:
29250:
29246:
29241:
29237:
29214:
29210:
29206:
29203:
29200:
29180:
29173:
29169:
29164:
29158:
29155:
29152:
29148:
29144:
29137:
29133:
29128:
29120:
29116:
29112:
29107:
29103:
29098:
29092:
29089:
29086:
29082:
29078:
29071:
29067:
29062:
29056:
29053:
29050:
29046:
29038:
29034:
29030:
29025:
29021:
29016:
29012:
29007:
29003:
28999:
28978:
28972:
28968:
28964:
28959:
28955:
28951:
28946:
28942:
28937:
28930:
28926:
28921:
28917:
28896:
28893:
28890:
28887:
28866:
28845:
28823:
28817:
28812:
28808:
28804:
28800:
28797:
28794:
28791:
28786:
28782:
28777:
28773:
28753:
28750:
28747:
28744:
28741:
28738:
28735:
28732:
28715:
28714:
28706:
28705:
28693:
28690:
28687:
28684:
28664:
28661:
28656:
28652:
28631:
28628:
28623:
28619:
28598:
28595:
28590:
28586:
28565:
28562:
28559:
28556:
28536:
28533:
28530:
28510:
28507:
28504:
28501:
28481:
28478:
28475:
28472:
28469:
28466:
28463:
28460:
28457:
28452:
28448:
28427:
28407:
28387:
28384:
28381:
28378:
28356:
28352:
28348:
28343:
28339:
28316:
28312:
28308:
28303:
28299:
28272:
28268:
28241:
28238:
28235:
28230:
28225:
28221:
28217:
28196:
28184:
28180:
28163:
28150:
28145:
28141:
28117:
28111:
28108:
28105:
28101:
28078:
28074:
28053:
28049:
28044:
28038:
28035:
28032:
28029:
28026:
28022:
28018:
28013:
28006:
28002:
27996:
27993:
27990:
27986:
27980:
27974:
27970:
27965:
27961:
27941:
27929:
27925:
27902:
27898:
27877:
27873:
27867:
27864:
27861:
27858:
27855:
27851:
27847:
27842:
27835:
27831:
27825:
27822:
27819:
27815:
27809:
27801:
27791:
27787:
27782:
27759:
27755:
27749:
27746:
27743:
27739:
27735:
27728:
27724:
27719:
27702:
27677:
27673:
27668:
27662:
27659:
27656:
27652:
27623:
27619:
27614:
27604:Because every
27597:
27584:
27562:
27558:
27537:
27534:
27531:
27528:
27525:
27505:
27483:
27480:
27477:
27473:
27452:
27449:
27446:
27443:
27440:
27430:
27416:
27396:
27376:
27354:
27350:
27346:
27343:
27340:
27337:
27326:
27309:
27306:
27303:
27299:
27276:
27273:
27270:
27266:
27245:
27240:
27236:
27213:
27210:
27207:
27203:
27199:
27194:
27191:
27188:
27184:
27161:
27157:
27134:
27131:
27128:
27124:
27101:
27096:
27093:
27089:
27085:
27082:
27079:
27076:
27072:
27066:
27063:
27060:
27056:
27049:
27041:
27037:
27033:
27030:
27027:
27024:
27021:
27018:
27013:
26992:
26980:
26977:
26974:
26970:
26949:
26946:
26943:
26940:
26937:
26917:
26895:
26891:
26887:
26884:
26881:
26878:
26858:
26855:
26852:
26849:
26846:
26826:
26823:
26820:
26817:
26814:
26811:
26791:
26771:
26751:
26731:
26709:
26705:
26684:
26674:
26660:
26638:
26633:
26630:
26626:
26622:
26619:
26616:
26613:
26609:
26603:
26600:
26597:
26593:
26586:
26578:
26574:
26570:
26567:
26562:
26557:
26552:
26548:
26525:
26521:
26517:
26512:
26508:
26487:
26467:
26451:
26449:, etc.) space.
26428:
26424:
26403:
26399:
26394:
26390:
26387:
26382:
26378:
26373:
26369:
26366:
26361:
26357:
26352:
26325:
26321:
26294:
26290:
26267:
26263:
26259:
26254:
26250:
26232:
26231:
26229:
26226:
26225:
26224:
26218:
26212:
26203:
26197:James' theorem
26194:
26189:
26183:
26178:
26173:
26168:
26161:
26158:
26150:compact spaces
26125:is not empty.
26114:
26107:
26102:
26099:
26095:
26091:
26088:
26064:
26059:
26054:
26051:
26048:
26045:
26042:
26039:
26036:
26014:
26010:
25996:
25982:
25973:is a cover of
25960:
25948:
25940:
25918:
25898:
25893:
25889:
25865:
25845:
25824:
25821:
25817:
25782:
25766:
25730:
25727:
25726:
25725:
25709:
25706:
25703:
25700:
25676:
25673:
25670:
25667:
25664:
25636:
25633:
25630:
25627:
25616:
25609:
25600:
25597:
25596:
25595:
25583:
25580:
25577:
25572:
25568:
25547:
25544:
25541:
25521:
25516:
25512:
25508:
25505:
25502:
25499:
25494:
25490:
25486:
25483:
25474:
25471:
25467:
25464:
25461:
25458:
25455:
25452:
25448:
25441:
25437:
25432:
25428:
25408:
25405:
25402:
25380:
25375:
25372:
25369:
25364:
25357:
25353:
25348:
25344:
25322:
25319:
25299:
25296:
25291:
25287:
25283:
25278:
25274:
25253:
25250:
25245:
25242:
25237:
25232:
25229:
25207:
25187:
25184:
25181:
25178:
25175:
25155:
25152:
25149:
25144:
25140:
25116:
25092:
25072:
25048:
25037:
25034:James' theorem
25020:
25014:
25010:
25006:
25003:
24999:
24995:
24976:
24970:
24954:
24950:
24927:
24923:
24892:
24888:
24863:
24851:
24848:
24846:
24843:
24830:
24811:
24806:
24802:
24798:
24795:
24792:
24789:
24786:
24764:
24758:
24754:
24733:
24730:
24727:
24724:
24713:
24709:
24705:
24702:
24699:
24696:
24693:
24672:
24668:
24665:
24662:
24659:
24639:
24636:
24631: in
24625:
24621:
24617:
24612:
24609:
24606:
24601:
24597:
24594:
24591:
24586:
24582:
24577:
24570:
24566:
24545:
24542:
24539:
24519:
24497:
24494:
24491:
24486:
24482:
24479:
24476:
24471:
24467:
24462:
24440:
24437:
24432:
24429:
24426:
24421:
24416:
24412:
24408:
24403:
24398:
24394:
24364:
24358:
24354:
24327:
24322:
24319:
24316:
24312:
24308:
24305:
24302:
24299:
24296:
24293:
24291:
24288:
24284:
24281:
24278:
24275:
24272:
24269:
24266:
24262:
24257:
24251:
24247:
24243:
24239:
24236:
24233:
24232:
24212:
24192:
24187:
24181:
24177:
24156:
24151:
24147:
24123:
24097:
24093:
24066:
24062:
24056:
24053:
24050:
24046:
24042:
24039:
24019:
24016:
24013:
24010:
24007:
24003:
23999:
23995:
23991:
23987:
23983:
23980:
23977:
23974:
23969:
23965:
23944:
23941:
23938:
23935:
23921:
23900:
23896:
23892:
23887:
23883:
23879:
23876:
23854:
23850:
23822:
23802:
23799:
23779:
23759:
23756:
23736:
23733:
23711:
23707:
23703:
23700:
23697:
23692:
23688:
23684:
23679:
23675:
23654:
23651:
23630:
23626:
23621:
23617:
23613:
23610:
23571:
23551:
23548:
23526:
23522:
23501:
23498:
23495:
23492:
23489:
23465:
23461:
23456:
23452:
23448:
23445:
23442:
23439:
23436:
23432:
23428:
23425:
23420:
23416:
23411:
23407:
23403:
23400:
23397:
23394:
23391:
23387:
23378:
23375:
23371:
23364:
23359:
23356:
23353:
23349:
23345:
23342:
23339:
23336:
23333:
23330:
23327:
23308:
23305:
23302:
23299:
23296:
23274:
23269:
23266:
23263:
23258:
23253:
23249:
23245:
23240:
23235:
23231:
23210:
23190:
23185:
23181:
23160:
23140:
23113:
23110:
23098:
23095:
23071:
23049:
23045:
23025:
23020:
23015:
23011:
23008:
23003:
22999:
22994:
22989:
22984:
22980:
22957:
22953:
22949:
22946:
22943:
22940:
22937:
22932:
22928:
22907:
22883:
22859:
22856:
22853:
22850:
22847:
22844:
22841:
22838:
22833:
22830:
22827:
22822:
22817:
22813:
22809:
22788:
22776:
22772:
22743:
22740:
22717:
22713:
22684:
22680:
22675:
22669:
22666:
22663:
22659:
22655:
22648:
22644:
22639:
22635:
22613:
22608:
22604:
22581:
22577:
22573:
22551:
22547:
22543:
22538:
22534:
22530:
22527:
22520:
22516:
22511:
22507:
22487:
22483:
22475:
22471:
22466:
22462:
22459:
22456:
22450:
22440:
22436:
22431:
22427:
22423:
22416:
22409:
22403:
22399:
22395:
22390:
22386:
22379:
22369:
22365:
22360:
22356:
22352:
22332:
22322:
22317:
22313:
22293:
22290:
22268:
22264:
22234:
22230:
22209:
22205:
22199:
22195:
22191:
22186:
22182:
22178:
22173:
22169:
22164:
22160:
22157:
22152:
22148:
22128:
22125:
22122:
22119:
22110:and for every
22097:
22093:
22089:
22084:
22080:
22058:
22050:
22046:
22041:
22037:
22034:
22031:
22025:
22017:
22012:
22007:
22002:
21998:
21993:
21973:
21961:
21957:
21936:
21931:
21926:
21921:
21916:
21913:
21910:
21905:
21900:
21896:
21892:
21887:
21882:
21878:
21853:
21849:
21844:
21838:
21835:
21832:
21828:
21808:
21794:
21790:
21785:
21781:
21759:
21755:
21751:
21748:
21726:
21722:
21702:
21692:
21670:
21666:
21637:
21633:
21628:
21622:
21619:
21616:
21612:
21608:
21605:
21603:
21598:
21594:
21590:
21587:
21585:
21583:
21576:
21572:
21567:
21561:
21558:
21555:
21551:
21547:
21544:
21542:
21537:
21533:
21529:
21526:
21524:
21522:
21515:
21511:
21507:
21504:
21502:
21498:
21494:
21490:
21489:
21467:
21464:
21461:
21456:
21451:
21447:
21443:
21438:
21433:
21429:
21408:
21386:
21381:
21377:
21365:weak-* compact
21351:
21345:
21341:
21337:
21334:
21331:
21328:
21325:
21322:
21317:
21313:
21309:
21306:
21302:
21282:
21270:
21266:
21245:
21221:
21218:
21215:
21212:
21209:
21206:
21203:
21200:
21197:
21194:
21191:
21188:
21169:
21159:
21156:
21153:
21150:
21147:
21144:
21141:
21138:
21118:
21115:
21112:
21097:
21096:
21084:
21063:
21058:
21054:
21051:
21046:
21042:
21037:
21033:
21030:
21025:
21021:
21016:
20995:
20991:
20985:
20982:
20979:
20975:
20971:
20966:
20961:
20935:
20921:
20920:
20917:
20912:
20904:locally convex
20884:
20879:
20875:
20872:
20867:
20863:
20858:
20854:
20851:
20846:
20842:
20837:
20812:
20808:
20805:
20800:
20796:
20791:
20787:
20767:
20745:
20741:
20710:
20706:
20693:
20678:
20674:
20648:
20642:
20639:
20636:
20632:
20614:
20613:
20601:
20581:
20578:
20575:
20572:
20569:
20566:
20563:
20560:
20557:
20554:
20551:
20548:
20545:
20542:
20539:
20536:
20533:
20513:
20510:
20507:
20504:
20501:
20498:
20495:
20492:
20489:
20486:
20483:
20480:
20477:
20472:
20468:
20447:
20444:
20441:
20438:
20435:
20432:
20429:
20426:
20423:
20420:
20417:
20414:
20411:
20408:
20405:
20402:
20397:
20393:
20372:
20369:
20366:
20363:
20360:
20355:
20351:
20347:
20342:
20339:
20336:
20331:
20327:
20324:
20321:
20318:
20315:
20310:
20306:
20301:
20296:
20291:
20288:
20285:
20280:
20276:
20273:
20270:
20265:
20261:
20257:
20254:
20251:
20248:
20243:
20239:
20234:
20229:
20224:
20221:
20218:
20213:
20208:
20204:
20201:
20198:
20193:
20189:
20185:
20182:
20179:
20176:
20171:
20167:
20162:
20158:
20154:
20149:
20144:
20141:
20138:
20133:
20128:
20123:
20119:
20115:
20111:
20107:
20102:
20097:
20093:
20089:
20086:
20067:
20056:
20051:
20047:
20043:
20039:
20019:
20016:
20013:
20010:
20007:
20004:
20001:
19998:
19995:
19992:
19989:
19986:
19983:
19980:
19977:
19974:
19971:
19968:
19965:
19962:
19959:
19956:
19935:
19914:
19911:
19908:
19905:
19902:
19899:
19896:
19893:
19890:
19887:
19884:
19881:
19878:
19874:
19869:
19865:
19861:
19857:
19837:
19817:
19814:
19811:
19808:
19789:
19779:
19776:
19773:
19770:
19767:
19764:
19743:
19739:
19735:
19731:
19727:
19723:
19720:
19700:
19696:
19692:
19688:
19667:
19664:
19661:
19658:
19655:
19652:
19649:
19646:
19643:
19640:
19637:
19634:
19629:
19625:
19604:
19601:
19598:
19595:
19592:
19589:
19584:
19581:
19578:
19573:
19569:
19566:
19563:
19558:
19554:
19549:
19544:
19541:
19538:
19535:
19530:
19526:
19505:
19502:
19499:
19496:
19493:
19488:
19485:
19482:
19477:
19473:
19470:
19467:
19462:
19458:
19453:
19448:
19445:
19442:
19439:
19434:
19430:
19409:
19406:
19403:
19400:
19379:
19375:
19372:
19369:
19364:
19360:
19356:
19353:
19350:
19347:
19342:
19338:
19333:
19313:
19301:
19297:
19275:
19271:
19267:
19263:
19260:
19257:
19252:
19249:
19246:
19241:
19236:
19232:
19228:
19223:
19218:
19214:
19193:
19190:
19187:
19184:
19181:
19178:
19175:
19172:
19169:
19166:
19163:
19160:
19157:
19154:
19151:
19148:
19145:
19121:
19118:
19115:
19112:
19109:
19106:
19103:
19100:
19097:
19094:
19091:
19088:
19067:
19057:and limits in
19046:
19043:
19040:
19037:
19034:
19031:
19028:
19025:
19022:
19019:
19014:
19010:
18989:
18986:
18983:
18980:
18977:
18974:
18971:
18968:
18965:
18962:
18959:
18954:
18950:
18920:
18917:
18914:
18911:
18908:
18903:
18899:
18892:
18890:
18887:
18886:
18881:
18877:
18868:
18863:
18860:
18857:
18852:
18848:
18845:
18842:
18839:
18834:
18830:
18825:
18817:
18815:
18812:
18811:
18808:
18805:
18802:
18797:
18793:
18789:
18770:
18759:
18755:
18752:
18749:
18744:
18740:
18735:
18731:
18728:
18723:
18720:
18717:
18712:
18708:
18705:
18702:
18697:
18693:
18689:
18685:
18677:
18675:
18672:
18671:
18667:
18663:
18660:
18657:
18652:
18649:
18646:
18641:
18637:
18634:
18631:
18626:
18622:
18617:
18612:
18609:
18606:
18603:
18598:
18594:
18585:
18571:
18568:
18565:
18560:
18555:
18551:
18548:
18545:
18540:
18536:
18531:
18527:
18523:
18515:
18513:
18510:
18509:
18501:
18498:
18495:
18492:
18487:
18483:
18479:
18476:
18470:
18455:
18447:
18443:
18440:
18437:
18432:
18428:
18423:
18419:
18416:
18415:
18395:
18392:
18389:
18386:
18383:
18380:
18377:
18374:
18371:
18368:
18365:
18362:
18359:
18339:
18336:
18333:
18330:
18327:
18324:
18321:
18318:
18314:
18310:
18307:
18304:
18299:
18295:
18290:
18286:
18266:
18262:
18241:
18238:
18235:
18232:
18229:
18226:
18223:
18220:
18215:
18211:
18190:
18170:
18167:
18164:
18145:
18135:
18132:
18129:
18126:
18106:
18085:
18081:
18077:
18073:
18070:
18050:
18047:
18044:
18041:
18038:
18035:
18032:
18029:
18026:
18023:
18020:
18017:
17994:
17991:
17988:
17985:
17982:
17979:
17976:
17973:
17970:
17967:
17964:
17959:
17955:
17944:
17941:
17938:
17935:
17932:
17929:
17926:
17923:
17920:
17917:
17914:
17911:
17908:
17903:
17899:
17894:
17891:
17888:
17885:
17882:
17879:
17876:
17873:
17870:
17865:
17861:
17856:
17853:
17850:
17847:
17844:
17841:
17838:
17835:
17832:
17827:
17823:
17803:
17799:
17778:
17775:
17755:
17752:
17749:
17746:
17738:
17735:
17732:
17729:
17726:
17723:
17720:
17700:
17697:
17694:
17691:
17670:
17649:
17646:
17643:
17640:
17637:
17634:
17631:
17628:
17623:
17619:
17598:
17593:
17588:
17566:
17563:
17558:
17554:
17533:
17528:
17525:
17522:
17517:
17513:
17510:
17507:
17502:
17498:
17493:
17472:
17462:
17459:
17456:
17451:
17447:
17437:
17426:
17404:
17400:
17377:
17373:
17370:
17367:
17364:
17361:
17358:
17353:
17349:
17328:
17325:
17322:
17319:
17297:
17294:
17291:
17288:
17285:
17282:
17262:
17242:
17222:
17217:
17213:
17209:
17206:
17186:
17181:
17176:
17154:
17132:
17128:
17105:
17102:
17099:
17094:
17089:
17085:
17081:
17076:
17071:
17067:
17044:
17039:
17034:
17031:
17018:
17017:
17015:Proof of lemma
17014:
17009:
16995:
16974:
16968:
16965:
16962:
16958:
16954:
16949:
16944:
16921:
16899:
16877:
16855:
16834:
16812:
16808:
16777:
16772:
16748:
16744:
16731:
16730:
16729:
16718:
16698:
16695:
16692:
16689:
16686:
16683:
16680:
16660:
16657:
16637:
16617:
16614:
16611:
16608:
16605:
16600:
16596:
16592:
16589:
16586:
16583:
16578:
16574:
16553:
16533:
16530:
16527:
16524:
16521:
16516:
16513:
16510:
16505:
16501:
16498:
16495:
16490:
16486:
16481:
16459:
16456:
16453:
16450:
16430:
16427:
16424:
16421:
16418:
16415:
16412:
16392:
16387:
16383:
16379:
16376:
16356:
16353:
16331:
16327:
16304:
16301:
16298:
16293:
16288:
16284:
16280:
16256:
16252:
16248:
16245:
16230:
16229:
16215:
16211:
16189:
16185:
16182:
16179:
16176:
16173:
16170:
16167:
16162:
16158:
16154:
16151:
16147:
16127:
16115:
16111:
16090:
16087:
16063:
16060:
16057:
16037:
16034:
16031:
16005:
16000:
15996:
15992:
15989:
15968:
15964:
15961:
15940:
15934:
15930:
15926:
15923:
15920:
15917:
15914:
15911:
15906:
15901:
15896:
15893:
15889:
15885:
15878:
15874:
15869:
15845:
15825:
15820:
15815:
15791:
15787:
15783:
15778:
15774:
15770:
15763:
15759:
15754:
15733:
15729:
15706:
15701:
15696:
15692:
15686:
15683:
15680:
15676:
15645:
15640:
15637:
15634:
15629: if
15626:
15623:
15619:
15618:
15615:
15612:
15609:
15604: if
15601:
15597:
15593:
15589:
15588:
15586:
15565:
15553:
15549:
15536:
15532:
15526:
15523:
15520:
15516:
15512:
15509:
15507:
15505:
15500:
15495:
15492:
15489:
15479:
15475:
15471:
15463:
15459:
15448:
15440:
15434:
15431:
15428:
15424:
15420:
15415:
15412:
15409:
15404:
15399:
15395:
15391:
15384:
15379:
15376:
15374:
15372:
15367:
15362:
15359:
15356:
15346:
15342:
15338:
15335:
15332:
15329:
15326:
15323:
15305:
15300:
15295:
15291:
15259:
15254:
15251:
15249:
15247:
15242:
15237:
15234:
15230:
15226:
15223:
15220:
15217:
15213:
15207:
15204:
15201:
15197:
15193:
15175:
15170:
15165:
15159:
15127:
15108:
15100:
15098:
15092:
15088:
15083:
15079:
15078:
15053:
15047:
15044:
15041:
15037:
15033:
15028:
15023:
14999:
14995:
14962:
14955:
14951:
14946:
14942:
14939:
14936:
14933:
14930:
14910:
14905:
14901:
14897:
14893:
14889:
14886:
14883:
14880:
14876:
14853:
14850:
14846:
14842:
14839:
14836:
14833:
14829:
14823:
14820:
14817:
14813:
14809:
14805:
14800:
14795:
14791:
14787:
14783:
14779:
14775:
14771:
14766:
14762:
14755:
14751:
14747:
14741:
14737:
14733:
14729:
14725:
14721:
14718:
14715:
14712:
14705:
14701:
14697:
14691:
14687:
14683:
14679:
14676:
14673:
14670:
14666:
14658:
14654:
14650:
14628:
14625:
14622:
14618:
14614:
14611:
14608:
14605:
14601:
14595:
14592:
14589:
14585:
14559:
14538:
14535:
14532:
14529:
14521:
14517:
14513:
14492:
14480:
14476:
14454:
14451:
14448:
14428:
14425:
14420:
14416:
14395:
14390:
14386:
14382:
14379:
14359:
14356:
14351:
14347:
14326:
14319:
14315:
14310:
14306:
14303:
14300:
14297:
14294:
14274:
14271:
14268:
14265:
14262:
14258:
14254:
14251:
14248:
14243:
14236:
14216:
14211:
14207:
14203:
14200:
14180:
14177:
14174:
14154:
14151:
14148:
14145:
14121:
14117:
14112:
14108:
14104:
14099:
14095:
14091:
14085:
14078:
14058:
14051:
14047:
14042:
14036:
14033:
14030:
14026:
14022:
14017:
14013:
13988:
13967:
13963:
13959:
13953:
13950:
13947:
13943:
13939:
13934:
13927:
13907:
13904:
13901:
13898:
13881:
13880:
13869:
13864:
13859:
13854:
13850:
13844:
13841:
13838:
13834:
13808:
13804:
13793:
13782:
13775:
13771:
13766:
13760:
13757:
13754:
13750:
13746:
13741:
13737:
13713:
13706:
13702:
13697:
13691:
13688:
13685:
13681:
13659:
13653:
13649:
13645:
13642:
13639:
13636:
13633:
13627:
13619:
13615:
13611:
13608:
13604:
13597:
13589:
13584:
13581:
13577:
13573:
13570:
13567:
13564:
13560:
13554:
13551:
13548:
13544:
13537:
13529:
13525:
13521:
13518:
13513:
13492:
13480:
13476:
13448:
13443:
13439:
13435:
13431:
13427:
13423:
13419:
13415:
13411:
13408:
13405:
13386:
13372:
13368:
13363:
13349:is compact by
13334:
13330:
13325:
13319:
13316:
13313:
13309:
13291:
13278:
13256:
13252:
13248:
13245:
13221:
13201:
13198:
13178:
13157:
13154:
13151:
13147:
13143:
13140:
13137:
13134:
13130:
13124:
13121:
13118:
13114:
13090:
13086:
13082:
13079:
13059:
13054:
13050:
13046:
13041:
13037:
13016:
13011:
13007:
13003:
12998:
12994:
12990:
12985:
12981:
12959:
12954:
12950:
12946:
12941:
12937:
12913:
12908:
12904:
12900:
12895:
12891:
12884:
12876:
12871:
12868:
12864:
12860:
12857:
12854:
12851:
12847:
12841:
12838:
12835:
12831:
12824:
12816:
12812:
12808:
12805:
12800:
12779:
12767:
12763:
12740:
12736:
12715:
12710:
12706:
12702:
12697:
12693:
12666:
12662:
12641:
12638:
12633:
12629:
12625:
12622:
12602:
12599:
12594:
12590:
12569:
12566:
12563:
12560:
12540:
12537:
12524:it is also an
12513:
12510:
12490:
12470:
12448:
12444:
12419:
12405:
12392:
12368:
12342:
12338:
12301:
12297:
12292:
12286:
12283:
12280:
12276:
12243:
12239:
12218:
12215:
12210:
12206:
12202:
12199:
12179:
12176:
12171:
12167:
12147:
12144:
12141:
12138:
12116:
12111:
12106:
12103:
12099:
12095:
12092:
12089:
12086:
12082:
12076:
12073:
12070:
12066:
12059:
12051:
12047:
12043:
12040:
12035:
12014:
12002:
11998:
11984:has its usual
11972:
11951:
11940:usual topology
11927:
11924:
11921:
11917:
11913:
11909:
11905:
11901:
11897:
11894:
11891:
11872:
11860:
11856:
11835:
11832:
11811:
11807:
11803:
11797: or
11793:
11789:
11785:
11763:
11742:
11722:
11705:
11692:
11682:
11674:
11673:
11660:
11640:
11621:
11611:
11587:
11567:
11557:
11536:
11533:
11529:
11525:
11522:
11519:
11516:
11512:
11506:
11503:
11500:
11496:
11473:
11470:
11466:
11462:
11459:
11456:
11453:
11449:
11443:
11440:
11437:
11433:
11411:
11408:
11405:
11401:
11397:
11393:
11389:
11385:
11381:
11378:
11375:
11353:
11349:
11328:
11325:
11322:
11319:
11316:
11313:
11310:
11307:
11304:
11301:
11298:
11279:
11269:
11266:
11263:
11260:
11235:
11212:
11208:
11204:
11201:
11198:
11195:
11192:
11182:
11179:
11175:
11171:
11168:
11165:
11162:
11158:
11152:
11149:
11146:
11142:
11122:
11119:
11096:
11093:
11089:
11085:
11082:
11079:
11076:
11072:
11066:
11063:
11060:
11056:
11032:
11028:
11022:
11019:
11016:
11012:
10989:
10985:
10979:
10976:
10973:
10969:
10948:
10945:
10942:
10939:
10918:
10895:
10891:
10870:
10866:
10860:
10857:
10854:
10850:
10840:inherits from
10827:
10823:
10817:
10814:
10811:
10807:
10780:
10776:
10770:
10767:
10764:
10760:
10738:
10712:
10708:
10703:
10699:
10679:
10676:
10673:
10670:
10644:
10641:
10621:
10601:
10579:
10576:
10573:
10568:
10563:
10559:
10555:
10542:
10541:
10530:
10526:
10505:
10502:
10499:
10496:
10493:
10488:
10485:
10482:
10477:
10473:
10470:
10467:
10462:
10458:
10453:
10431:
10428:
10425:
10422:
10398:
10393:
10390:
10387:
10382:
10378:
10375:
10372:
10367:
10363:
10358:
10353:
10348:
10345:
10342:
10337:
10332:
10327:
10323:
10319:
10313:
10306:
10302:
10281:
10270:
10265:
10262:
10259:
10254:
10249:
10245:
10241:
10236:
10230:
10223:
10202:
10199:
10196:
10193:
10190:
10187:
10184:
10181:
10176:
10169:
10159:where because
10157:
10156:
10145:
10141:
10120:
10117:
10114:
10109:
10102:
10099:
10095:
10090:
10087:
10084:
10079:
10074:
10070:
10066:
10061:
10055:
10048:
10028:
10025:
10022:
10019:
9992:
9989:
9984:
9981:
9978:
9973:
9968:
9964:
9960:
9938:
9934:
9928:
9925:
9922:
9918:
9914:
9909:
9904:
9880:
9876:
9855:
9835:
9830:
9827:
9824:
9819:
9814:
9810:
9806:
9781:
9752:
9746:
9743:
9740:
9736:
9732:
9727:
9722:
9693:
9690:
9687:
9684:
9681:
9678:
9673:
9669:
9665:
9662:
9642:
9639:
9636:
9633:
9630:
9626:
9621:
9617:
9613:
9609:
9589:
9586:
9583:
9580:
9577:
9572:
9569:
9566:
9561:
9556:
9551:
9547:
9543:
9539:
9535:
9513:
9510:
9490:
9487:
9482:
9478:
9457:
9454:
9451:
9448:
9445:
9425:
9420:
9416:
9412:
9409:
9390:
9378:
9375:
9372:
9367:
9362:
9357:
9353:
9349:
9345:
9341:
9336:
9332:
9327:
9323:
9319:
9315:
9293:
9289:
9266:
9261:
9258:
9255:
9250:
9245:
9240:
9236:
9232:
9228:
9224:
9200:
9197:
9194:
9189:
9184:
9179:
9175:
9171:
9167:
9163:
9141:
9138:
9135:
9132:
9129:
9124:
9120:
9116:
9113:
9093:
9071:
9067:
9046:
9043:
9040:
9037:
9034:
9021:Similarly for
9010:
9007:
9004:
9001:
8996:
8992:
8969:
8965:
8942:
8938:
8928:is denoted by
8917:
8914:
8911:
8889:
8885:
8864:
8861:
8858:
8854:
8833:
8830:
8806:
8803:
8800:
8797:
8794:
8791:
8768:
8765:
8762:
8759:
8754:
8750:
8729:
8707:
8704:
8701:
8696:
8691:
8687:
8683:
8678:
8673:
8669:
8638:
8630:
8624:
8621:
8618:
8612:
8606:
8602:
8597:
8591:
8585:
8581:
8577:
8574:
8570:
8567:
8564:
8561:
8556:
8553:
8550:
8545:
8540:
8535:
8530:
8525:
8522:
8518:
8513:
8508:
8503:
8499:
8496:
8493:
8492:
8463:
8457:
8454:
8451:
8448:
8445:
8440:
8435:
8431:
8427:
8421:
8416:
8413:
8410:
8405:
8400:
8396:
8392:
8386:
8382:
8379:
8375:
8370:
8367:
8364:
8359:
8354:
8350:
8346:
8341:
8338:
8334:
8328:
8325:
8322:
8319:
8316:
8312:
8308:
8304:
8299:
8293:
8290:
8287:
8283:
8278:
8274:
8271:
8267:
8263:
8257:
8254:
8251:
8247:
8243:
8239:
8236:
8233:
8232:
8212:
8209:
8206:
8203:
8200:
8196:
8192:
8189:
8186:
8166:
8144:
8141:
8138:
8116:
8113:
8110:
8107:
8083:
8079:
8074:
8070:
8067:
8064:
8061:
8058:
8036:
8031:
8026:
8023:
7999:
7995:
7990:
7984:
7981:
7978:
7974:
7953:
7948:
7943:
7938:
7934:
7928:
7925:
7922:
7918:
7914:
7907:
7903:
7898:
7892:
7889:
7886:
7882:
7859:
7856:
7853:
7848:
7843:
7839:
7835:
7808:
7801:
7781:
7761:
7741:
7721:
7718:
7698:
7678:
7675:
7672:
7669:
7666:
7647:
7637:
7634:
7631:
7626:
7619:
7598:
7594:
7591:
7588:
7585:
7563:
7556:
7534:
7530:
7526:
7521:
7518:
7515:
7510:
7505:
7501:
7497:
7492:
7487:
7483:
7471:
7467:
7463:
7457:
7454:
7451:
7447:
7443:
7438:
7431:
7411:
7408:
7405:
7386:
7354:
7349:
7326:
7320:
7317:
7314:
7310:
7288:
7282:
7279:
7276:
7272:
7268:
7263:
7258:
7245:
7244:
7233:
7228:
7225:
7222:
7217:
7212:
7208:
7204:
7180:
7176:
7156:
7146:
7143:
7140:
7137:
7116:
7112:
7109:
7106:
7103:
7082:
7076:
7073:
7070:
7066:
7043:
7040:
7037:
7032:
7027:
7023:
7019:
7014:
7009:
7005:
6994:
6982:
6970:
6959:
6954:
6951:
6948:
6944:
6940:
6937:
6934:
6931:
6928:
6909:
6897:
6893:
6882:
6868:
6846:
6841:
6818:
6814:
6811:
6808:
6805:
6795:
6783:
6758:
6754:
6750:
6747:
6725:
6720:
6696:
6693:
6690:
6685:
6680:
6676:
6672:
6667:
6662:
6658:
6634:
6613:
6607:
6604:
6601:
6597:
6569:
6566:
6563:
6560:
6540:
6537:
6534:
6531:
6528:
6525:
6522:
6519:
6500:
6490:
6487:
6467:
6447:
6427:
6424:
6421:
6417:
6413:
6409:
6405:
6401:
6397:
6394:
6391:
6372:
6360:
6356:
6336:
6333:
6320:
6319:
6316:
6311:
6298:
6291:
6287:
6282:
6276:
6273:
6270:
6266:
6243:
6239:
6203:
6199:
6193:
6190:
6187:
6183:
6158:
6154:
6151:
6129:
6124:
6084:duality theory
6078:The following
6075:
6072:
6050:
6027:
6023:
6000:
5996:
5985:
5971:
5949:
5945:
5910:
5906:
5885:
5868:
5851:
5829:
5825:
5804:
5784:
5764:
5740:
5728:
5727:
5715:
5692:
5688:
5667:
5663:
5658:
5654:
5651:
5646:
5642:
5637:
5633:
5630:
5625:
5621:
5616:
5593:
5589:
5563:
5558:
5554:
5551:
5546:
5542:
5537:
5533:
5530:
5525:
5521:
5516:
5495:
5490:
5486:
5464:
5460:
5457:
5452:
5448:
5443:
5439:
5417:
5413:
5401:
5390:
5385:
5381:
5359:
5355:
5352:
5347:
5343:
5338:
5334:
5312:
5308:
5287:
5283:
5278:
5274:
5271:
5266:
5262:
5257:
5253:
5250:
5245:
5241:
5236:
5226:inherits from
5213:
5209:
5186:
5182:
5160:
5156:
5153:
5148:
5144:
5139:
5135:
5115:
5110:
5106:
5084:
5080:
5077:
5072:
5068:
5063:
5059:
5037:
5033:
5023:Conclude that
5021:
5008:
5004:
5001:
4996:
4992:
4987:
4983:
4961:
4957:
4931:
4927:
4924:
4919:
4915:
4910:
4906:
4885:
4881:
4878:
4873:
4869:
4864:
4860:
4838:
4834:
4813:
4808:
4804:
4780:
4760:
4755:
4751:
4739:bounded subset
4725:
4721:
4718:
4713:
4709:
4704:
4700:
4678:
4674:
4653:
4648:
4645:
4641:
4620:
4617:
4593:
4571:
4568:
4564:
4560:
4557:
4533:
4528:
4524:
4498:
4494:
4491:
4486:
4482:
4477:
4473:
4451:
4447:
4435:
4421:
4417:
4396:
4391:
4387:
4364:
4360:
4356:
4351:
4347:
4343:
4338:
4334:
4330:
4325:
4321:
4317:
4312:
4308:
4287:
4282:
4278:
4274:
4271:
4249:
4245:
4241:
4238:
4214:
4194:
4174:
4153:
4150:
4147:
4143:
4139:
4136:
4133:
4130:
4126:
4120:
4117:
4114:
4110:
4086:
4082:
4078:
4075:
4054:
4049:
4045:
4041:
4036:
4032:
4008:
4004:
4000:
3995:
3991:
3981:The inclusion
3970:
3966:
3961:
3957:
3954:
3949:
3945:
3940:
3936:
3933:
3928:
3924:
3919:
3897:
3892:
3888:
3885:
3880:
3876:
3871:
3867:
3864:
3859:
3855:
3850:
3827:
3823:
3800:
3796:
3792:
3787:
3783:
3771:
3760:
3757:
3753:
3749:
3746:
3743:
3740:
3736:
3715:
3712:
3709:
3706:
3686:
3682:
3678:
3675:
3671:
3667:
3663:
3659:
3655:
3651:
3648:
3644:
3622:
3601:
3598:
3595:
3590:
3586:
3564:
3543:
3540:
3537:
3534:
3531:
3528:
3525:
3522:
3517:
3513:
3492:
3489:
3486:
3483:
3463:
3460:
3456:
3452:
3449:
3446:
3443:
3439:
3418:
3413:
3409:
3405:
3402:
3382:
3378:
3373:
3369:
3366:
3361:
3357:
3352:
3348:
3345:
3340:
3336:
3331:
3310:
3288:
3284:
3261:
3258:
3255:
3250:
3245:
3241:
3237:
3232:
3227:
3223:
3200:
3196:
3192:
3189:
3169:
3164:
3160:
3138:
3134:
3131:
3126:
3122:
3117:
3113:
3091:
3087:
3063:
3058:
3054:
3050:
3045:
3042:
3039:
3035:
3023:
3022:
3011:
3006:
3002:
2998:
2993:
2989:
2985:
2980:
2976:
2955:
2951:
2945:
2941:
2937:
2934:
2930:
2909:
2888:
2884:
2881:
2877:
2873:
2870:
2867:
2864:
2860:
2854:
2851:
2848:
2844:
2837:
2829:
2825:
2821:
2818:
2814:
2810:
2805:
2801:
2790:
2777:
2771:
2767:
2763:
2760:
2756:
2734:
2711:
2707:
2704:
2700:
2696:
2693:
2690:
2687:
2683:
2675:
2671:
2667:
2664:
2660:
2653:
2647:
2644:
2641:
2637:
2633:
2628:
2625:
2621:
2610:
2597:
2591:
2587:
2583:
2580:
2576:
2555:
2534:
2530:
2527:
2523:
2519:
2516:
2513:
2510:
2506:
2500:
2497:
2494:
2490:
2483:
2475:
2471:
2467:
2464:
2460:
2456:
2451:
2447:
2423:
2403:
2381:
2377:
2371:
2367:
2363:
2360:
2356:
2332:
2328:
2322:
2318:
2314:
2311:
2307:
2281:
2275:
2271:
2267:
2264:
2260:
2236:
2216:
2194:
2190:
2165:
2160:
2156:
2152:
2149:
2129:
2125:
2121:
2118:
2113:
2109:
2104:
2100:
2079:
2074:
2070:
2067:
2062:
2058:
2053:
2049:
2046:
2041:
2037:
2032:
2022:inherits from
2009:
2005:
1976:
1972:
1967:
1963:
1960:
1955:
1951:
1946:
1942:
1939:
1934:
1930:
1925:
1903:
1898:
1894:
1891:
1886:
1882:
1877:
1873:
1870:
1865:
1861:
1856:
1830:
1825:
1821:
1818:
1813:
1809:
1804:
1800:
1797:
1792:
1788:
1783:
1762:
1758:
1753:
1749:
1746:
1741:
1737:
1732:
1728:
1725:
1720:
1716:
1711:
1699:locally convex
1683:
1679:
1675:
1672:
1667:
1663:
1658:
1654:
1629:
1625:
1589:
1585:
1560:
1536:
1532:
1511:
1497:
1495:
1492:
1479:
1475:
1470:
1466:
1463:
1458:
1454:
1449:
1445:
1442:
1437:
1433:
1428:
1406:
1400:
1396:
1392:
1389:
1385:
1364:
1342:
1338:
1317:
1312:
1308:
1286:
1282:
1279:
1274:
1270:
1265:
1261:
1238:
1218:
1194:
1190:
1187:
1183:
1179:
1176:
1173:
1170:
1166:
1160:
1157:
1154:
1150:
1143:
1135:
1131:
1127:
1124:
1120:
1116:
1111:
1107:
1083:
1078:
1074:
1059:locally convex
1052:
1038:
1017:
1005:
1002:
999:
996:
993:
971:
968:
965:
960:
956:
953:
950:
945:
941:
936:
914:
894:
872:
868:
847:
842:
839:
836:
831:
826:
822:
818:
813:
808:
804:
779:
755:
751:
747:
744:
739:
735:
730:
726:
704:
700:
678:
674:
671:
666:
662:
657:
653:
631:
627:
600:
596:
592:
587:
583:
562:
557:
553:
524:
511:
492:
487:
483:
480:
477:
473:
469:
464:
460:
456:
453:
449:
428:
425:
422:
419:
400:
389:
385:
382:
379:
375:
353:
349:
344:
340:
336:
333:
330:
326:
322:
319:
316:
312:
302:
301:evaluation map
285:
259:
255:
233:
212:
186:
183:
120:
117:
114:
111:
108:
105:
102:
99:
85:
79:
76:
56:weak* topology
15:
9:
6:
4:
3:
2:
32265:
32254:
32251:
32249:
32246:
32244:
32241:
32239:
32236:
32235:
32233:
32218:
32210:
32209:
32206:
32200:
32197:
32195:
32192:
32190:
32187:
32185:
32181:
32177:
32175:) convex
32174:
32171:
32169:
32166:
32164:
32160:
32158:
32155:
32153:
32150:
32148:
32147:Semi-complete
32145:
32143:
32140:
32138:
32135:
32133:
32129:
32126:
32124:
32120:
32118:
32115:
32113:
32110:
32108:
32105:
32103:
32100:
32098:
32095:
32093:
32090:
32088:
32085:
32083:
32080:
32078:
32075:
32073:
32070:
32068:
32065:
32063:
32062:Infrabarreled
32060:
32058:
32055:
32053:
32050:
32046:
32043:
32042:
32041:
32038:
32036:
32033:
32031:
32028:
32026:
32023:
32021:
32020:Distinguished
32018:
32016:
32013:
32011:
32008:
32006:
32003:
32001:
31998:
31996:
31992:
31988:
31986:
31983:
31981:
31977:
31973:
31971:
31968:
31966:
31963:
31961:
31958:
31957:
31955:
31953:Types of TVSs
31951:
31945:
31941:
31937:
31935:
31932:
31930:
31927:
31925:
31922:
31920:
31917:
31915:
31911:
31907:
31905:
31902:
31901:
31899:
31895:
31889:
31886:
31884:
31881:
31879:
31876:
31874:
31873:Prevalent/Shy
31871:
31869:
31866:
31864:
31863:Extreme point
31861:
31859:
31853:
31851:
31845:
31843:
31840:
31838:
31835:
31833:
31830:
31828:
31825:
31823:
31820:
31818:
31815:
31813:
31810:
31808:
31805:
31803:
31800:
31799:
31797:
31795:Types of sets
31793:
31787:
31784:
31782:
31779:
31777:
31774:
31772:
31769:
31765:
31762:
31760:
31757:
31755:
31752:
31751:
31750:
31747:
31745:
31742:
31738:
31737:Discontinuous
31735:
31733:
31730:
31728:
31725:
31723:
31720:
31718:
31715:
31713:
31710:
31708:
31705:
31704:
31703:
31700:
31696:
31693:
31692:
31691:
31688:
31687:
31685:
31681:
31675:
31672:
31668:
31665:
31664:
31663:
31660:
31657:
31654:
31652:
31648:
31645:
31643:
31640:
31638:
31635:
31633:
31630:
31628:
31625:
31624:
31622:
31620:
31616:
31610:
31607:
31605:
31602:
31600:
31597:
31595:
31594:Metrizability
31592:
31590:
31587:
31585:
31582:
31580:
31579:Fréchet space
31577:
31575:
31572:
31570:
31567:
31565:
31562:
31560:
31557:
31556:
31554:
31550:
31545:
31538:
31533:
31531:
31526:
31524:
31519:
31518:
31515:
31503:
31495:
31494:
31491:
31485:
31482:
31480:
31477:
31475:
31474:Weak topology
31472:
31470:
31467:
31465:
31462:
31460:
31457:
31456:
31454:
31450:
31443:
31439:
31436:
31434:
31431:
31429:
31426:
31424:
31421:
31419:
31416:
31414:
31411:
31409:
31406:
31404:
31401:
31399:
31398:Index theorem
31396:
31394:
31391:
31389:
31386:
31384:
31381:
31380:
31378:
31374:
31368:
31365:
31363:
31360:
31359:
31357:
31355:Open problems
31353:
31347:
31344:
31342:
31339:
31337:
31334:
31332:
31329:
31327:
31324:
31322:
31319:
31318:
31316:
31312:
31306:
31303:
31301:
31298:
31296:
31293:
31291:
31288:
31286:
31283:
31281:
31278:
31276:
31273:
31271:
31268:
31266:
31263:
31261:
31258:
31257:
31255:
31251:
31245:
31242:
31240:
31237:
31235:
31232:
31230:
31227:
31225:
31222:
31220:
31217:
31215:
31212:
31210:
31207:
31205:
31202:
31201:
31199:
31197:
31193:
31183:
31180:
31178:
31175:
31173:
31170:
31167:
31163:
31159:
31156:
31154:
31151:
31149:
31146:
31145:
31143:
31139:
31133:
31130:
31128:
31125:
31123:
31120:
31118:
31115:
31113:
31110:
31108:
31105:
31103:
31100:
31098:
31095:
31093:
31090:
31088:
31085:
31084:
31081:
31078:
31074:
31069:
31065:
31061:
31054:
31049:
31047:
31042:
31040:
31035:
31034:
31031:
31019:
31016:
31015:
31013:
31009:
31003:
31000:
30998:
30995:
30994:
30992:
30988:
30982:
30979:
30978:
30976:
30972:
30966:
30963:
30961:
30958:
30957:
30955:
30951:
30945:
30942:
30938:
30935:
30933:
30930:
30929:
30928:
30925:
30923:
30920:
30916:
30913:
30911:
30908:
30906:
30903:
30902:
30901:
30898:
30896:
30893:
30889:
30886:
30885:
30884:
30883:Norm topology
30881:
30880:
30878:
30876:
30872:
30866:
30863:
30861:
30858:
30856:
30853:
30851:
30848:
30846:
30843:
30841:
30840:Dual topology
30838:
30836:
30833:
30831:
30828:
30827:
30825:
30821:
30816:
30812:
30805:
30800:
30798:
30793:
30791:
30786:
30785:
30782:
30767:
30763:
30759:
30753:
30746:
30745:
30740:
30739:Lax, Peter D.
30736:
30732:
30728:
30724:
30718:
30714:
30710:
30706:
30702:
30698:
30697:
30686:
30682:
30678:
30672:
30668:
30664:
30660:
30656:
30652:
30648:
30642:
30638:
30634:
30630:
30626:
30622:
30618:
30612:
30608:
30604:
30600:
30596:
30591:
30587:
30583:
30577:
30573:
30569:
30568:
30563:
30562:Rudin, Walter
30559:
30555:
30551:
30547:
30541:
30537:
30532:
30528:
30526:0-19-851485-9
30522:
30518:
30513:
30509:
30505:
30501:
30497:
30493:
30487:
30483:
30479:
30475:
30474:
30453:
30449:
30445:
30441:
30434:
30427:
30425:
30417:
30412:
30405:
30400:
30393:
30388:
30386:
30384:
30382:
30380:
30378:
30370:
30365:
30363:
30361:
30353:
30348:
30344:
30319:
30312:as desired.
30299:
30296:
30293:
30288:
30284:
30280:
30269:
30263:
30238:
30231:
30227:
30222:
30218:
30214:
30206:
30202:
30197:
30191:
30188:
30185:
30181:
30176:
30170:
30160:
30154:
30146:
30136:
30130:
30124:
30104:
30097:
30093:
30088:
30084:
30081:
30076:
30072:
30068:
30065:
30062:
30042:
30039:
30028:
30022:
29997:
29994:
29991:
29971:
29968:
29965:
29954:
29948:
29938:
29935:
29932:
29901:
29881:
29871:
29868:
29865:
29845:
29836:
29832:
29828:
29822:
29818:
29814:
29810:
29806:
29803:
29777:
29773:
29769:
29763:
29759:
29755:
29751:
29744:
29732:
29728:
29724:
29718:
29714:
29710:
29706:
29699:
29693:
29673:
29670:
29667:
29664:
29644:
29641:
29636:
29632:
29611:
29608:
29605:
29602:
29582:
29579:
29574:
29570:
29547:
29544:
29541:
29536:
29531:
29527:
29523:
29499:
29495:
29472:
29468:
29445:
29441:
29432:
29414:
29410:
29402:
29385:
29382:
29359:
29355:
29346:
29345:least element
29328:
29324:
29320:
29300:
29295:
29291:
29287:
29282:
29278:
29274:
29252:
29248:
29244:
29239:
29235:
29212:
29208:
29204:
29201:
29198:
29178:
29171:
29167:
29162:
29156:
29153:
29150:
29146:
29142:
29135:
29131:
29126:
29118:
29114:
29110:
29105:
29101:
29096:
29090:
29087:
29084:
29080:
29076:
29069:
29065:
29060:
29054:
29051:
29048:
29044:
29036:
29032:
29028:
29023:
29019:
29014:
29010:
29005:
29001:
28997:
28976:
28970:
28966:
28962:
28957:
28953:
28949:
28944:
28940:
28935:
28924:
28919:
28915:
28894:
28891:
28888:
28885:
28843:
28821:
28810:
28806:
28802:
28798:
28795:
28792:
28789:
28784:
28780:
28775:
28771:
28764:the equality
28751:
28742:
28739:
28733:
28730:
28720:
28716:
28713:
28712:
28691:
28688:
28685:
28682:
28662:
28659:
28654:
28650:
28629:
28626:
28621:
28617:
28596:
28593:
28588:
28584:
28563:
28560:
28557:
28554:
28547:is such that
28534:
28531:
28528:
28508:
28505:
28502:
28499:
28479:
28470:
28467:
28458:
28455:
28450:
28446:
28425:
28405:
28385:
28382:
28379:
28376:
28354:
28350:
28346:
28341:
28337:
28314:
28310:
28306:
28301:
28297:
28288:
28287:partial order
28270:
28266:
28257:
28256:least element
28239:
28236:
28233:
28228:
28223:
28219:
28215:
28194:
28182:
28178:
28167:
28161:
28148:
28139:
28109:
28106:
28103:
28099:
28091:is closed in
28072:
28051:
28047:
28036:
28030:
28027:
28024:
28020:
28016:
28004:
28000:
27994:
27991:
27988:
27984:
27972:
27968:
27959:
27939:
27923:
27896:
27875:
27865:
27859:
27856:
27853:
27849:
27845:
27833:
27829:
27823:
27820:
27817:
27813:
27799:
27789:
27785:
27780:
27757:
27753:
27747:
27744:
27741:
27737:
27733:
27726:
27722:
27717:
27706:
27699:
27695:
27675:
27671:
27666:
27660:
27657:
27654:
27650:
27641:
27621:
27617:
27612:
27601:
27582:
27556:
27532:
27529:
27526:
27503:
27481:
27478:
27475:
27471:
27447:
27444:
27441:
27428:
27414:
27394:
27374:
27344:
27341:
27338:
27327:
27324:
27307:
27304:
27301:
27297:
27274:
27271:
27268:
27264:
27243:
27234:
27211:
27208:
27205:
27201:
27197:
27192:
27189:
27186:
27182:
27159:
27155:
27132:
27129:
27126:
27122:
27094:
27091:
27080:
27074:
27064:
27061:
27058:
27047:
27031:
27028:
27025:
27019:
27016:
26990:
26978:
26975:
26972:
26968:
26944:
26941:
26938:
26915:
26885:
26882:
26879:
26853:
26850:
26847:
26824:
26818:
26815:
26812:
26789:
26769:
26749:
26729:
26703:
26682:
26672:
26671:is dependent
26658:
26631:
26628:
26617:
26611:
26601:
26598:
26595:
26584:
26572:
26568:
26565:
26555:
26550:
26546:
26519:
26515:
26510:
26506:
26485:
26465:
26455:
26448:
26444:
26422:
26401:
26397:
26392:
26388:
26385:
26376:
26371:
26367:
26364:
26355:
26350:
26341:
26338:is given the
26319:
26310:
26307:is given the
26288:
26261:
26257:
26248:
26237:
26233:
26222:
26219:
26216:
26215:Mazur's lemma
26213:
26207:
26204:
26198:
26195:
26193:
26190:
26187:
26184:
26182:
26179:
26177:
26174:
26172:
26169:
26167:
26164:
26163:
26157:
26155:
26151:
26147:
26143:
26139:
26135:
26131:
26126:
26112:
26100:
26097:
26093:
26089:
26086:
26078:
26052:
26049:
26046:
26043:
26040:
26037:
26008:
25999:
25998:weak-* closed
25994:
25980:
25942:
25934:
25932:
25916:
25896:
25887:
25879:
25863:
25843:
25823:
25819:
25815:
25813:
25809:
25805:
25801:
25797:
25793:
25789:
25784:
25780:
25778:
25774:
25770:
25764:
25762:
25758:
25754:
25750:
25746:
25742:
25736:
25723:
25704:
25698:
25690:
25674:
25668:
25662:
25654:
25650:
25631:
25625:
25617:
25614:
25610:
25607:
25603:
25602:
25578:
25570:
25566:
25545:
25542:
25539:
25519:
25510:
25506:
25500:
25492:
25488:
25484:
25481:
25472:
25469:
25465:
25462:
25459:
25453:
25450:
25446:
25439:
25435:
25430:
25426:
25406:
25403:
25400:
25373:
25370:
25367:
25362:
25355:
25351:
25346:
25342:
25320:
25317:
25297:
25294:
25289:
25285:
25281:
25276:
25272:
25251:
25248:
25243:
25240:
25235:
25230:
25227:
25205:
25182:
25179:
25176:
25173:
25150:
25142:
25138:
25130:
25127:is the space
25114:
25106:
25090:
25070:
25062:
25046:
25038:
25035:
25018:
25008:
25004:
25001:
24997:
24993:
24985:
24981:
24977:
24974:
24968:
24948:
24921:
24912:
24911:
24910:
24908:
24886:
24877:
24861:
24841:
24828:
24809:
24804:
24800:
24796:
24790:
24784:
24756:
24752:
24731:
24728:
24725:
24722:
24711:
24707:
24703:
24697:
24691:
24663:
24660:
24657:
24637:
24634:
24623:
24619:
24610:
24607:
24604:
24599:
24592:
24584:
24580:
24575:
24568:
24543:
24540:
24537:
24517:
24495:
24492:
24489:
24484:
24477:
24469:
24465:
24460:
24438:
24435:
24430:
24427:
24424:
24419:
24414:
24410:
24406:
24401:
24396:
24392:
24382:
24380:
24356:
24352:
24342:
24325:
24320:
24317:
24314:
24303:
24297:
24292:
24279:
24270:
24249:
24245:
24237:
24234:
24210:
24190:
24179:
24175:
24154:
24145:
24135:
24134:is compact.
24121:
24113:
24095:
24091:
24082:
24064:
24060:
24054:
24051:
24048:
24044:
24040:
24037:
24011:
24005:
23997:
23989:
23981:
23978:
23972:
23967:
23963:
23942:
23939:
23936:
23933:
23920:
23918:
23914:
23885:
23881:
23877:
23874:
23848:
23838:
23836:
23820:
23800:
23797:
23777:
23757:
23754:
23734:
23731:
23705:
23701:
23698:
23695:
23690:
23686:
23682:
23677:
23673:
23652:
23649:
23615:
23611:
23608:
23600:
23596:
23591:
23589:
23585:
23569:
23549:
23546:
23520:
23496:
23493:
23490:
23463:
23454:
23450:
23446:
23443:
23440:
23437:
23430:
23426:
23423:
23418:
23409:
23405:
23401:
23398:
23395:
23392:
23385:
23376:
23373:
23369:
23357:
23354:
23351:
23347:
23343:
23337:
23334:
23331:
23325:
23306:
23303:
23300:
23297:
23294:
23267:
23264:
23261:
23256:
23251:
23247:
23243:
23238:
23233:
23229:
23208:
23188:
23179:
23158:
23138:
23129:
23127:
23123:
23119:
23109:
23096:
23093:
23085:
23069:
23047:
23043:
23023:
23018:
23013:
23009:
23006:
23001:
22997:
22992:
22987:
22982:
22978:
22947:
22944:
22941:
22935:
22926:
22905:
22897:
22881:
22873:
22851:
22848:
22839:
22836:
22831:
22828:
22825:
22820:
22815:
22811:
22807:
22786:
22774:
22770:
22761:
22757:
22741:
22738:
22715:
22711:
22702:
22701:least element
22682:
22678:
22673:
22667:
22664:
22661:
22657:
22653:
22646:
22642:
22637:
22633:
22624:
22611:
22606:
22602:
22579:
22575:
22571:
22549:
22545:
22541:
22536:
22532:
22528:
22525:
22518:
22514:
22509:
22505:
22485:
22481:
22473:
22469:
22464:
22460:
22457:
22454:
22448:
22438:
22434:
22429:
22425:
22421:
22414:
22407:
22401:
22397:
22393:
22388:
22384:
22377:
22367:
22363:
22358:
22354:
22350:
22330:
22320:
22315:
22311:
22291:
22288:
22266:
22262:
22252:
22250:
22232:
22228:
22207:
22203:
22197:
22193:
22189:
22184:
22180:
22176:
22171:
22167:
22162:
22155:
22150:
22146:
22126:
22123:
22120:
22117:
22095:
22091:
22087:
22082:
22078:
22056:
22048:
22044:
22039:
22035:
22032:
22029:
22023:
22015:
22005:
22000:
21996:
21991:
21971:
21959:
21955:
21934:
21929:
21919:
21914:
21911:
21908:
21903:
21898:
21894:
21890:
21885:
21880:
21876:
21851:
21847:
21842:
21836:
21833:
21830:
21826:
21806:
21792:
21788:
21783:
21779:
21753:
21749:
21746:
21724:
21720:
21700:
21690:
21668:
21664:
21635:
21631:
21626:
21620:
21617:
21614:
21610:
21606:
21604:
21592:
21588:
21586:
21574:
21570:
21565:
21559:
21556:
21553:
21549:
21545:
21543:
21531:
21527:
21525:
21509:
21505:
21503:
21496:
21492:
21465:
21462:
21459:
21454:
21449:
21445:
21441:
21436:
21431:
21427:
21406:
21397:
21384:
21375:
21366:
21349:
21343:
21339:
21335:
21329:
21323:
21320:
21311:
21307:
21304:
21300:
21280:
21264:
21243:
21235:
21232:(such as any
21216:
21213:
21210:
21207:
21204:
21201:
21198:
21195:
21192:
21189:
21167:
21157:
21148:
21145:
21139:
21136:
21116:
21113:
21110:
21095:
21082:
21075:is complete.
21061:
21056:
21052:
21049:
21040:
21035:
21031:
21028:
21019:
21014:
20993:
20983:
20980:
20977:
20973:
20969:
20964:
20950:
20949:product space
20923:
20922:
20916:
20915:
20910:
20908:
20905:
20902:
20899:
20882:
20877:
20873:
20870:
20861:
20856:
20852:
20849:
20840:
20835:
20827:
20810:
20806:
20803:
20794:
20789:
20785:
20765:
20739:
20731:
20704:
20692:
20672:
20663:
20640:
20637:
20634:
20630:
20621:
20612:
20599:
20592:as desired.
20579:
20573:
20567:
20564:
20558:
20552:
20549:
20543:
20540:
20537:
20531:
20511:
20505:
20502:
20499:
20493:
20484:
20481:
20478:
20470:
20466:
20458:Because also
20445:
20439:
20433:
20430:
20424:
20418:
20409:
20406:
20403:
20395:
20391:
20367:
20364:
20361:
20353:
20349:
20345:
20340:
20337:
20334:
20329:
20322:
20319:
20316:
20308:
20304:
20299:
20294:
20289:
20286:
20283:
20278:
20271:
20263:
20259:
20255:
20249:
20241:
20237:
20232:
20227:
20222:
20219:
20216:
20211:
20206:
20199:
20191:
20187:
20183:
20177:
20169:
20165:
20160:
20156:
20152:
20147:
20142:
20139:
20136:
20131:
20126:
20121:
20117:
20113:
20109:
20105:
20100:
20095:
20091:
20087:
20084:
20065:
20054:
20049:
20045:
20041:
20037:
20014:
20008:
20005:
19999:
19993:
19990:
19981:
19975:
19972:
19966:
19960:
19954:
19906:
19900:
19897:
19891:
19885:
19879:
19872:
19867:
19863:
19859:
19855:
19848:implies that
19835:
19815:
19812:
19809:
19806:
19787:
19774:
19771:
19768:
19762:
19729:
19721:
19718:
19698:
19690:
19659:
19653:
19650:
19644:
19638:
19627:
19623:
19602:
19596:
19590:
19582:
19579:
19576:
19571:
19564:
19556:
19552:
19547:
19542:
19536:
19528:
19524:
19500:
19494:
19486:
19483:
19480:
19475:
19468:
19460:
19456:
19451:
19446:
19440:
19432:
19428:
19407:
19404:
19401:
19398:
19377:
19370:
19362:
19358:
19354:
19348:
19340:
19336:
19331:
19311:
19299:
19295:
19269:
19258:
19255:
19250:
19247:
19244:
19239:
19234:
19230:
19226:
19221:
19216:
19212:
19204:Define a net
19191:
19185:
19179:
19176:
19170:
19164:
19161:
19155:
19152:
19149:
19143:
19133:
19132:as desired.
19119:
19113:
19110:
19104:
19101:
19095:
19089:
19086:
19041:
19038:
19032:
19023:
19020:
19012:
19008:
19000:Because also
18987:
18981:
18975:
18972:
18963:
18960:
18952:
18948:
18912:
18909:
18901:
18897:
18888:
18879:
18875:
18861:
18858:
18855:
18850:
18843:
18840:
18832:
18828:
18823:
18813:
18803:
18795:
18791:
18787:
18768:
18757:
18750:
18742:
18738:
18733:
18729:
18721:
18718:
18715:
18710:
18703:
18695:
18691:
18687:
18683:
18673:
18658:
18655:
18650:
18647:
18644:
18639:
18632:
18624:
18620:
18615:
18610:
18604:
18596:
18592:
18569:
18566:
18563:
18558:
18553:
18546:
18538:
18534:
18529:
18525:
18521:
18511:
18493:
18485:
18481:
18477:
18474:
18453:
18445:
18438:
18430:
18426:
18421:
18417:
18390:
18384:
18381:
18378:
18369:
18363:
18357:
18331:
18325:
18319:
18312:
18305:
18297:
18293:
18288:
18284:
18264:
18236:
18230:
18221:
18213:
18209:
18188:
18168:
18165:
18162:
18143:
18130:
18124:
18104:
18071:
18068:
18048:
18042:
18036:
18033:
18030:
18024:
18021:
18015:
18005:
17992:
17986:
17983:
17977:
17968:
17965:
17957:
17953:
17942:
17936:
17933:
17930:
17924:
17915:
17912:
17909:
17901:
17897:
17892:
17886:
17880:
17871:
17863:
17859:
17854:
17848:
17842:
17833:
17825:
17821:
17801:
17776:
17773:
17753:
17750:
17747:
17744:
17736:
17733:
17730:
17727:
17724:
17721:
17718:
17698:
17695:
17692:
17689:
17644:
17638:
17629:
17621:
17617:
17596:
17591:
17564:
17556:
17552:
17531:
17526:
17523:
17520:
17515:
17508:
17500:
17496:
17491:
17470:
17457:
17449:
17445:
17424:
17402:
17398:
17390:
17368:
17365:
17359:
17351:
17347:
17326:
17323:
17320:
17317:
17308:
17295:
17292:
17289:
17286:
17283:
17280:
17260:
17240:
17220:
17211:
17207:
17204:
17184:
17179:
17152:
17126:
17103:
17100:
17097:
17092:
17087:
17083:
17079:
17074:
17069:
17065:
17042:
17032:
17029:
17020:
17019:
17013:
17012:
17007:
16993:
16966:
16963:
16960:
16956:
16952:
16947:
16845:over a field
16832:
16806:
16798:
16775:
16761:is closed in
16742:
16716:
16696:
16693:
16690:
16684:
16678:
16658:
16655:
16635:
16615:
16612:
16606:
16598:
16594:
16590:
16584:
16576:
16572:
16551:
16528:
16522:
16514:
16511:
16508:
16503:
16496:
16488:
16484:
16479:
16457:
16454:
16451:
16448:
16428:
16425:
16422:
16416:
16410:
16390:
16385:
16381:
16377:
16374:
16354:
16351:
16329:
16325:
16302:
16299:
16296:
16291:
16286:
16282:
16278:
16254:
16250:
16246:
16243:
16235:
16232:
16231:
16213:
16209:
16187:
16183:
16180:
16174:
16168:
16165:
16160:
16156:
16152:
16149:
16145:
16125:
16113:
16109:
16088:
16085:
16077:
16061:
16058:
16055:
16035:
16032:
16029:
16021:
16018:
16017:
16016:
16003:
15998:
15994:
15990:
15987:
15962:
15959:
15938:
15932:
15928:
15924:
15918:
15912:
15909:
15904:
15894:
15891:
15887:
15883:
15876:
15872:
15867:
15856:
15843:
15823:
15818:
15785:
15781:
15772:
15768:
15761:
15757:
15752:
15731:
15704:
15694:
15684:
15681:
15678:
15674:
15665:
15638:
15635:
15632:
15613:
15610:
15607:
15595:
15591:
15584:
15563:
15551:
15547:
15534:
15530:
15524:
15521:
15518:
15514:
15510:
15508:
15493:
15490:
15487:
15477:
15473:
15469:
15461:
15457:
15446:
15432:
15429:
15426:
15422:
15418:
15413:
15410:
15407:
15402:
15397:
15393:
15389:
15377:
15375:
15360:
15357:
15354:
15344:
15340:
15336:
15330:
15324:
15321:
15303:
15293:
15289:
15252:
15250:
15235:
15232:
15221:
15215:
15205:
15202:
15199:
15191:
15173:
15163:
15157:
15106:
15099:
15090:
15086:
15081:
15068:
15045:
15042:
15039:
15035:
15031:
15026:
14993:
14985:
14980:
14978:
14974:
14973:as desired.
14960:
14953:
14949:
14944:
14940:
14934:
14928:
14908:
14903:
14899:
14895:
14884:
14878:
14864:
14851:
14848:
14837:
14831:
14821:
14818:
14815:
14807:
14803:
14798:
14793:
14789:
14785:
14781:
14777:
14773:
14769:
14764:
14760:
14753:
14749:
14745:
14739:
14735:
14731:
14727:
14723:
14716:
14710:
14703:
14699:
14695:
14689:
14685:
14674:
14668:
14656:
14652:
14648:
14626:
14623:
14620:
14609:
14603:
14593:
14590:
14587:
14573:
14557:
14536:
14533:
14530:
14527:
14519:
14515:
14511:
14490:
14478:
14474:
14452:
14449:
14446:
14426:
14423:
14418:
14414:
14393:
14388:
14384:
14380:
14377:
14357:
14354:
14349:
14345:
14324:
14317:
14313:
14308:
14304:
14298:
14292:
14272:
14266:
14260:
14256:
14249:
14241:
14214:
14205:
14201:
14198:
14178:
14175:
14172:
14152:
14149:
14146:
14143:
14119:
14115:
14110:
14106:
14102:
14093:
14089:
14083:
14056:
14049:
14045:
14040:
14034:
14031:
14028:
14024:
14020:
14011:
14002:
13986:
13951:
13948:
13945:
13941:
13937:
13932:
13905:
13902:
13899:
13896:
13887:
13885:
13867:
13862:
13852:
13842:
13839:
13836:
13832:
13824:
13823:product space
13802:
13794:
13780:
13773:
13769:
13764:
13758:
13755:
13752:
13748:
13744:
13735:
13727:
13726:
13725:
13711:
13704:
13700:
13695:
13689:
13686:
13683:
13679:
13657:
13651:
13647:
13643:
13637:
13631:
13625:
13613:
13609:
13606:
13602:
13595:
13582:
13579:
13568:
13562:
13552:
13549:
13546:
13535:
13523:
13519:
13516:
13490:
13474:
13465:
13464:compact space
13462:
13441:
13437:
13433:
13425:
13417:
13409:
13406:
13384:
13370:
13366:
13361:
13352:
13332:
13328:
13323:
13317:
13314:
13311:
13307:
13299:
13298:product space
13289:
13276:
13250:
13246:
13243:
13235:
13219:
13199:
13196:
13176:
13155:
13152:
13149:
13138:
13132:
13122:
13119:
13116:
13084:
13080:
13077:
13057:
13048:
13044:
13035:
13014:
13005:
13001:
12992:
12988:
12983:
12979:
12970:
12957:
12948:
12944:
12939:
12935:
12924:
12911:
12902:
12898:
12889:
12882:
12869:
12866:
12855:
12849:
12839:
12836:
12833:
12822:
12810:
12806:
12803:
12777:
12765:
12761:
12738:
12734:
12713:
12708:
12704:
12700:
12691:
12682:
12660:
12639:
12636:
12631:
12627:
12623:
12620:
12600:
12597:
12592:
12588:
12567:
12564:
12561:
12558:
12551:so for every
12538:
12535:
12527:
12511:
12508:
12488:
12468:
12442:
12433:
12417:
12404:
12390:
12382:
12366:
12356:
12336:
12327:
12323:
12319:
12299:
12295:
12290:
12284:
12281:
12278:
12274:
12266:
12265:product space
12262:
12259:
12237:
12216:
12213:
12208:
12204:
12200:
12197:
12177:
12174:
12169:
12165:
12145:
12142:
12139:
12136:
12129:If for every
12127:
12114:
12104:
12101:
12090:
12084:
12074:
12071:
12068:
12057:
12045:
12041:
12038:
12012:
11996:
11987:
11949:
11941:
11922:
11919:
11911:
11903:
11895:
11892:
11870:
11858:
11854:
11833:
11830:
11805:
11787:
11740:
11720:
11704:
11690:
11680:
11672:
11658:
11638:
11619:
11609:
11602:
11565:
11555:
11534:
11531:
11520:
11514:
11504:
11501:
11498:
11471:
11468:
11457:
11451:
11441:
11438:
11435:
11406:
11403:
11395:
11387:
11379:
11376:
11351:
11347:
11326:
11320:
11317:
11314:
11311:
11305:
11299:
11277:
11264:
11258:
11250:
11210:
11206:
11202:
11196:
11190:
11180:
11177:
11166:
11160:
11150:
11147:
11144:
11120:
11117:
11108:
11107:
11094:
11091:
11080:
11074:
11064:
11061:
11058:
11043:
11030:
11020:
11017:
11014:
11010:
10987:
10983:
10977:
10974:
10971:
10967:
10946:
10943:
10940:
10937:
10908:is closed in
10893:
10889:
10868:
10858:
10855:
10852:
10848:
10825:
10821:
10815:
10812:
10809:
10805:
10797:that the set
10796:
10778:
10774:
10768:
10765:
10762:
10758:
10727:
10706:
10701:
10697:
10677:
10674:
10671:
10668:
10660:
10655:
10642:
10639:
10632:pointwise on
10619:
10599:
10592:converges to
10577:
10574:
10571:
10566:
10561:
10557:
10553:
10528:
10516:converges in
10500:
10494:
10486:
10483:
10480:
10475:
10468:
10460:
10456:
10451:
10429:
10426:
10423:
10420:
10412:
10411:
10410:
10396:
10391:
10388:
10385:
10380:
10373:
10365:
10361:
10356:
10351:
10346:
10343:
10340:
10335:
10330:
10325:
10321:
10317:
10311:
10300:
10279:
10268:
10263:
10260:
10257:
10252:
10247:
10243:
10239:
10234:
10228:
10197:
10191:
10188:
10182:
10174:
10143:
10131:converges in
10115:
10107:
10093:
10088:
10085:
10082:
10077:
10072:
10068:
10064:
10059:
10053:
10026:
10023:
10020:
10017:
10009:
10008:
10007:
10005:
9990:
9982:
9979:
9976:
9971:
9966:
9962:
9958:
9949:then the net
9936:
9926:
9923:
9920:
9916:
9912:
9907:
9878:
9874:
9853:
9833:
9828:
9825:
9822:
9817:
9812:
9808:
9804:
9795:
9779:
9771:
9767:
9744:
9741:
9738:
9734:
9730:
9725:
9709:
9708:
9704:
9691:
9685:
9679:
9671:
9667:
9663:
9660:
9637:
9631:
9624:
9619:
9615:
9611:
9607:
9584:
9578:
9570:
9567:
9564:
9559:
9554:
9549:
9545:
9541:
9537:
9533:
9511:
9508:
9488:
9480:
9476:
9455:
9449:
9446:
9443:
9423:
9418:
9414:
9410:
9407:
9388:
9376:
9373:
9370:
9365:
9360:
9355:
9351:
9347:
9343:
9339:
9334:
9330:
9325:
9321:
9317:
9313:
9291:
9287:
9259:
9256:
9253:
9248:
9243:
9238:
9234:
9230:
9226:
9222:
9198:
9195:
9192:
9187:
9182:
9177:
9173:
9169:
9165:
9161:
9139:
9136:
9130:
9127:
9122:
9118:
9114:
9111:
9091:
9069:
9065:
9044:
9038:
9035:
9032:
9024:
9008:
9002:
8994:
8990:
8967:
8963:
8940:
8936:
8915:
8912:
8909:
8887:
8883:
8862:
8859:
8831:
8828:
8820:
8804:
8798:
8795:
8792:
8782:
8766:
8760:
8757:
8752:
8748:
8727:
8705:
8702:
8699:
8694:
8689:
8685:
8681:
8676:
8671:
8667:
8659:
8654:
8653:
8649:
8636:
8628:
8622:
8616:
8604:
8600:
8595:
8583:
8579:
8565:
8554:
8548:
8538:
8533:
8511:
8497:
8494:
8482:
8481:homeomorphism
8461:
8455:
8449:
8446:
8443:
8438:
8433:
8429:
8425:
8419:
8414:
8411:
8408:
8403:
8398:
8394:
8390:
8384:
8368:
8365:
8362:
8357:
8352:
8348:
8344:
8326:
8320:
8317:
8314:
8310:
8306:
8302:
8291:
8288:
8285:
8281:
8276:
8255:
8252:
8249:
8245:
8237:
8234:
8207:
8201:
8194:
8190:
8187:
8184:
8164:
8157:
8142:
8139:
8136:
8127:
8114:
8111:
8108:
8105:
8081:
8077:
8072:
8068:
8062:
8056:
8034:
8024:
8021:
7997:
7993:
7988:
7982:
7979:
7976:
7972:
7951:
7946:
7936:
7926:
7923:
7920:
7916:
7912:
7905:
7901:
7896:
7890:
7887:
7884:
7880:
7857:
7854:
7851:
7846:
7841:
7837:
7833:
7822:
7806:
7779:
7759:
7739:
7719:
7716:
7709:and function
7696:
7676:
7670:
7664:
7645:
7632:
7624:
7589:
7586:
7583:
7561:
7532:
7528:
7519:
7516:
7513:
7508:
7503:
7499:
7495:
7490:
7485:
7481:
7455:
7452:
7449:
7445:
7441:
7436:
7409:
7406:
7403:
7388:
7384:
7382:
7381:inverse limit
7378:
7374:
7370:
7352:
7318:
7315:
7312:
7308:
7280:
7277:
7274:
7270:
7266:
7261:
7231:
7226:
7223:
7220:
7215:
7210:
7206:
7202:
7178:
7174:
7154:
7141:
7135:
7107:
7104:
7101:
7074:
7071:
7068:
7064:
7041:
7038:
7035:
7030:
7025:
7021:
7017:
7012:
7007:
7003:
6972:
6971:
6957:
6952:
6949:
6946:
6935:
6929:
6907:
6895:
6891:
6880:
6866:
6844:
6830:belonging to
6809:
6806:
6803:
6796:: A function
6773:
6772:
6771:
6770:
6756:
6745:
6723:
6694:
6691:
6688:
6683:
6678:
6674:
6670:
6665:
6660:
6656:
6648:
6632:
6605:
6602:
6599:
6595:
6585:
6584:
6580:
6567:
6564:
6561:
6558:
6535:
6532:
6529:
6526:
6523:
6520:
6498:
6488:
6485:
6465:
6445:
6422:
6419:
6411:
6403:
6395:
6392:
6370:
6358:
6354:
6334:
6331:
6322:
6321:
6315:
6314:
6310:
6296:
6289:
6285:
6280:
6274:
6271:
6268:
6264:
6237:
6229:
6225:
6221:
6217:
6201:
6191:
6188:
6185:
6181:
6173:
6149:
6127:
6112:
6108:
6104:
6100:
6096:
6092:
6089:
6085:
6081:
6071:
6069:
6066:
6062:
6058:
6054:
6048:
6045:
6043:
6021:
5994:
5983:
5969:
5943:
5932:
5930:
5926:
5925:operator norm
5904:
5883:
5867:
5865:
5849:
5823:
5802:
5782:
5762:
5754:
5738:
5726:
5713:
5690:
5686:
5665:
5661:
5656:
5652:
5649:
5640:
5635:
5631:
5628:
5619:
5614:
5591:
5587:
5578:
5561:
5556:
5552:
5549:
5540:
5535:
5531:
5528:
5519:
5514:
5493:
5484:
5462:
5458:
5455:
5446:
5441:
5437:
5415:
5411:
5402:
5388:
5379:
5357:
5353:
5350:
5341:
5336:
5332:
5310:
5306:
5285:
5281:
5276:
5272:
5269:
5260:
5255:
5251:
5248:
5239:
5234:
5207:
5180:
5158:
5154:
5151:
5142:
5137:
5133:
5113:
5104:
5082:
5078:
5075:
5066:
5061:
5057:
5035:
5031:
5022:
5006:
5002:
4999:
4990:
4985:
4981:
4959:
4955:
4946:
4929:
4925:
4922:
4913:
4908:
4904:
4883:
4879:
4876:
4867:
4862:
4858:
4832:
4811:
4802:
4793:
4778:
4758:
4749:
4740:
4723:
4719:
4716:
4707:
4702:
4698:
4676:
4672:
4651:
4646:
4643:
4639:
4618:
4615:
4607:
4591:
4569:
4566:
4562:
4558:
4555:
4547:
4531:
4522:
4513:
4496:
4492:
4489:
4480:
4475:
4471:
4449:
4445:
4436:
4419:
4415:
4394:
4385:
4362:
4358:
4354:
4345:
4341:
4336:
4332:
4328:
4319:
4315:
4306:
4285:
4280:
4276:
4272:
4269:
4243:
4239:
4236:
4228:
4212:
4192:
4172:
4151:
4148:
4145:
4134:
4128:
4118:
4115:
4112:
4080:
4076:
4073:
4052:
4047:
4043:
4039:
4030:
4002:
3998:
3993:
3989:
3968:
3964:
3959:
3955:
3952:
3943:
3938:
3934:
3931:
3922:
3917:
3895:
3890:
3886:
3883:
3874:
3869:
3865:
3862:
3853:
3848:
3825:
3821:
3798:
3794:
3790:
3781:
3772:
3758:
3755:
3744:
3738:
3710:
3704:
3684:
3680:
3676:
3673:
3665:
3657:
3649:
3646:
3642:
3596:
3588:
3584:
3538:
3532:
3523:
3515:
3511:
3490:
3487:
3484:
3481:
3461:
3458:
3447:
3441:
3416:
3407:
3403:
3400:
3380:
3376:
3371:
3367:
3364:
3355:
3350:
3346:
3343:
3334:
3329:
3308:
3282:
3259:
3256:
3253:
3248:
3243:
3239:
3235:
3230:
3225:
3221:
3194:
3190:
3187:
3167:
3158:
3136:
3132:
3129:
3120:
3115:
3111:
3085:
3076:
3075:
3074:
3061:
3056:
3052:
3048:
3043:
3040:
3037:
3033:
3009:
3000:
2996:
2987:
2983:
2978:
2974:
2953:
2949:
2939:
2935:
2932:
2928:
2907:
2886:
2882:
2879:
2868:
2862:
2852:
2849:
2846:
2835:
2823:
2819:
2816:
2812:
2808:
2799:
2791:
2775:
2765:
2761:
2758:
2754:
2745:
2732:
2709:
2705:
2702:
2691:
2685:
2673:
2669:
2665:
2662:
2651:
2645:
2642:
2639:
2635:
2631:
2626:
2623:
2619:
2611:
2595:
2585:
2581:
2578:
2574:
2553:
2532:
2528:
2525:
2514:
2508:
2498:
2495:
2492:
2481:
2469:
2465:
2462:
2458:
2454:
2449:
2445:
2437:
2436:
2435:
2421:
2401:
2392:
2379:
2375:
2365:
2361:
2358:
2354:
2346:
2330:
2326:
2316:
2312:
2309:
2305:
2296:
2279:
2269:
2265:
2262:
2258:
2248:
2234:
2214:
2207:converges to
2188:
2179:
2163:
2154:
2150:
2147:
2127:
2123:
2119:
2116:
2107:
2102:
2098:
2077:
2072:
2068:
2065:
2056:
2051:
2047:
2044:
2035:
2030:
2003:
1994:
1990:
1974:
1970:
1965:
1961:
1958:
1949:
1944:
1940:
1937:
1928:
1923:
1901:
1896:
1892:
1889:
1880:
1875:
1871:
1868:
1859:
1854:
1845:
1828:
1823:
1819:
1816:
1807:
1802:
1798:
1795:
1786:
1781:
1760:
1756:
1751:
1747:
1744:
1735:
1730:
1726:
1723:
1714:
1709:
1700:
1697:
1681:
1677:
1673:
1670:
1661:
1656:
1652:
1645:
1623:
1613:
1611:
1607:
1603:
1587:
1575:
1550:
1534:
1509:
1490:
1477:
1473:
1468:
1464:
1461:
1452:
1447:
1443:
1440:
1431:
1426:
1404:
1394:
1390:
1387:
1383:
1362:
1340:
1336:
1315:
1306:
1284:
1280:
1277:
1268:
1263:
1259:
1252:
1236:
1229:of origin in
1216:
1209:
1192:
1188:
1185:
1174:
1168:
1158:
1155:
1152:
1141:
1129:
1125:
1122:
1118:
1114:
1109:
1105:
1097:
1081:
1072:
1064:
1060:
1056:
1050:
1036:
1028:
1016:
1003:
997:
991:
969:
966:
963:
958:
951:
943:
939:
934:
912:
892:
885:converges to
870:
866:
845:
840:
837:
834:
829:
824:
820:
816:
811:
806:
802:
793:
777:
769:
753:
749:
745:
742:
733:
728:
724:
698:
676:
672:
669:
660:
655:
651:
625:
616:
594:
590:
581:
560:
551:
542:
538:
522:
513:
509:
507:
490:
485:
481:
478:
475:
471:
467:
458:
454:
451:
447:
423:
417:
398:
387:
383:
380:
377:
373:
338:
334:
331:
328:
324:
320:
317:
314:
310:
300:
299:
283:
275:
253:
210:
200:
196:
192:
182:
180:
176:
172:
168:
164:
160:
156:
151:
149:
145:
141:
138:
134:
133:Stefan Banach
112:
109:
106:
97:
89:
83:
75:
71:
69:
65:
61:
57:
53:
49:
45:
41:
38:
34:
30:
26:
22:
32123:Polynomially
32052:Grothendieck
32045:tame Fréchet
31995:Bornological
31855:Linear cone
31847:Convex cone
31822:Banach disks
31764:Sesquilinear
31631:
31619:Main results
31609:Vector space
31564:Completeness
31559:Banach space
31464:Balanced set
31438:Distribution
31376:Applications
31243:
31229:Krein–Milman
31214:Closed graph
30965:Mackey–Arens
30959:
30953:Main results
30769:. Retrieved
30743:
30704:
30666:
30636:
30602:
30566:
30535:
30516:
30481:
30455:. Retrieved
30443:
30439:
30411:
30399:
30347:
28719:
28710:
28709:
28166:
27705:
27600:
27323:
26454:
26236:
26128:
25825:
25803:
25799:
25785:
25760:
25756:
25748:
25738:
24980:Banach space
24876:normed space
24854:Assume that
24853:
24845:Consequences
24383:
24343:
24136:
23925:
23839:
23592:
23130:
23115:
22625:
22254:In fact, if
22253:
21398:
21101:
20924:
20694:
20618:
19134:
18006:
17766:in place of
17309:
17118:is a net in
17021:
16732:
16317:is a net in
16233:
16019:
15858:
14981:
14977:Proof of (2)
14976:
14975:
14865:
14337:Recall that
13999:coordinate (
13888:
13884:Proof of (1)
13883:
13882:
13295:
12926:
12925:
12410:Assume that
12409:
12358:
12128:
11706:
11678:
11601:identity map
11247:denotes the
11109:
11045:
11044:
10656:
10543:
10158:
9710:
9706:
9705:
8902:at an index
8781:directed set
8655:
8651:
8650:
8129:If a subset
8128:
7823:
7389:
7246:
6586:
6582:
6581:
6458:centered at
6323:
6077:
6046:
5934:
5869:
5730:
5705:is compact.
5577:complete TVS
5172:topology on
4824:a subset of
3274:is a net in
3024:
2393:
2295:dual pairing
2249:
2091:is equal to
1844:complete TVS
1842:is always a
1614:
1549:real numbers
1501:
1208:neighborhood
1053:necessarily
1018:
514:
365:defined by
202:
178:
174:
170:
154:
152:
140:normed space
81:
72:
32:
28:
18:
32117:Quasinormed
32030:FK-AK space
31924:Linear span
31919:Convex hull
31904:Affine hull
31707:Almost open
31647:Hahn–Banach
31393:Heat kernel
31383:Hardy space
31290:Trace class
31204:Hahn–Banach
31166:Topological
30944:Ultrastrong
30927:Strong dual
30835:Dual system
28289:defined by
28170:This tuple
27328:dual space
26130:Compactness
26000:subsets of
25949:: whenever
25689:trace class
24684:defined by
24556:such that
24530:indexed by
24167:denoted by
22699:the unique
21480:such that
19287:by letting
19136:Proof that
18008:Proof that
16020:Observation
12927:Proof that
11988:). Define
11708:Proposition
7385:projections
7128:defined by
6879:-indexed) "
6091:convergence
2725:bipolar of
2250:The triple
1846:; however,
1606:dual system
508:called the
506:dual system
195:Dual system
25:mathematics
32232:Categories
32157:Stereotype
32015:(DF)-space
32010:Convenient
31749:Functional
31717:Continuous
31702:Linear map
31642:F. Riesz's
31584:Linear map
31326:C*-algebra
31141:Properties
30875:Topologies
30830:Dual space
30470:References
30404:Köthe 1983
30352:Rudin 1991
29657:for every
29595:for every
28675:for every
28369:for every
27638:is also a
27407:such that
27325:continuous
26027:such that
25771:case (see
25733:See also:
25419:such that
24451:and a net
23926:For every
23126:metrizable
21367:subset of
19391:for every
17682:for every
14136:for every
13236:(that is,
12613:such that
10930:for every
10413:for every
10010:for every
9866:and every
8156:partitions
8098:for every
8049:such that
7732:"plugging
6995:: A tuple
5984:impossible
5862:(with the
5050:is also a
4514:subset of
4437:Show that
4229:(that is,
3773:Show that
3474:for every
3077:Show that
2966:Note that
2434:and let:
1773:The space
1694:then this
1328:Moreover,
189:See also:
44:dual space
32173:Uniformly
32132:Reflexive
31980:Barrelled
31976:Countably
31888:Symmetric
31786:Transpose
31300:Unbounded
31295:Transpose
31253:Operators
31182:Separable
31177:Reflexive
31162:Algebraic
31148:Barrelled
31002:Total set
30888:Dual norm
30855:Polar set
30685:853623322
30665:(2006) .
30655:175294365
30625:840278135
30554:144216834
30508:840293704
30480:(1983) .
30339:Citations
30320:◼
30294:≤
30281:≤
30189:∈
30182:∏
30161:∈
30098:∙
30085:∏
30069:∩
30066:∈
30040:≤
29995:∈
29966:≤
29936:∈
29906:#
29869:∈
29841:#
29833:∩
29815:∩
29807:∈
29782:#
29774:∩
29756:∩
29745:⊆
29737:′
29729:∩
29711:∩
29700:⊆
29668:∈
29642:≤
29606:∈
29545:∈
29383:⊆
29321:∩
29288:∈
29275:∩
29245:∈
29240:∙
29205:∩
29202:⊆
29154:∈
29147:∏
29111:∈
29106:∙
29097:⋂
29088:∈
29081:∏
29052:∈
29045:∏
29029:∈
29024:∙
29015:⋂
28998:∩
28963:∈
28958:∙
28889:∈
28796:∈
28772:∩
28746:∞
28734:⊆
28686:∈
28660:≤
28594:≤
28558:∈
28503:∈
28474:∞
28462:→
28451:∙
28380:∈
28347:≤
28315:∙
28307:≤
28302:∙
28237:∈
28183:∙
28144:#
28107:∈
28100:∏
28077:#
28034:∖
28028:∈
28021:∏
28017:×
27992:∈
27985:∏
27969:∩
27964:#
27928:#
27901:#
27863:∖
27857:∈
27850:∏
27846:×
27821:∈
27814:∏
27745:∈
27738:∏
27658:∈
27651:∏
27583:ν
27561:#
27533:ν
27482:ν
27476:∘
27448:ν
27375:ν
27353:′
27345:σ
27308:σ
27302:∘
27275:σ
27269:∘
27239:#
27212:σ
27206:∘
27193:τ
27187:∘
27160:∘
27133:τ
27127:∘
27092:≤
27062:∈
27040:′
27032:σ
27020:∈
26979:σ
26973:∘
26945:σ
26894:′
26886:σ
26854:σ
26819:σ
26750:σ
26730:τ
26708:#
26629:≤
26599:∈
26577:′
26569:∈
26551:∘
26524:#
26511:∘
26466:τ
26427:′
26381:′
26368:σ
26360:′
26324:′
26293:′
26266:′
26258:⊆
26253:′
26101:∈
26094:⋂
26090:∩
26053:∈
26041:∩
26013:′
25892:′
25781:Hausdorff
25769:separable
25606:reflexive
25579:μ
25515:′
25501:μ
25485:∈
25473:μ
25460:∫
25457:→
25454:μ
25427:∫
25404:∈
25379:∞
25298:…
25186:∞
25151:μ
25013:′
24994:σ
24984:reflexive
24953:′
24926:′
24907:dual norm
24891:′
24829:◼
24805:∙
24801:λ
24763:′
24726:∈
24708:λ
24667:→
24624:∙
24620:λ
24616:→
24608:∈
24541:∈
24493:∈
24436:∈
24428:∈
24411:λ
24397:∙
24393:λ
24363:′
24318:∈
24287:↦
24265:→
24256:′
24186:′
24150:′
24052:∈
24045:∏
24015:‖
24009:‖
24006:≤
23982:∈
23937:∈
23853:′
23710:′
23702:∈
23699:…
23625:→
23620:′
23525:′
23500:⟩
23497:⋅
23491:⋅
23488:⟨
23480:in which
23460:⟩
23441:−
23435:⟨
23415:⟩
23396:−
23390:⟨
23374:−
23363:∞
23348:∑
23326:ρ
23304:∈
23273:∞
23234:∙
23184:′
23118:separable
23048:∙
23019:∘
23007:
22983:∘
22956:#
22945:
22931:#
22918:(because
22855:∞
22843:→
22829:∈
22775:∙
22739:⊆
22665:∈
22658:∏
22647:∙
22634:∏
22572:⋂
22542:∈
22529:∩
22519:∙
22506:∏
22474:∙
22461:∏
22458:⊆
22439:∙
22426:∏
22394:∈
22389:∙
22368:∙
22355:∏
22321:
22233:∙
22190:∈
22185:∙
22121:∈
22088:∈
22083:∙
22049:∙
22036:∏
22033:⊆
22006:∈
22001:∙
21920:∈
21912:∈
21881:∙
21834:∈
21827:∏
21793:∙
21780:∏
21758:#
21725:∘
21669:∙
21618:∈
21611:∏
21607:∩
21597:′
21557:∈
21550:∏
21546:∩
21536:#
21514:#
21497:∘
21463:∈
21432:∙
21380:#
21336:⊆
21316:#
21308:∈
21269:#
21214:∈
21152:∞
21114:⊆
21083:◼
21045:#
21032:σ
21024:#
20981:∈
20974:∏
20901:Hausdorff
20866:#
20853:σ
20845:#
20799:#
20786:σ
20744:#
20728:When the
20709:#
20677:#
20638:∈
20631:∏
20600:◼
20491:→
20471:∙
20416:→
20396:∙
20354:∙
20338:∈
20287:∈
20220:∈
20140:∈
20096:∙
20088:∘
20050:∙
19877:→
19868:∙
19738:→
19730:×
19691:×
19633:→
19628:∙
19588:→
19580:∈
19529:∙
19492:→
19484:∈
19433:∙
19402:∈
19270:×
19262:→
19248:∈
19217:∙
19030:→
19013:∙
18970:→
18953:∙
18902:∙
18859:∈
18719:∈
18662:→
18648:∈
18597:∙
18567:∈
18486:∙
18478:∘
18431:∙
18317:→
18298:∙
18228:→
18214:∙
18080:→
17975:→
17958:∙
17922:→
17902:∙
17878:→
17864:∙
17840:→
17826:∙
17711:By using
17693:∈
17636:→
17622:∙
17562:→
17557:∙
17524:∈
17450:∙
17403:∙
17372:→
17352:∙
17321:∈
17290:∈
17216:#
17208:∈
17131:#
17101:∈
17070:∙
17033:∈
16964:∈
16957:∏
16811:#
16747:#
16717:◼
16691:∈
16648:) subset
16613:⊆
16591:∈
16520:→
16512:∈
16452:∈
16423:⊆
16378:∈
16300:∈
16247:∈
16181:⊆
16153:∈
16059:⊆
16033:⊆
15925:⊆
15895:∈
15844:◼
15790:#
15777:#
15769:∩
15682:∈
15675:∏
15611:∈
15522:∈
15515:∏
15491:∈
15470:∈
15430:∈
15423:∏
15419:∈
15411:∈
15358:∈
15337:∈
15294:∈
15233:≤
15203:∈
15164:∈
15043:∈
15036:∏
14998:#
14941:∈
14896:≤
14849:≤
14819:∈
14808:≤
14621:≤
14591:∈
14531:∈
14450:∈
14381:∈
14305:∈
14210:#
14202:∈
14176:∈
14147:∈
14107:⊆
14098:#
14032:∈
14025:∏
14021:⊆
14016:#
13962:→
13949:∈
13942:∏
13900:∈
13840:∈
13833:∏
13807:#
13756:∈
13749:∏
13745:⊆
13740:#
13687:∈
13680:∏
13644:⊆
13618:#
13610:∈
13580:≤
13550:∈
13528:#
13520:∈
13479:#
13461:Hausdorff
13434:≤
13410:∈
13315:∈
13308:∏
13277:◼
13255:′
13247:∈
13150:≤
13120:∈
13089:#
13081:∈
13053:′
13045:⊆
13040:#
13010:′
13002:∩
12997:#
12984:∘
12953:#
12940:∘
12907:′
12899:∩
12894:#
12867:≤
12837:∈
12815:′
12807:∈
12766:∘
12739:∘
12709:∘
12696:#
12665:#
12624:∈
12562:∈
12461:and that
12447:′
12341:#
12282:∈
12275:∏
12242:#
12201:∈
12140:∈
12102:≤
12072:∈
12050:#
12042:∈
12001:#
11938:with its
11920:≤
11896:∈
11502:∈
11469:≤
11439:∈
11380:∈
11318:∈
11203:⊆
11178:≤
11148:∈
11092:≤
11062:∈
11018:∈
11011:∏
10975:∈
10968:∏
10941:∈
10856:∈
10849:∏
10813:∈
10806:∏
10766:∈
10759:∏
10707:⊆
10672:∈
10575:∈
10492:→
10484:∈
10424:∈
10389:∈
10344:∈
10261:∈
10098:→
10086:∈
10021:∈
10004:converges
9988:→
9980:∈
9924:∈
9917:∏
9826:∈
9742:∈
9735:∏
9677:→
9672:∙
9664:∘
9629:→
9620:∙
9576:→
9568:∈
9486:→
9481:∙
9453:→
9419:∙
9411:∘
9374:∈
9326:∙
9292:∙
9265:∞
9196:∈
9134:→
9123:∙
9115:∘
9070:∙
9042:→
8995:∙
8913:∈
8888:∙
8857:→
8799:≤
8764:→
8753:∙
8703:∈
8672:∙
8620:∖
8573:↦
8552:∖
8539:×
8521:→
8453:∖
8447:∈
8412:∈
8378:↦
8366:∈
8324:∖
8318:∈
8311:∏
8307:×
8289:∈
8282:∏
8270:→
8253:∈
8246:∏
8205:∖
8195:∪
8140:⊆
8109:∈
8069:∈
8025:∈
7980:∈
7973:∏
7924:∈
7917:∏
7913:⊆
7888:∈
7881:∏
7855:∈
7593:→
7525:↦
7517:∈
7486:∙
7466:→
7453:∈
7446:∏
7407:∈
7316:∈
7309:∏
7278:∈
7271:∏
7224:∈
7111:→
7072:∈
7065:∏
7039:∈
7008:∙
6981:→
6950:∈
6896:∙
6813:→
6782:→
6774:Function
6749:→
6692:∈
6661:∙
6645:-indexed
6603:∈
6596:∏
6562:⊆
6533:∈
6420:≤
6396:∈
6272:∈
6265:∏
6242:#
6189:∈
6182:∏
6153:→
6068:Hausdorff
6063:that all
6026:′
5999:′
5948:′
5909:′
5828:′
5714:◼
5691:∘
5645:#
5632:σ
5624:#
5592:∘
5545:#
5532:σ
5524:#
5489:′
5451:′
5438:σ
5416:∘
5384:#
5346:#
5333:σ
5311:∘
5265:#
5252:σ
5244:#
5212:′
5185:′
5147:′
5134:σ
5109:#
5071:#
5058:σ
5036:∘
4995:′
4982:σ
4960:∘
4918:′
4905:σ
4872:′
4859:σ
4837:′
4807:′
4754:′
4712:′
4699:σ
4677:∘
4647:∘
4644:∘
4570:∘
4567:∘
4559:⊆
4527:′
4485:′
4472:σ
4450:∘
4420:∘
4390:′
4363:∘
4350:′
4342:∩
4337:∘
4324:′
4316:∩
4311:#
4281:∘
4273:∈
4262:) and so
4248:′
4240:∈
4146:≤
4116:∈
4085:#
4077:∈
4048:∘
4040:⊆
4035:#
4007:#
3999:⊆
3994:∘
3948:′
3935:σ
3927:′
3879:#
3866:σ
3858:#
3826:∘
3799:∘
3786:#
3756:≤
3674:≤
3650:∈
3634:) subset
3530:→
3485:∈
3459:≤
3412:#
3404:∈
3360:#
3347:σ
3339:#
3287:#
3257:∈
3226:∙
3199:#
3191:∈
3163:#
3125:#
3112:σ
3090:#
3057:∘
3049:⊆
3044:∘
3041:∘
3038:∘
3005:′
2997:∩
2992:#
2979:∘
2944:#
2880:≤
2850:∈
2828:#
2820:∈
2804:#
2770:′
2703:≤
2674:∘
2666:∈
2643:∈
2627:∘
2624:∘
2590:′
2526:≤
2496:∈
2474:′
2466:∈
2450:∘
2370:′
2321:#
2274:′
2193:′
2159:′
2151:∈
2112:′
2099:σ
2061:#
2048:σ
2040:#
2008:′
1954:#
1941:σ
1933:#
1885:′
1872:σ
1864:′
1812:#
1799:σ
1791:#
1740:#
1727:σ
1719:#
1696:Hausdorff
1666:#
1653:σ
1628:#
1602:polar set
1457:#
1444:σ
1436:#
1399:#
1341:∘
1311:′
1273:′
1260:σ
1186:≤
1156:∈
1134:′
1126:∈
1110:∘
1077:′
1055:Hausdorff
967:∈
871:∙
838:∈
807:∙
738:′
725:σ
703:′
665:#
652:σ
630:#
599:#
591:⊆
586:′
556:′
482:⋅
476:⋅
463:#
348:→
343:#
335:×
321:⋅
315:⋅
258:#
199:Polar set
185:Statement
137:separable
40:unit ball
32217:Category
32168:Strictly
32142:Schwartz
32082:LF-space
32077:LB-space
32035:FK-space
32005:Complete
31985:BK-space
31910:Relative
31857:(subset)
31849:(subset)
31776:Seminorm
31759:Bilinear
31502:Category
31314:Algebras
31196:Theorems
31153:Complete
31122:Schwartz
31068:glossary
30937:operator
30910:operator
30771:July 22,
30766:47767143
30741:(2002).
30731:21195908
30703:(1990).
30635:(1996).
30590:21163277
30564:(1991).
28492:For any
28129:because
27888:The set
27114:so that
26160:See also
26132:implies
26075:has the
25720:has the
25218:satisfy
25198:and let
25129:Lp space
24744:lies in
24377:has the
24030:and let
22872:seminorm
22760:balanced
21868:for any
21256:), then
20898:complete
19420:Because
18181:Because
17544:Because
17310:For any
16470:because
16441:For any
15636:∉
14550:Because
14227:Because
14191:and let
13889:For any
12971:Because
12261:subspace
11599:and the
11249:supremum
10724:is some
10442:the net
10039:the net
9711:The set
9707:Topology
8819:sequence
7373:category
6993:Function
6551:for any
5506:Because
4974:is also
4771:Because
4099:so that
3503:Because
2950:⟩
2929:⟨
2776:⟩
2755:⟨
2596:⟩
2575:⟨
2376:⟩
2355:⟨
2327:⟩
2306:⟨
2280:⟩
2259:⟨
1405:⟩
1384:⟨
1025:For any
858:the net
794:of maps
504:forms a
491:⟩
486:⟩
472:⟨
448:⟨
388:⟩
374:⟨
325:⟩
311:⟨
298:bilinear
159:Bourbaki
32182:)
32130:)
32072:K-space
32057:Hilbert
32040:Fréchet
32025:F-space
32000:Brauner
31993:)
31978:)
31960:Asplund
31942:)
31912:)
31832:Bounded
31727:Compact
31712:Bounded
31649: (
31305:Unitary
31285:Nuclear
31270:Compact
31265:Bounded
31260:Adjoint
31234:Min–max
31127:Sobolev
31112:Nuclear
31102:Hilbert
31097:Fréchet
31062: (
30990:Subsets
30922:Mackey
30845:Duality
30811:Duality
30500:0248498
29984:so let
29431:nullary
28254:is the
26445:(resp.
26443:compact
26079:, then
25393:and an
23084:closure
22249:infimum
17389:denote
16867:(where
16734:Lemma (
14165:So fix
12263:of the
12258:compact
11775:(where
10881:And if
9213:(or by
7371:in the
4544:By the
4205:; thus
2723:be the
2345:pairing
1206:of any
1061:) with
173:or the
78:History
60:product
54:in the
52:compact
42:of the
32194:Webbed
32180:Quasi-
32102:Montel
32092:Mackey
31991:Ultra-
31970:Banach
31878:Radial
31842:Convex
31812:Affine
31754:Linear
31722:Closed
31546:(TVSs)
31280:Normal
31117:Orlicz
31107:Hölder
31087:Banach
31076:Spaces
31064:topics
30815:linear
30764:
30754:
30729:
30719:
30683:
30673:
30653:
30643:
30623:
30613:
30588:
30578:
30552:
30542:
30523:
30506:
30498:
30488:
30457:26 Dec
29898:
29875:
29748:
29742:
29703:
29697:
29401:family
28711:Proofs
28211:
28188:
27956:
27933:
27803:
27797:
27051:
27045:
27007:
26984:
26588:
26582:
26154:convex
26138:family
25995:convex
25931:weak-*
25166:where
23201:Since
22803:
22780:
22756:convex
22564:where
22498:then
22452:
22446:
22418:
22412:
22381:
22375:
22347:
22324:
22027:
22021:
21988:
21965:
21823:
21800:
21717:
21694:
21297:
21274:
21184:
21161:
20082:
20059:
19804:
19781:
19328:
19305:
18894:
18819:
18785:
18762:
18679:
18581:
18578:
18575:
18517:
18472:
18160:
18137:
17487:
17464:
16236:: Let
16142:
16119:
16076:closed
15580:
15557:
15467:
15454:
15450:
15444:
15319:
15316:
15312:
15309:
15287:
15284:
15280:
15277:
15273:
15270:
15266:
15263:
15189:
15186:
15182:
15179:
15161:
15155:
15152:
15148:
15145:
15141:
15138:
15134:
15131:
14507:
14484:
13629:
13623:
13599:
13593:
13539:
13533:
13507:
13484:
13401:
13378:
12886:
12880:
12826:
12820:
12794:
12771:
12061:
12055:
12029:
12006:
11887:
11864:
11636:
11613:
11582:
11559:
11294:
11271:
11225:where
10296:
10273:
9846:where
9792:and a
9405:
9382:
8817:Every
7662:
7639:
7171:
7148:
6973:Tuple
6924:
6901:
6647:tuples
6515:
6492:
6387:
6364:
6222:, and
6218:, the
4604:is an
2839:
2833:
2655:
2649:
2485:
2479:
1608:, and
1145:
1139:
1029:(TVS)
790:and a
573:where
415:
392:
197:, and
37:closed
27:, the
32152:Smith
32137:Riesz
32128:Semi-
31940:Quasi
31934:Polar
31092:Besov
30905:polar
30748:(PDF)
30436:(PDF)
30055:From
29191:From
28576:then
26695:(and
26441:is a
26414:then
26228:Notes
25909:Then
25773:above
25059:is a
24874:is a
23923:Proof
23840:When
23539:with
22942:cobal
22870:is a
22070:then
21363:is a
20896:is a
16101:then
16074:is a
16022:: If
15744:Thus
14866:Thus
14570:is a
13459:is a
13232:is a
13212:thus
13103:then
12430:is a
12379:is a
12229:then
10959:then
10544:Thus
10214:and
9084:into
9025:, if
8177:into
7792:into
7752:into
6794:Tuple
5751:is a
5575:is a
5430:is a
5325:is a
4691:is a
4464:is a
4225:is a
3104:is a
2293:is a
1995:that
1499:Proof
1096:polar
535:is a
245:then
46:of a
31771:Norm
31695:form
31683:Maps
31440:(or
31158:Dual
30974:Maps
30900:Weak
30817:maps
30773:2020
30762:OCLC
30752:ISBN
30727:OCLC
30717:ISBN
30681:OCLC
30671:ISBN
30651:OCLC
30641:ISBN
30621:OCLC
30611:ISBN
30586:OCLC
30576:ISBN
30550:OCLC
30540:ISBN
30521:ISBN
30504:OCLC
30486:ISBN
30459:2021
29460:and
29343:the
29267:and
28907:let
28642:and
28532:>
26960:by
26673:only
25836:Let
25822:);
25794:for
25687:the
25264:Let
25183:<
25177:<
24777:and
23955:let
22758:and
21739:(so
21202:>
19711:Let
19516:and
18061:Let
17339:let
17022:Let
16795:The
15980:and
14982:The
14355:>
13918:let
13296:The
12598:>
12175:>
11713:Let
11671:).
11532:<
11404:<
11251:and
7821:".
7390:The
7377:sets
6478:and
6109:and
6061:Weil
5579:and
4066:let
3909:and
3180:Let
2394:Let
1094:the
153:The
30607:GTM
30448:doi
30073:Box
29929:sup
29889:def
29819:Box
29760:Box
29715:Box
29469:Box
29411:Box
29356:Box
29347:of
29325:Box
29292:Box
29279:Box
29209:Box
29002:Box
28929:inf
28816:inf
28521:if
28258:of
28202:def
27947:def
27496:of
27429:not
27427:is
27055:sup
26998:def
26869:by
26675:on
26651:of
26592:sup
26458:If
25993:by
25941:or
25765:not
25761:ZFC
25039:If
24982:is
24969:not
24909:.
24565:lim
23086:in
22898:of
22794:def
22712:Box
22703:of
22603:Box
22576:Box
22546:Box
22533:Box
22338:def
22312:Box
22263:Box
22159:inf
21979:def
21947:if
21814:def
21708:def
21288:def
21236:of
21175:def
20073:def
19925:in
19795:def
19678:in
19319:def
18776:def
18461:def
18252:in
18151:def
17660:in
17577:in
17478:def
17165:in
16911:or
16889:is
16544:in
16133:def
15666:on
15571:def
15196:sup
15114:def
14979::
14812:sup
14584:sup
14498:def
13886::
13543:sup
13498:def
13392:def
13113:sup
13070:If
12830:sup
12785:def
12528:of
12065:sup
12020:def
11878:def
11681:not
11651:on
11627:def
11573:def
11495:sup
11432:sup
11285:def
11234:sup
11141:sup
11055:sup
10728:of
10287:def
9794:net
9653:or
9501:in
9396:def
9279:if
8821:in
8720:in
8658:net
7653:def
7610:to
7387:.
7375:of
7162:def
7056:in
6915:def
6506:def
6378:def
6093:in
6088:net
6049:not
5962:of
5876:If
5842:of
5731:If
4851:is
4794:of
4741:of
4608:of
4109:sup
3321:in
2843:sup
2659:sup
2489:sup
2180:in
2178:net
1612:.
1572:or
1522:by
1298:on
1149:sup
1057:or
1051:not
792:net
717:by
644:by
617:on
515:If
512:.
406:def
276:of
203:If
165:on
161:to
84:the
50:is
19:In
32234::
31066:–
30760:.
30725:.
30715:.
30707:.
30679:.
30649:.
30619:.
30605:.
30584:.
30574:.
30548:.
30502:.
30496:MR
30494:.
30444:77
30442:.
30438:.
30423:^
30376:^
30359:^
30164:Pr
30140:Pr
30043:1.
25933:)
25804:HB
25800:HB
25757:ZF
25749:ZF
25608:).
25252:1.
24978:A
23919:.
23590:.
22998:cl
20909:.
15991::=
15963::=
14852:1.
14424::=
14235:Pr
14077:Pr
13926:Pr
11639:Id
10305:Pr
10222:Pr
10168:Pr
10101:Pr
10047:Pr
8656:A
7800:Pr
7618:Pr
7555:Pr
7430:Pr
6883:"
5931:.
4548:,
3759:1.
2176:a
1604:,
193:,
181:.
32178:(
32163:B
32161:(
32121:(
31989:(
31974:(
31938:(
31908:(
31658:)
31536:e
31529:t
31522:v
31444:)
31168:)
31164:/
31160:(
31070:)
31052:e
31045:t
31038:v
30803:e
30796:t
30789:v
30775:.
30733:.
30687:.
30657:.
30627:.
30592:.
30556:.
30529:.
30510:.
30461:.
30450::
30300:,
30297:1
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30285:m
30277:|
30273:)
30270:u
30267:(
30264:f
30260:|
30239:,
30232:u
30228:m
30223:B
30219:=
30215:)
30207:x
30203:m
30198:B
30192:X
30186:x
30177:(
30171:u
30158:)
30155:f
30152:(
30147:u
30137:=
30134:)
30131:u
30128:(
30125:f
30105:,
30094:m
30089:B
30082:=
30077:P
30063:f
30036:|
30032:)
30029:u
30026:(
30023:f
30019:|
29998:U
29992:u
29972:,
29969:1
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29958:)
29955:u
29952:(
29949:f
29945:|
29939:U
29933:u
29902:U
29882:=
29872:P
29866:f
29846:.
29837:X
29829:)
29823:P
29811:(
29804:f
29778:X
29770:)
29764:P
29752:(
29733:X
29725:)
29719:P
29707:(
29694:P
29674:.
29671:U
29665:u
29645:1
29637:u
29633:m
29612:,
29609:U
29603:u
29583:1
29580:=
29575:u
29571:r
29548:X
29542:x
29537:)
29532:x
29528:r
29524:(
29500:P
29496:T
29473:P
29446:P
29442:T
29415:P
29386:.
29360:P
29329:P
29301:,
29296:P
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29249:T
29236:m
29213:P
29199:P
29179:.
29172:x
29168:m
29163:B
29157:X
29151:x
29143:=
29136:x
29132:R
29127:B
29119:P
29115:T
29102:R
29091:X
29085:x
29077:=
29070:x
29066:R
29061:B
29055:X
29049:x
29037:P
29033:T
29020:R
29011:=
29006:P
28977:}
28971:P
28967:T
28954:R
28950::
28945:x
28941:R
28936:{
28925:=
28920:x
28916:m
28895:,
28892:X
28886:x
28865:K
28844:0
28822:A
28811:B
28807:=
28803:}
28799:A
28793:a
28790::
28785:a
28781:B
28776:{
28752:,
28749:)
28743:,
28740:0
28737:[
28731:A
28692:.
28689:U
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28630:0
28627:=
28622:0
28618:m
28597:r
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28535:0
28529:r
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28468:0
28465:[
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28216:(
28195:=
28179:m
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27100:}
27095:1
27088:|
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