12103:
11388:
3631:(TVS) is a linear operator so all of the properties described above for continuous linear operators apply to them. However, because of their specialized nature, we can say even more about continuous linear functionals than we can about more general continuous linear operators.
1056:
7980:
8225:
2126:
is bounded on some (equivalently, on every) non-degenerate open or closed ball (not necessarily centered at the origin, and of any radius) if and only if it is bounded on the closed unit ball centered at the origin
2825:
equivalent concepts. For this reason, this article will avoid the term "locally bounded" and instead say "locally bounded at every point" (there is no disagreement about the definition of "locally bounded
5235:
2539:
8435:
4973:
5514:
4657:
5367:
1741:
526:
3052:
that is not bounded on any neighborhood of any point. Thus although every linear map that is bounded on a neighborhood is necessarily continuous, the converse is not guaranteed in general.
1324:
960:
10795:
7606:
8337:
5717:
4253:
3132:
is a bounded neighborhood of the origin in a (locally bounded) TVS then its image under any continuous linear map will be a bounded set (so this map is thus bounded on this neighborhood
7392:
5932:
5651:
4568:
3374:
In general, without additional information about either the linear map or its domain or codomain, the map being "bounded" is not equivalent to it being "bounded on a neighborhood". If
2932:
10578:
4374:
8037:
10668:
8656:
5441:
2817:" is sometimes used to refer to a map that is locally bounded at every point of its domain, but some functional analysis authors define "locally bounded" to instead be a synonym of "
2178:
8503:
7061:
6986:
6907:
4446:
7087:
6439:
6339:
6313:
4118:
7817:
4614:
4420:
4189:
3745:
2964:
2418:
1837:
1978:
1471:
7848:
5968:
1928:
10705:
8604:
8081:
394:
8552:
4474:
2092:
1681:
5787:
4907:
4030:
7843:
3683:
2272:
1528:
1372:
1350:
955:
587:
10744:
10512:
5844:
2054:
557:
10871:
6202:
1172:
899:
7752:
4724:
7781:
6829:
5566:
4746:
3459:
3404:
2589:
2380:
2223:
2124:
1560:
1405:
1212:
113:
7461:
5746:
5396:
5281:
4507:
2004:
198:
8363:
7140:
6260:
4849:
3985:
3243:
2623:
1863:
929:
4313:
4282:
4147:
3959:
2348:
5540:
5039:
4878:
3528:
3248:
Any linear map from a TVS into a locally bounded TVS (such as any linear functional) is continuous if and only if it is bounded on a neighborhood. Conversely, if
2722:
2301:
1616:
6074:
8086:
7716:
7562:
7488:
7033:
6933:
6854:
6366:
6121:
6046:
5871:
5304:
4698:
3888:
3840:
3772:
3571:
2987:
2749:
2441:
1643:
822:
319:
242:
8455:
8245:
7693:
7673:
7653:
7629:
7539:
7515:
7432:
7412:
7278:
7254:
7234:
7214:
7187:
7160:
7010:
6800:
6780:
6756:
6732:
6708:
6682:
6656:
6629:
6601:
6573:
6538:
6510:
6483:
6459:
6409:
6389:
6285:
6224:
6163:
6143:
6094:
6023:
5996:
5807:
5590:
5255:
5117:
5093:
5013:
4993:
4823:
4796:
4772:
4073:
4053:
4005:
3932:
3909:
3861:
3817:
3795:
3703:
3657:
3611:
3591:
3548:
3499:
3479:
3427:
3310:
3286:
3266:
3214:
3194:
3174:
3150:
3130:
3034:
3010:
2801:
2781:
2693:
2673:
2650:
2461:
2321:
1781:
1761:
1584:
1491:
1425:
1266:
1143:
1119:
1099:
1079:
865:
845:
799:
768:
744:
724:
700:
669:
641:
610:
464:
437:
417:
359:
339:
296:
265:
219:
166:
144:
3335:. But importantly, in the most general setting of a linear operator between arbitrary topological vector spaces, it is possible for a linear operator to be
9622:
11424:
11277:
10902:
9956:
10601:
3312:
must be a locally bounded TVS. In particular, a linear functional on a arbitrary TVS is continuous if and only if it is bounded on a neighborhood.
2323:
is a normed (or seminormed) space then it suffices to check this condition for the open or closed unit ball centered at the origin. Explicitly, if
11113:
10078:
9223:. Grundlehren der mathematischen Wissenschaften. Vol. 159. Translated by Garling, D.J.H. New York: Springer Science & Business Media.
11934:
9610:
10940:
10897:
11551:
11526:
11103:
10011:
2885:(and thus also bounded) but not bounded on any neighborhood. In particular, it demonstrates that being "bounded on a neighborhood" is
11508:
11230:
11085:
9983:
8250:
5128:
2469:
7309:
The proof uses the facts that the translation of an open set in a linear topological space is again an open set, and the equality
2303:
is a bounded subset of it codomain; or said more briefly, if it is bounded on every bounded subset of its domain. When the domain
12142:
11976:
11478:
11417:
11061:
9617:
9449:
8717:
8368:
7289:
6632:
3710:
3363:
3077:
3045:
2935:
268:
7312:
4912:
11721:
11545:
10330:
7303:
7257:
5446:
4881:
3101:
2725:
2246:
1619:
1237:
1231:
10272:
9410:
9380:
9350:
9323:
9297:
9228:
9198:
9171:
9137:
9083:
9011:
8984:
2990:
10088:
7142:
between TVS that is identically equal to zero is a linear map that is continuous, bounded, and bounded on the neighborhood
4619:
10171:
11986:
11483:
11453:
9638:
8732:
7292:
6608:
6541:
5071:; that is, for a linear functional on a normed space, being "bounded" is equivalent to being "bounded on a neighborhood".
3347:
3081:
672:
617:
1686:
476:
12106:
11757:
11410:
10953:
9263:
5309:
11894:
11042:
10933:
9812:
9595:
9110:
9045:
2842:" (of some point) if and only if it is locally bounded at every point of its domain, in which case it is necessarily
1278:
10760:
7567:
11799:
11312:
10407:
9730:
9573:
6685:
6513:
644:
9037:
10957:
10260:
10196:
9747:
7098:
5658:
4194:
3112:
is a locally bounded TVS since the unit ball centered at the origin is a bounded neighborhood of the origin. If
10041:
5876:
5595:
4512:
3152:). Consequently, a linear map from a locally bounded TVS into any other TVS is continuous if and only if it is
2905:
11829:
10534:
10452:
10255:
9934:
9713:
8699:
4318:
3060:
To summarize the discussion below, for a linear map on a normed (or seminormed) space, being continuous, being
2882:
2843:
63:
33:
7985:
11961:
11563:
11540:
11108:
10631:
10293:
10071:
10026:
10016:
9315:
9071:
8688:
5401:
2130:
8610:
2869:. The converse statements are not true in general but they are both true when the linear map's domain is a
12127:
12012:
11391:
11164:
11098:
10926:
10128:
10118:
10046:
9973:
9849:
9518:
7037:
6938:
6859:
4425:
10123:
8463:
7066:
6418:
6318:
6292:
4078:
11833:
11128:
10466:
10456:
10056:
9442:
8682:
7788:
4577:
4383:
4152:
4033:
3716:
3072:
codomain is normable (or seminormable) is continuous if and only if it bounded on a neighborhood. And a
2949:
2385:
1786:
1051:{\displaystyle {\text{ for all }}x,y\in X,{\text{ if }}\|x-y\|<\delta {\text{ then }}\|Fx-Fy\|<r.}
12069:
11521:
11516:
11373:
11327:
11251:
11133:
10825:
10627:
10289:
10103:
9996:
9991:
9886:
9859:
9676:
9569:
8723:
2653:
1933:
1430:
5937:
1868:
12132:
11865:
11675:
11368:
11184:
10683:
10583:
10277:
10250:
10233:
10051:
9896:
9565:
8042:
5060:
5048:
364:
8558:
4451:
2059:
1648:
12137:
11638:
11633:
11626:
11621:
11493:
11433:
11220:
11118:
11021:
10800:
10093:
10083:
10001:
9939:
9866:
9820:
9735:
9560:
8694:
8509:
7110:
5751:
4887:
4010:
3660:
3628:
2808:
2226:
1219:
120:
67:
40:
7822:
3666:
2251:
1507:
1355:
1333:
934:
562:
11899:
11880:
11556:
11536:
11317:
11093:
10722:
10485:
10066:
10006:
8667:
7166:(although it is possible for the constant zero map to be its only continuous linear functional).
7094:
6659:
6576:
5819:
5052:
3355:
3336:
3332:
3073:
3061:
2966:
is bounded on a neighborhood if and only if there exists a bounded neighborhood of the origin in
2943:
2890:
2866:
2855:
2851:
2818:
2542:
2239:
2231:
2188:
2009:
703:
535:
51:
10834:
6168:
1148:
878:
12088:
12078:
12062:
11762:
11711:
11611:
11596:
11348:
11292:
11256:
10108:
10021:
9802:
9718:
9554:
9548:
9435:
9155:
7721:
4707:
10348:
7757:
6805:
5545:
4729:
3432:
3377:
2562:
2353:
2196:
2097:
1533:
1380:
1330:
if it is norm-bounded (or equivalently, von
Neumann bounded). For example, the scalar field (
1185:
86:
12057:
11744:
11726:
11691:
11531:
11055:
10892:
10887:
10362:
10310:
10267:
10191:
10144:
9881:
9543:
9510:
9483:
7437:
7163:
7090:
5810:
5722:
5372:
5260:
4700:). This is one of several reasons why many definitions involving linear functionals, such as
4479:
3864:
1983:
174:
169:
36:
11051:
10181:
8342:
7119:
6229:
4828:
3964:
3222:
2602:
1842:
908:
12073:
12017:
11996:
11331:
10830:
10036:
10031:
9742:
9626:
9532:
9368:
9238:
4291:
4260:
4125:
3937:
2326:
10918:
5519:
5018:
4854:
3504:
2698:
2277:
1592:
8:
11956:
11951:
11909:
11488:
11297:
11235:
10949:
10673:
10474:
10431:
10245:
9968:
9698:
9505:
9075:
7975:{\displaystyle f(U):=\{f(u):u\in U\}\quad {\text{ and }}\quad |f(U)|:=\{|f(u)|:u\in U\},}
6053:
3320:
3065:
17:
10619:
7698:
7544:
7470:
7015:
6915:
6836:
6348:
6103:
6028:
5853:
5286:
4662:
3870:
3822:
3754:
3553:
2969:
2731:
2423:
1625:
804:
301:
224:
11941:
11884:
11818:
11803:
11670:
11660:
11322:
11189:
10907:
10818:
10441:
10411:
10228:
10186:
9793:
9703:
9648:
9495:
9307:
9160:
8737:
8708:
8440:
8230:
7678:
7658:
7638:
7614:
7608:(where in particular, one side is infinite if and only if the other side is infinite).
7524:
7500:
7417:
7397:
7263:
7239:
7219:
7199:
7172:
7145:
7113:(TVS) is continuous. This is not true if the finite-dimensional TVS is not Hausdorff.
6995:
6785:
6765:
6741:
6717:
6693:
6667:
6641:
6614:
6604:
6586:
6558:
6523:
6495:
6468:
6444:
6394:
6374:
6270:
6209:
6148:
6128:
6079:
6008:
5981:
5792:
5575:
5240:
5102:
5078:
4998:
4978:
4808:
4781:
4757:
4058:
4038:
3990:
3917:
3894:
3846:
3802:
3780:
3688:
3642:
3622:
3596:
3576:
3533:
3484:
3464:
3412:
3359:
3295:
3271:
3251:
3199:
3179:
3159:
3135:
3115:
3085:
3019:
3013:
2995:
2786:
2766:
2678:
2658:
2635:
2446:
2306:
1930:
and any translation of a bounded set is again bounded) if and only if it is bounded on
1766:
1746:
1569:
1476:
1410:
1270:
1251:
1128:
1104:
1084:
1064:
850:
830:
784:
771:
753:
729:
709:
685:
654:
626:
595:
449:
422:
402:
344:
324:
281:
250:
204:
151:
129:
10749:
10719:
10680:
9216:
3156:. Moreover, any TVS with this property must be a locally bounded TVS. Explicitly, if
11653:
11579:
11302:
10588:
10061:
9842:
9785:
9765:
9416:
9406:
9386:
9376:
9356:
9346:
9329:
9319:
9293:
9286:
9269:
9259:
9242:
9224:
9204:
9194:
9177:
9167:
9143:
9133:
9116:
9106:
9089:
9079:
9051:
9041:
9017:
9007:
8990:
8980:
7193:
3748:
3217:
2554:
467:
6579:(that is, it maps bounded subsets of its domain to bounded subsets of its codomain).
12046:
11616:
11601:
11402:
11307:
11225:
11194:
11174:
11159:
11154:
11149:
10218:
10213:
10201:
10113:
10098:
9961:
9901:
9876:
9807:
9797:
9660:
9318:. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer.
9029:
8676:
6486:
5068:
3109:
1245:
872:
613:
78:
11929:
11468:
10986:
9605:
9166:. Translated by Chaljub, Orlando. New York: Gordon and Breach Science Publishers.
6545:
3319:
the codomain of a linear map is normable or seminormable, then continuity will be
12021:
11869:
11169:
11123:
11071:
11066:
11037:
10238:
10223:
10149:
9951:
9944:
9911:
9871:
9837:
9829:
9757:
9725:
9590:
9522:
9398:
9234:
8976:
5043:
Importantly, a linear functional being "bounded on a neighborhood" is in general
3706:
2939:
1496:
Any translation, scalar multiple, and subset of a bounded set is again bounded.
440:
116:
10996:
1236:
The notion of a "bounded set" for a topological vector space is that of being a
12052:
12001:
11716:
11358:
11210:
11011:
10808:
10756:
10416:
10282:
9929:
9919:
9538:
9490:
9063:
5971:
3037:
2833:
1375:
529:
3176:
is a TVS such that every continuous linear map (into any TVS) whose domain is
2056:
and any scalar multiple of a bounded set is again bounded). Consequently, if
12121:
12036:
11946:
11889:
11849:
11777:
11752:
11696:
11648:
11584:
11363:
11287:
11016:
11001:
10991:
10813:
10426:
10380:
10315:
10166:
10161:
10154:
9775:
9708:
9681:
9500:
9473:
9420:
9390:
9333:
9273:
9258:. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press.
9246:
9040:. Translated by Eggleston, H.G.; Madan, S. Berlin New York: Springer-Verlag.
9021:
8994:
2900:: A continuous and bounded linear map that is not bounded on any neighborhood
2464:
9360:
9147:
9120:
9093:
9055:
5970:
Polar sets, and so also this particular inequality, play important roles in
4704:
for example, involve closed (rather than open) neighborhoods and non-strict
3268:
is a TVS such that every continuous linear map (from any TVS) with codomain
2783:
in its domain at which it is locally bounded, in which case this linear map
12083:
12031:
11991:
11981:
11859:
11706:
11701:
11498:
11448:
11353:
11006:
10976:
10325:
10320:
9780:
9770:
9643:
9633:
9478:
9458:
9281:
9208:
8220:{\displaystyle \sup |f(U)|~=~\sup\{|f(u)|:u\in U\}~=~\sup _{u\in U}|f(u)|.}
7518:
7464:
6735:
6549:
5064:
3407:
3351:
3105:
3049:
2876:
2870:
2847:
1493:
is contained in some open (or closed) ball centered at the origin (zero).
1241:
868:
775:
676:
47:
9181:
3358:
if and only if it is continuous. The same is true of a linear map from a
12041:
12026:
11919:
11813:
11808:
11793:
11772:
11736:
11643:
11463:
11282:
11272:
11179:
10981:
10614:
10530:
10436:
10421:
10401:
10375:
10340:
9891:
9854:
9527:
3040:). This shows that it is possible for a linear map to be continuous but
21:
11854:
11767:
11731:
11591:
11473:
11215:
11047:
10370:
10351: ((cs, lcs)-closed, (cs, bcs)-complete, (lower) ideally convex, (H
10335:
10176:
9924:
9686:
8795:
3934:
said to be continuous at the origin if for every open (or closed) ball
1215:
9427:
8852:
8850:
8848:
8846:
8844:
8793:
8791:
8789:
8787:
8785:
8783:
8781:
8779:
8777:
8775:
8762:
8760:
8758:
8756:
8754:
12006:
11823:
10446:
9691:
9655:
6411:
are complex vector spaces then this list may be extended to include:
6342:
5847:
5051:" because (as described above) it is possible for a linear map to be
4701:
471:
8917:
8905:
7299:
if and only if every bounded linear functional on it is continuous.
3044:
bounded on any neighborhood. Indeed, this example shows that every
11971:
11966:
11924:
11904:
11874:
11665:
10712:
10596:
10522:
10482:
10385:
10208:
9006:. Graduate Texts in Mathematics. Vol. 15. New York: Springer.
8841:
8772:
8751:
7632:
7296:
4285:
276:
8973:
Topological Vector Spaces: The Theory
Without Convexity Conditions
8720: – A vector space with a topology defined by convex open sets
8702: – A vector space with a topology defined by convex open sets
7216:
is necessarily continuous if and only if every vector subspace of
3092:
Guaranteeing that "continuous" implies "bounded on a neighborhood"
2881:
The next example shows that it is possible for a linear map to be
2463:
is also a (semi)normed space then this happens if and only if the
11914:
10517:
9600:
3613:
of the origin, which (as mentioned above) guarantees continuity.
443:
locally convex spaces then this list may be extended to include:
8975:. Lecture Notes in Mathematics. Vol. 639. Berlin New York:
8953:
7256:
is necessarily a bounded linear functional if and only if every
7109:
Every linear map whose domain is a finite-dimensional
Hausdorff
8817:
8670: – Linear transformation between topological vector spaces
3634:
3370:
Guaranteeing that "bounded" implies "bounded on a neighborhood"
8711: – Linear map from a vector space to its field of scalars
7492:
1121:
finite-dimensional then this list may be extended to include:
5230:{\displaystyle \sup _{x\in sU}|f(x)|=|s|\sup _{u\in U}|f(u)|}
3550:
is a bounded linear map) and a neighborhood of the origin in
2534:{\displaystyle \|F\|:=\sup _{\|x\|\leq 1}\|F(x)\|<\infty }
2834:
Bounded on a neighborhood implies continuous implies bounded
10948:
8883:
8881:
8879:
8877:
8430:{\displaystyle B_{\leq r}:=\{c\in \mathbb {F} :|c|\leq r\}}
6758:
is real-valued) then this list may be extended to include:
4968:{\textstyle \displaystyle \sup _{u\in U}|f(u)|<\infty .}
4805:
Explicitly, this means that there exists some neighborhood
8740: – Linear operator defined on a dense linear subspace
8457:
centered at the origin then the following are equivalent:
2938:
then this linear map is always continuous (indeed, even a
2865:
then it is continuous, and if it is continuous then it is
5509:{\displaystyle \displaystyle \sup _{n\in N_{r}}|f(n)|=r.}
5306:
will be neighborhood of the origin. So in particular, if
3461:
is necessarily continuous; this is because any open ball
1683:
is a normed (or seminormed) space happens if and only if
8874:
8862:
8829:
7611:
Every non-trivial continuous linear functional on a TVS
7162:
of the origin. In particular, every TVS has a non-empty
6516:
at some (or equivalently, at every) point of its domain.
2877:
Continuous and bounded but not bounded on a neighborhood
2873:. Examples and additional details are now given below.
2549:
Function bounded on a neighborhood and local boundedness
647:
at some (or equivalently, at every) point of its domain.
8713:
Pages displaying short descriptions of redirect targets
8704:
Pages displaying short descriptions of redirect targets
8672:
Pages displaying short descriptions of redirect targets
5369:
is a positive real number then for every positive real
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is contained in a finite-dimensional vector subspace.
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4652:{\displaystyle X=\mathbb {R} ,f=\operatorname {Id} ,}
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A subset of a normed (or seminormed) space is called
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9373:
Topological Vector Spaces, Distributions and
Kernels
8971:
Adasch, Norbert; Ernst, Bruno; Keim, Dieter (1978).
8728:
Pages displaying wikidata descriptions as a fallback
54:if and only if it is a continuous linear operator.
9004:
Lectures in
Functional Analysis and Operator Theory
2094:is a normed or seminormed space, then a linear map
801:maps some neighborhood of 0 to a bounded subset of
11278:Spectral theory of ordinary differential equations
10865:
10789:
10738:
10699:
10662:
10572:
10506:
9285:
9159:
9105:(in Romanian). New York: Interscience Publishers.
8685: – Mathematical method in functional analysis
8650:
8598:
8546:
8497:
8449:
8429:
8357:
8331:
8239:
8219:
8075:
8031:
7974:
7845:is a non-empty subset, then by defining the sets
7837:
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7647:
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7600:
7556:
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7509:
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7081:
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7004:
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6702:
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6403:
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5990:
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5801:
5781:
5740:
5711:
5645:
5584:
5560:
5534:
5508:
5435:
5390:
5362:{\textstyle R:=\displaystyle \sup _{u\in U}|f(u)|}
5361:
5298:
5275:
5249:
5229:
5111:
5087:
5033:
5007:
4987:
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3260:
3237:
3208:
3188:
3168:
3144:
3124:
3028:
3004:
2981:
2958:
2926:
2854:(because a continuous linear operator is always a
2795:
2775:
2743:
2716:
2687:
2667:
2644:
2617:
2583:
2533:
2455:
2435:
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2217:
2172:
2118:
2086:
2048:
1998:
1972:
1922:
1857:
1831:
1775:
1755:
1736:{\displaystyle \sup _{s\in S}\|F(s)\|<\infty .}
1735:
1675:
1637:
1610:
1578:
1554:
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1485:
1465:
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521:{\displaystyle {}^{t}F:Y^{\prime }\to X^{\prime }}
520:
458:
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411:
388:
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8726: – ordered vector space with a partial order
5096:
12119:
8562:
8513:
8304:
8254:
8177:
8124:
8090:
8047:
7991:
6688:(or equivalently, at every) point of its domain.
5881:
5663:
5600:
5452:
5321:
5190:
5133:
4918:
4517:
4323:
4199:
3616:
3327:Guaranteeing that "bounded" implies "continuous"
2486:
1691:
1435:
1283:
8970:
5748:which shows that the positive scalar multiples
3196:is necessarily bounded on a neighborhood, then
1319:{\displaystyle \sup _{s\in S}\|s\|<\infty .}
10790:{\displaystyle S\left(\mathbb {R} ^{n}\right)}
9405:. Mineola, New York: Dover Publications, Inc.
9254:Narici, Lawrence; Beckenstein, Edward (2011).
7655:is a linear functional on a real vector space
7601:{\displaystyle \|f\|=\|\operatorname {Re} f\|}
4775:
3289:
3153:
3100:if there exists a neighborhood that is also a
3064:, and being bounded on a neighborhood are all
2862:
2839:
57:
11418:
10934:
10898:Mathematical formulation of quantum mechanics
9443:
9306:
8332:{\displaystyle \sup |f(sU)|~=~|s|\sup |f(U)|}
6552:) then this list may be extended to include:
3501:is both a bounded subset (which implies that
1177:
901:) then this list may be extended to include:
778:) then this list may be extended to include:
620:) then this list may be extended to include:
9154:
9130:Functional Analysis: Theory and Applications
8691: – Bounded operators with sub-unit norm
8424:
8388:
8164:
8127:
7966:
7929:
7894:
7867:
7595:
7583:
7577:
7571:
6975:
6942:
6896:
6863:
5776:
5755:
3635:Characterizing continuous linear functionals
2522:
2507:
2496:
2490:
2479:
2473:
2382:is a bounded linear operator if and only if
2164:
2155:
2149:
2134:
2078:
2072:
1967:
1946:
1826:
1802:
1721:
1706:
1667:
1661:
1304:
1298:
1268:is von Neumann bounded if and only if it is
1036:
1018:
1004:
992:
888:
882:
9403:Modern Methods in Topological Vector Spaces
7493:Properties of continuous linear functionals
6635:then this list may be extended to include:
6489:then this list may be extended to include:
5712:{\displaystyle \sup _{x\in rU}|f(x)|\leq r}
4248:{\displaystyle \sup _{u\in U}|f(u)|\leq r.}
3216:must be a locally bounded TVS (because the
271:then this list may be extended to include:
11425:
11411:
10941:
10927:
9450:
9436:
8679: – Type of continuous linear operator
7387:{\displaystyle F^{-1}(D)+x=F^{-1}(D+F(x))}
6226:is continuous if and only if the seminorm
5927:{\displaystyle \sup _{u\in U}|f(u)|\leq 1}
5646:{\displaystyle \sup _{u\in U}|f(u)|\leq 1}
4563:{\displaystyle \sup _{u\in U}|f(u)|\leq r}
3048:that is not seminormable has a linear TVS-
2927:{\displaystyle \operatorname {Id} :X\to X}
10773:
10573:{\displaystyle B_{p,q}^{s}(\mathbb {R} )}
10563:
9074:. Vol. 96 (2nd ed.). New York:
9001:
8398:
8072:
8046:
8028:
7989:
7805:
6710:is sequentially continuous at the origin.
4892:
4737:
4733:
4715:
4711:
4630:
4462:
4369:{\displaystyle \sup _{u\in U}|f(u)|<r}
4015:
3733:
3671:
3331:A continuous linear operator is always a
3055:
1360:
1338:
1101:are Hausdorff locally convex spaces with
11231:Group algebra of a locally compact group
9397:
9292:. McGraw-Hill Science/Engineering/Math.
9028:
8947:
8935:
8899:
8887:
8868:
8835:
8811:
8032:{\displaystyle \,\sup _{u\in U}|f(u)|\,}
1473:is finite, which happens if and only if
123:(TVSs). The following are equivalent:
10663:{\displaystyle L^{\lambda ,p}(\Omega )}
9457:
9188:
9127:
9100:
9034:Topological Vector Spaces: Chapters 1–5
8718:Locally convex topological vector space
8651:{\textstyle f(rU)\subseteq B_{\leq r}.}
5436:{\displaystyle N_{r}:={\tfrac {r}{R}}U}
5120:
5097:bounded on a neighborhood of the origin
4799:
2936:locally convex topological vector space
2173:{\displaystyle \{x\in X:\|x\|\leq 1\}.}
12120:
11564:Uniform boundedness (Banach–Steinhaus)
10903:Ordinary Differential Equations (ODEs)
10017:Banach–Steinhaus (Uniform boundedness)
9367:
9343:An introduction to Functional Analysis
9340:
9062:
7236:is closed. Every linear functional on
4422:to be true (consider for example when
1232:Bounded set (topological vector space)
11406:
10922:
9431:
9375:. Mineola, N.Y.: Dover Publications.
9280:
9215:
8498:{\textstyle f(U)\subseteq B_{\leq 1}}
7056:{\displaystyle \operatorname {Re} f,}
6981:{\displaystyle \{x\in X:f(x)\leq r\}}
6902:{\displaystyle \{x\in X:f(x)\leq r\}}
5655:This inequality holds if and only if
4509:), whereas the non-strict inequality
4441:{\displaystyle f=\operatorname {Id} }
3593:is thus bounded on this neighborhood
3036:is Hausdorff, is the same as being a
1240:. If the space happens to also be a
706:(that is, it maps bounded subsets of
7082:{\displaystyle \operatorname {Im} f}
7012:is complex then either all three of
6434:{\displaystyle \operatorname {Im} f}
6334:{\displaystyle \operatorname {Re} f}
6308:{\displaystyle \operatorname {Re} f}
5999:
5443:is a neighborhood of the origin and
5059:continuous. However, continuity and
4975:This supremum over the neighborhood
4149:is a closed ball then the condition
4113:{\displaystyle f(U)\subseteq B_{r}.}
3406:is a bounded linear operator from a
3323:to being bounded on a neighborhood.
3245:is always a continuous linear map).
3080:will be continuous if its domain is
8733:Topologies on spaces of linear maps
7812:{\displaystyle f:X\to \mathbb {F} }
7293:metrizable topological vector space
6762:There exists a continuous seminorm
6738:(which in particular, implies that
6125:There exists a continuous seminorm
4778:(of some point). Said differently,
4609:{\displaystyle f(U)\subseteq B_{r}}
4415:{\displaystyle f(U)\subseteq B_{r}}
4184:{\displaystyle f(U)\subseteq B_{r}}
3740:{\displaystyle f:X\to \mathbb {F} }
2959:{\displaystyle \operatorname {Id} }
2413:{\displaystyle F\left(B_{1}\right)}
1832:{\displaystyle x+S:=\{x+s:s\in S\}}
675:or metrizable (such as a normed or
321:there exists a continuous seminorm
13:
10731:
10692:
10654:
10498:
8039:can be written more succinctly as
7302:A continuous linear operator maps
5063:are equivalent if the domain is a
4958:
2803:is necessarily locally bounded at
2759:" (of some point) if there exists
2528:
1727:
1310:
571:
544:
513:
500:
14:
12154:
10395:Subsets / set operations
10172:Differentiation in Fréchet spaces
4616:to be true (consider for example
1973:{\displaystyle cS:=\{cs:s\in S\}}
1466:{\displaystyle \sup _{s\in S}|s|}
679:) then we may add to this list:
12102:
12101:
11387:
11386:
11313:Topological quantum field theory
9132:. New York: Dover Publications.
5963:{\displaystyle f\in U^{\circ }.}
4288:characterization. Assuming that
2807:point of its domain. The term "
1923:{\displaystyle F(x+S)=F(x)+F(S)}
1783:if and only if it is bounded on
875:(with both seminorms denoted by
12089:With the approximation property
10700:{\displaystyle \ell ^{\infty }}
9068:A course in functional analysis
9002:Berberian, Sterling K. (1974).
8599:{\textstyle \sup |f(rU)|\leq r}
8076:{\displaystyle \,\sup |f(U)|\,}
7903:
7897:
6686:sequentially continuous at some
5809:will satisfy the definition of
5572:There exists some neighborhood
5542:proves the next statement when
1407:is a normed space, so a subset
389:{\displaystyle q\circ F\leq p.}
221:is continuous at the origin in
73:Characterizations of continuity
12143:Theory of continuous functions
11552:Open mapping (Banach–Schauder)
10860:
10841:
10657:
10651:
10567:
10559:
10501:
10495:
10089:Lomonosov's invariant subspace
10012:Banach–Schauder (open mapping)
8700:Finest locally convex topology
8626:
8617:
8586:
8582:
8573:
8566:
8547:{\textstyle \sup |f(U)|\leq 1}
8534:
8530:
8524:
8517:
8476:
8470:
8414:
8406:
8325:
8321:
8315:
8308:
8300:
8292:
8278:
8274:
8265:
8258:
8210:
8206:
8200:
8193:
8148:
8144:
8138:
8131:
8111:
8107:
8101:
8094:
8068:
8064:
8058:
8051:
8024:
8020:
8014:
8007:
7950:
7946:
7940:
7933:
7922:
7918:
7912:
7905:
7879:
7873:
7861:
7855:
7801:
7734:
7726:
7381:
7378:
7372:
7360:
7335:
7329:
7126:
6966:
6960:
6887:
6881:
6542:metrizable or pseudometrizable
6248:
6240:
6181:
6173:
6000:locally bounded at every point
5914:
5910:
5904:
5897:
5699:
5695:
5689:
5682:
5633:
5629:
5623:
5616:
5492:
5488:
5482:
5475:
5354:
5350:
5344:
5337:
5223:
5219:
5213:
5206:
5185:
5177:
5169:
5165:
5159:
5152:
5121:locally bounded at the origin.
4951:
4947:
4941:
4934:
4867:
4861:
4687:
4672:
4590:
4584:
4550:
4546:
4540:
4533:
4469:{\displaystyle X=\mathbb {F} }
4396:
4390:
4356:
4352:
4346:
4339:
4315:is instead an open ball, then
4232:
4228:
4222:
4215:
4165:
4159:
4091:
4085:
3774:The following are equivalent:
3729:
3517:
3511:
3445:
3390:
3229:
2918:
2889:always synonymous with being "
2711:
2705:
2575:
2519:
2513:
2366:
2290:
2284:
2209:
2110:
2087:{\displaystyle (X,\|\cdot \|)}
2081:
2063:
2043:
2037:
2025:
2016:
1917:
1911:
1902:
1896:
1887:
1875:
1718:
1712:
1676:{\displaystyle (Y,\|\cdot \|)}
1670:
1652:
1605:
1599:
1546:
1459:
1451:
1393:
1385:
1198:
505:
99:
64:Continuous function (topology)
1:
11109:Uniform boundedness principle
9072:Graduate Texts in Mathematics
8960:Narici & Beckenstein 2011
8924:Narici & Beckenstein 2011
8912:Narici & Beckenstein 2011
8857:Narici & Beckenstein 2011
8824:Narici & Beckenstein 2011
8800:Narici & Beckenstein 2011
8767:Narici & Beckenstein 2011
8744:
8689:Contraction (operator theory)
8437:is the closed ball of radius
7283:
5782:{\displaystyle \{rU:r>0\}}
4902:{\displaystyle \mathbb {F} ;}
4800:locally bounded at some point
4025:{\displaystyle \mathbb {F} ,}
3911:is continuous at the origin.
3627:Every linear functional on a
3617:Continuous linear functionals
3346:A linear map whose domain is
3068:. A linear map whose domain
2861:For any linear map, if it is
2846:(even if its domain is not a
2545:linear operator is bounded.
559:to equicontinuous subsets of
9974:Singular value decomposition
9310:; Wolff, Manfred P. (1999).
7838:{\displaystyle U\subseteq X}
5789:of this single neighborhood
4659:and the closed neighborhood
3678:{\displaystyle \mathbb {F} }
2934:is the identity map on some
2595:bounded on a neighborhood of
2267:{\displaystyle B\subseteq X}
2247:(von Neumann) bounded subset
2193:By definition, a linear map
1523:{\displaystyle S\subseteq X}
1367:{\displaystyle \mathbb {C} }
1345:{\displaystyle \mathbb {R} }
950:{\displaystyle \delta >0}
582:{\displaystyle X^{\prime }.}
7:
11773:Radially convex/Star-shaped
11758:Pre-compact/Totally bounded
10739:{\displaystyle L^{\infty }}
10507:{\displaystyle ba(\Sigma )}
10376:Radially convex/Star-shaped
9221:Topological Vector Spaces I
9193:. Stuttgart: B.G. Teubner.
9128:Edwards, Robert E. (1995).
8683:Continuous linear extension
8661:
7819:is a linear functional and
7189:is any Hausdorff TVS. Then
7104:
7101:(respectively, unbounded).
6734:is a vector space over the
5839:{\displaystyle U^{\circ },}
4570:is instead a necessary but
3819:is uniformly continuous on
2049:{\displaystyle F(cS)=cF(S)}
552:{\displaystyle Y^{\prime }}
58:Continuous linear operators
10:
12159:
11459:Continuous linear operator
11252:Invariant subspace problem
10866:{\displaystyle W(X,L^{p})}
8724:Positive linear functional
7541:is a linear functional on
6197:{\displaystyle |f|\leq p.}
3620:
3481:centered at the origin in
2552:
2186:
1980:for every non-zero scalar
1427:is bounded if and only if
1229:
1178:Continuity and boundedness
1167:{\displaystyle X\times Y.}
894:{\displaystyle \|\cdot \|}
76:
61:
26:continuous linear operator
12097:
11842:
11804:Algebraic interior (core)
11786:
11684:
11572:
11546:Vector-valued Hahn–Banach
11507:
11441:
11434:Topological vector spaces
11382:
11341:
11265:
11244:
11203:
11142:
11084:
11030:
10972:
10965:
10880:
10465:
10412:Algebraic interior (core)
10394:
10303:
10137:
10027:Cauchy–Schwarz inequality
9982:
9910:
9756:
9670:Function space Topologies
9669:
9583:
9466:
9312:Topological Vector Spaces
9256:Topological Vector Spaces
9162:Topological Vector Spaces
7747:{\displaystyle |f|\leq p}
7463:which is true due to the
7097:), or else all three are
5816:By definition of the set
5049:bounded linear functional
4776:bounded on a neighborhood
4719:{\displaystyle \,\leq \,}
4284:be a closed ball in this
3290:bounded on a neighborhood
3154:bounded on a neighborhood
2863:bounded on a neighborhood
2840:bounded on a neighborhood
2821:", which are related but
2755:bounded on a neighborhood
1500:Function bounded on a set
1220:topological vector spaces
121:topological vector spaces
41:topological vector spaces
30:continuous linear mapping
11634:Topological homomorphism
11494:Topological vector space
11221:Spectrum of a C*-algebra
9341:Swartz, Charles (1992).
9101:Dunford, Nelson (1988).
9038:Éléments de mathématique
8695:Discontinuous linear map
7776:{\displaystyle f\leq p.}
7111:topological vector space
6824:{\displaystyle f\leq p.}
6760:
6637:
6554:
6491:
6413:
5811:continuity at the origin
5592:of the origin such that
5561:{\displaystyle R\neq 0.}
4741:{\displaystyle \,<\,}
3865:continuous at some point
3661:topological vector space
3629:topological vector space
3454:{\displaystyle F:X\to Y}
3399:{\displaystyle F:X\to Y}
2584:{\displaystyle F:X\to Y}
2375:{\displaystyle F:X\to Y}
2218:{\displaystyle F:X\to Y}
2119:{\displaystyle F:X\to Y}
1555:{\displaystyle F:X\to Y}
1400:{\displaystyle |\cdot |}
1207:{\displaystyle F:X\to Y}
1123:
170:continuous at some point
108:{\displaystyle F:X\to Y}
68:Discontinuous linear map
46:An operator between two
11318:Noncommutative geometry
9345:. New York: M. Dekker.
9156:Grothendieck, Alexander
8668:Bounded linear operator
7456:{\displaystyle x\in X,}
6660:bounded linear operator
6577:bounded linear operator
6514:sequentially continuous
5741:{\displaystyle r>0,}
5391:{\displaystyle r>0,}
5276:{\displaystyle s\neq 0}
5047:equivalent to being a "
4502:{\displaystyle U=B_{r}}
4448:is the identity map on
3333:bounded linear operator
3074:bounded linear operator
2856:bounded linear operator
2819:bounded linear operator
2543:sequentially continuous
2420:is a bounded subset of
2350:denotes this ball then
2240:bounded linear operator
2189:Bounded linear operator
1999:{\displaystyle c\neq 0}
1238:von Neumann bounded set
903:
780:
704:bounded linear operator
681:
645:sequentially continuous
622:
445:
273:
193:{\displaystyle x\in X.}
52:bounded linear operator
11692:Absolutely convex/disk
11374:Tomita–Takesaki theory
11349:Approximation property
11293:Calculus of variations
10867:
10791:
10740:
10701:
10664:
10574:
10508:
9677:Banach–Mazur compactum
9467:Types of Banach spaces
9189:Jarchow, Hans (1981).
8652:
8600:
8548:
8499:
8451:
8431:
8359:
8358:{\displaystyle r>0}
8333:
8241:
8221:
8077:
8033:
7976:
7839:
7813:
7777:
7748:
7712:
7689:
7669:
7649:
7625:
7602:
7558:
7535:
7511:
7484:
7457:
7428:
7408:
7388:
7274:
7250:
7230:
7210:
7183:
7156:
7136:
7135:{\displaystyle X\to Y}
7083:
7057:
7029:
7006:
6982:
6929:
6903:
6850:
6825:
6796:
6776:
6752:
6728:
6704:
6678:
6652:
6625:
6597:
6569:
6534:
6506:
6479:
6455:
6435:
6405:
6385:
6362:
6335:
6309:
6281:
6256:
6255:{\displaystyle p:=|f|}
6220:
6198:
6159:
6139:
6117:
6090:
6076:or else the kernel of
6070:
6042:
6019:
5992:
5964:
5928:
5867:
5840:
5803:
5783:
5742:
5713:
5647:
5586:
5562:
5536:
5510:
5437:
5392:
5363:
5300:
5277:
5251:
5237:holds for all scalars
5231:
5113:
5089:
5035:
5009:
4989:
4969:
4903:
4874:
4845:
4844:{\displaystyle x\in X}
4819:
4792:
4768:
4742:
4720:
4694:
4653:
4610:
4564:
4503:
4470:
4442:
4416:
4370:
4309:
4278:
4249:
4185:
4143:
4114:
4069:
4049:
4026:
4001:
3981:
3980:{\displaystyle r>0}
3955:
3928:
3905:
3884:
3857:
3836:
3813:
3791:
3768:
3741:
3699:
3679:
3653:
3607:
3587:
3567:
3544:
3524:
3495:
3475:
3455:
3423:
3400:
3306:
3282:
3262:
3239:
3238:{\displaystyle X\to X}
3210:
3190:
3170:
3146:
3126:
3056:Guaranteeing converses
3030:
3006:
2983:
2960:
2928:
2797:
2777:
2745:
2718:
2689:
2669:
2646:
2619:
2618:{\displaystyle x\in X}
2585:
2535:
2457:
2437:
2414:
2376:
2344:
2317:
2297:
2268:
2219:
2174:
2120:
2088:
2050:
2000:
1974:
1924:
1859:
1858:{\displaystyle x\in X}
1833:
1777:
1757:
1737:
1677:
1639:
1612:
1580:
1556:
1524:
1487:
1467:
1421:
1401:
1368:
1346:
1320:
1262:
1208:
1168:
1139:
1115:
1095:
1075:
1052:
951:
925:
924:{\displaystyle r>0}
895:
861:
841:
818:
795:
764:
740:
726:to bounded subsets of
720:
696:
665:
637:
618:pseudometrizable space
606:
583:
553:
522:
460:
433:
413:
390:
355:
335:
315:
292:
261:
238:
215:
194:
162:
140:
109:
11727:Complemented subspace
11541:hyperplane separation
11369:Banach–Mazur distance
11332:Generalized functions
10893:Finite element method
10888:Differential operator
10868:
10792:
10741:
10702:
10665:
10575:
10509:
10349:Convex series related
10145:Abstract Wiener space
10072:hyperplane separation
9627:Minkowski functionals
9511:Polarization identity
9191:Locally convex spaces
8653:
8601:
8549:
8500:
8452:
8432:
8365:is a real number and
8360:
8334:
8242:
8222:
8078:
8034:
7977:
7840:
7814:
7778:
7749:
7713:
7690:
7670:
7650:
7626:
7603:
7559:
7536:
7512:
7485:
7458:
7429:
7409:
7389:
7275:
7251:
7231:
7211:
7184:
7164:continuous dual space
7157:
7137:
7116:Every (constant) map
7084:
7058:
7030:
7007:
6983:
6930:
6904:
6851:
6826:
6797:
6777:
6753:
6729:
6705:
6679:
6653:
6626:
6598:
6570:
6535:
6507:
6480:
6456:
6436:
6406:
6386:
6363:
6336:
6315:is continuous, where
6310:
6282:
6257:
6221:
6199:
6160:
6140:
6118:
6091:
6071:
6043:
6020:
5993:
5965:
5934:holds if and only if
5929:
5868:
5841:
5804:
5784:
5743:
5714:
5648:
5587:
5563:
5537:
5511:
5438:
5393:
5364:
5301:
5278:
5252:
5232:
5114:
5090:
5036:
5010:
4990:
4970:
4904:
4875:
4846:
4820:
4793:
4769:
4743:
4721:
4695:
4654:
4611:
4565:
4504:
4471:
4443:
4417:
4371:
4310:
4308:{\displaystyle B_{r}}
4279:
4277:{\displaystyle B_{r}}
4257:It is important that
4250:
4191:holds if and only if
4186:
4144:
4142:{\displaystyle B_{r}}
4115:
4070:
4050:
4027:
4002:
3982:
3956:
3954:{\displaystyle B_{r}}
3929:
3906:
3885:
3858:
3837:
3814:
3792:
3769:
3742:
3700:
3680:
3663:(TVS) over the field
3654:
3608:
3588:
3568:
3545:
3525:
3496:
3476:
3456:
3424:
3401:
3315:Thus when the domain
3307:
3283:
3263:
3240:
3211:
3191:
3171:
3147:
3127:
3104:. For example, every
3031:
3007:
2984:
2961:
2929:
2798:
2778:
2746:
2719:
2690:
2670:
2647:
2620:
2586:
2536:
2458:
2438:
2415:
2377:
2345:
2343:{\displaystyle B_{1}}
2318:
2298:
2269:
2220:
2175:
2121:
2089:
2051:
2001:
1975:
1925:
1860:
1834:
1778:
1758:
1738:
1678:
1640:
1613:
1581:
1557:
1525:
1488:
1468:
1422:
1402:
1369:
1347:
1321:
1263:
1209:
1169:
1140:
1116:
1096:
1076:
1053:
952:
926:
896:
862:
842:
819:
796:
765:
741:
721:
697:
666:
638:
607:
584:
554:
523:
461:
434:
414:
391:
356:
336:
316:
293:
275:for every continuous
262:
239:
216:
195:
163:
141:
110:
37:linear transformation
20:and related areas of
11977:Locally convex space
11527:Closed graph theorem
11479:Locally convex space
11114:Kakutani fixed-point
11099:Riesz representation
10835:
10761:
10723:
10684:
10632:
10535:
10486:
10475:Absolute continuity
10129:Schauder fixed-point
10119:Riesz representation
10079:Kakutani fixed-point
10047:Freudenthal spectral
9533:L-semi-inner product
8611:
8559:
8510:
8464:
8441:
8369:
8343:
8251:
8231:
8087:
8043:
7986:
7849:
7823:
7789:
7758:
7722:
7699:
7679:
7659:
7639:
7615:
7568:
7545:
7525:
7501:
7471:
7438:
7418:
7398:
7313:
7264:
7240:
7220:
7200:
7173:
7146:
7120:
7067:
7038:
7016:
6996:
6939:
6916:
6860:
6837:
6806:
6786:
6766:
6742:
6718:
6694:
6668:
6642:
6615:
6609:pseudometrizable TVS
6587:
6559:
6524:
6496:
6469:
6445:
6419:
6395:
6375:
6349:
6319:
6293:
6271:
6230:
6210:
6169:
6149:
6129:
6104:
6080:
6054:
6029:
6009:
5982:
5938:
5877:
5854:
5846:which is called the
5820:
5793:
5752:
5723:
5659:
5596:
5576:
5546:
5535:{\displaystyle r:=1}
5520:
5447:
5402:
5373:
5310:
5287:
5261:
5241:
5129:
5103:
5099:. Said differently,
5079:
5034:{\displaystyle f=0.}
5019:
4999:
4979:
4913:
4888:
4873:{\displaystyle f(U)}
4855:
4829:
4809:
4782:
4758:
4730:
4708:
4663:
4620:
4578:
4513:
4480:
4452:
4426:
4384:
4376:is a sufficient but
4319:
4292:
4261:
4195:
4153:
4126:
4079:
4059:
4039:
4011:
3991:
3965:
3938:
3918:
3895:
3871:
3847:
3823:
3803:
3781:
3755:
3717:
3689:
3667:
3643:
3597:
3577:
3554:
3534:
3523:{\displaystyle F(B)}
3505:
3485:
3465:
3433:
3413:
3378:
3364:locally convex space
3296:
3272:
3252:
3223:
3200:
3180:
3160:
3136:
3116:
3096:A TVS is said to be
3078:locally convex space
3046:locally convex space
3020:
2996:
2970:
2950:
2906:
2787:
2767:
2732:
2717:{\displaystyle F(U)}
2699:
2679:
2659:
2636:
2603:
2563:
2470:
2447:
2424:
2386:
2354:
2327:
2307:
2296:{\displaystyle F(B)}
2278:
2252:
2197:
2131:
2098:
2060:
2010:
1984:
1934:
1869:
1843:
1787:
1767:
1763:is bounded on a set
1747:
1687:
1649:
1626:
1611:{\displaystyle F(S)}
1593:
1570:
1534:
1508:
1477:
1431:
1411:
1381:
1356:
1334:
1279:
1252:
1186:
1149:
1129:
1105:
1085:
1065:
961:
935:
909:
879:
851:
831:
805:
785:
754:
730:
710:
686:
655:
627:
596:
563:
536:
477:
450:
423:
403:
365:
345:
325:
302:
282:
251:
225:
205:
175:
152:
130:
87:
12128:Functional analysis
11957:Interpolation space
11489:Operator topologies
11298:Functional calculus
11257:Mahler's conjecture
11236:Von Neumann algebra
10950:Functional analysis
10558:
10296:measurable function
10246:Functional calculus
10109:Parseval's identity
10022:Bessel's inequality
9969:Polar decomposition
9748:Uniform convergence
9506:Inner product space
9308:Schaefer, Helmut H.
9288:Functional analysis
8926:, pp. 225–273.
8914:, pp. 451–457.
8859:, pp. 441–457.
8802:, pp. 156–175.
8769:, pp. 126–128.
7306:into bounded sets.
6714:and if in addition
6415:The imaginary part
6069:{\displaystyle f=0}
5813:given in (4) above.
4909:that is, such that
4726:(rather than strict
3429:into some TVS then
2559:In contrast, a map
2183:Bounded linear maps
966: for all
18:functional analysis
11987:(Pseudo)Metrizable
11819:Minkowski addition
11671:Sublinear function
11323:Riemann hypothesis
11022:Topological vector
10908:Validated numerics
10863:
10819:Sobolev inequality
10787:
10736:
10697:
10660:
10589:Bounded variation
10570:
10538:
10523:Banach coordinate
10504:
10442:Minkowski addition
10104:M. Riesz extension
9584:Banach spaces are:
8738:Unbounded operator
8709:Linear functionals
8648:
8596:
8544:
8495:
8447:
8427:
8355:
8329:
8237:
8217:
8191:
8073:
8029:
8005:
7972:
7835:
7809:
7773:
7744:
7711:{\displaystyle X,}
7708:
7685:
7665:
7645:
7621:
7598:
7557:{\displaystyle X,}
7554:
7531:
7507:
7483:{\displaystyle F.}
7480:
7453:
7424:
7404:
7384:
7270:
7246:
7226:
7206:
7179:
7152:
7132:
7079:
7053:
7028:{\displaystyle f,}
7025:
7002:
6978:
6928:{\displaystyle r,}
6925:
6899:
6849:{\displaystyle r,}
6846:
6821:
6792:
6772:
6748:
6724:
6700:
6674:
6648:
6621:
6605:bornological space
6593:
6565:
6530:
6502:
6475:
6451:
6431:
6401:
6381:
6361:{\displaystyle f.}
6358:
6331:
6305:
6277:
6252:
6216:
6194:
6155:
6135:
6116:{\displaystyle X.}
6113:
6086:
6066:
6041:{\displaystyle X.}
6038:
6015:
5988:
5960:
5924:
5895:
5866:{\displaystyle U,}
5863:
5836:
5799:
5779:
5738:
5709:
5680:
5643:
5614:
5582:
5558:
5532:
5506:
5505:
5473:
5433:
5428:
5388:
5359:
5358:
5335:
5299:{\displaystyle sU}
5296:
5273:
5247:
5227:
5204:
5150:
5109:
5085:
5031:
5005:
4985:
4965:
4964:
4932:
4899:
4870:
4841:
4815:
4788:
4764:
4738:
4716:
4693:{\displaystyle U=}
4690:
4649:
4606:
4560:
4531:
4499:
4466:
4438:
4412:
4366:
4337:
4305:
4274:
4245:
4213:
4181:
4139:
4110:
4065:
4045:
4032:there exists some
4022:
3997:
3977:
3951:
3924:
3901:
3883:{\displaystyle X.}
3880:
3853:
3835:{\displaystyle X.}
3832:
3809:
3787:
3767:{\displaystyle X.}
3764:
3737:
3695:
3675:
3649:
3623:Sublinear function
3603:
3583:
3566:{\displaystyle X,}
3563:
3540:
3520:
3491:
3471:
3451:
3419:
3396:
3360:bornological space
3302:
3278:
3258:
3235:
3206:
3186:
3166:
3142:
3122:
3082:(pseudo)metrizable
3026:
3014:seminormable space
3002:
2982:{\displaystyle X,}
2979:
2956:
2924:
2793:
2773:
2744:{\displaystyle Y.}
2741:
2714:
2685:
2665:
2652:if there exists a
2642:
2629:locally bounded at
2615:
2581:
2531:
2506:
2453:
2436:{\displaystyle Y;}
2433:
2410:
2372:
2340:
2313:
2293:
2264:
2215:
2170:
2116:
2084:
2046:
1996:
1970:
1920:
1855:
1829:
1773:
1753:
1733:
1705:
1673:
1638:{\displaystyle Y,}
1635:
1608:
1576:
1552:
1520:
1483:
1463:
1449:
1417:
1397:
1364:
1342:
1316:
1297:
1258:
1204:
1164:
1135:
1111:
1091:
1071:
1048:
947:
931:there exists some
921:
891:
857:
837:
817:{\displaystyle Y.}
814:
791:
772:seminormable space
760:
736:
716:
692:
661:
633:
602:
579:
549:
518:
456:
429:
409:
386:
351:
331:
314:{\displaystyle Y,}
311:
288:
257:
237:{\displaystyle X.}
234:
211:
190:
158:
136:
105:
12115:
12114:
11834:Relative interior
11580:Bilinear operator
11464:Linear functional
11400:
11399:
11303:Integral operator
11080:
11079:
10916:
10915:
10628:Morrey–Campanato
10610:compact Hausdorff
10457:Relative interior
10311:Absolutely convex
10278:Projection-valued
9887:Strictly singular
9813:on Hilbert spaces
9574:of Hilbert spaces
9412:978-0-486-49353-4
9382:978-0-486-45352-1
9352:978-0-8247-8643-4
9325:978-1-4612-7155-0
9299:978-0-07-054236-5
9230:978-3-642-64988-2
9200:978-3-519-02224-4
9173:978-0-677-30020-7
9139:978-0-486-68143-6
9085:978-0-387-97245-9
9030:Bourbaki, Nicolas
9013:978-0-387-90081-0
8986:978-3-540-08662-8
8890:, pp. 53–55.
8871:, pp. 54–55.
8838:, pp. 47–50.
8450:{\displaystyle r}
8290:
8284:
8247:is a scalar then
8240:{\displaystyle s}
8176:
8175:
8169:
8123:
8117:
7990:
7901:
7695:is a seminorm on
7688:{\displaystyle p}
7668:{\displaystyle X}
7648:{\displaystyle f}
7624:{\displaystyle X}
7534:{\displaystyle f}
7510:{\displaystyle X}
7427:{\displaystyle Y}
7407:{\displaystyle D}
7273:{\displaystyle X}
7249:{\displaystyle X}
7229:{\displaystyle X}
7209:{\displaystyle X}
7194:linear functional
7182:{\displaystyle X}
7155:{\displaystyle X}
7005:{\displaystyle X}
6795:{\displaystyle X}
6775:{\displaystyle p}
6751:{\displaystyle f}
6727:{\displaystyle X}
6703:{\displaystyle f}
6677:{\displaystyle f}
6651:{\displaystyle f}
6624:{\displaystyle Y}
6596:{\displaystyle X}
6568:{\displaystyle f}
6533:{\displaystyle X}
6505:{\displaystyle f}
6478:{\displaystyle X}
6454:{\displaystyle f}
6404:{\displaystyle Y}
6384:{\displaystyle X}
6280:{\displaystyle f}
6219:{\displaystyle f}
6158:{\displaystyle X}
6138:{\displaystyle p}
6089:{\displaystyle f}
6018:{\displaystyle f}
5991:{\displaystyle f}
5880:
5802:{\displaystyle U}
5662:
5599:
5585:{\displaystyle U}
5451:
5427:
5320:
5250:{\displaystyle s}
5189:
5132:
5112:{\displaystyle f}
5088:{\displaystyle f}
5008:{\displaystyle 0}
4988:{\displaystyle U}
4917:
4818:{\displaystyle U}
4791:{\displaystyle f}
4767:{\displaystyle f}
4516:
4322:
4198:
4068:{\displaystyle X}
4055:of the origin in
4048:{\displaystyle U}
4000:{\displaystyle 0}
3927:{\displaystyle f}
3904:{\displaystyle f}
3856:{\displaystyle f}
3812:{\displaystyle f}
3790:{\displaystyle f}
3749:linear functional
3698:{\displaystyle X}
3652:{\displaystyle X}
3606:{\displaystyle B}
3586:{\displaystyle F}
3543:{\displaystyle F}
3530:is bounded since
3494:{\displaystyle X}
3474:{\displaystyle B}
3422:{\displaystyle X}
3305:{\displaystyle Y}
3281:{\displaystyle Y}
3261:{\displaystyle Y}
3218:identity function
3209:{\displaystyle X}
3189:{\displaystyle X}
3169:{\displaystyle X}
3145:{\displaystyle B}
3125:{\displaystyle B}
3029:{\displaystyle X}
3005:{\displaystyle X}
2838:A linear map is "
2796:{\displaystyle F}
2776:{\displaystyle x}
2688:{\displaystyle X}
2675:of this point in
2668:{\displaystyle U}
2645:{\displaystyle x}
2555:Local boundedness
2541:is finite. Every
2485:
2456:{\displaystyle Y}
2316:{\displaystyle X}
1776:{\displaystyle S}
1756:{\displaystyle F}
1690:
1579:{\displaystyle S}
1486:{\displaystyle S}
1434:
1420:{\displaystyle S}
1282:
1261:{\displaystyle S}
1138:{\displaystyle F}
1114:{\displaystyle Y}
1094:{\displaystyle Y}
1074:{\displaystyle X}
1016:
990:
967:
873:seminormed spaces
860:{\displaystyle Y}
840:{\displaystyle X}
794:{\displaystyle F}
763:{\displaystyle Y}
739:{\displaystyle Y}
719:{\displaystyle X}
695:{\displaystyle F}
664:{\displaystyle X}
636:{\displaystyle F}
605:{\displaystyle X}
468:weakly continuous
459:{\displaystyle F}
432:{\displaystyle Y}
412:{\displaystyle X}
354:{\displaystyle X}
334:{\displaystyle p}
291:{\displaystyle q}
260:{\displaystyle Y}
214:{\displaystyle F}
161:{\displaystyle F}
139:{\displaystyle F}
12150:
12133:Linear operators
12105:
12104:
12079:Uniformly smooth
11748:
11740:
11707:Balanced/Circled
11697:Absorbing/Radial
11427:
11420:
11413:
11404:
11403:
11390:
11389:
11308:Jones polynomial
11226:Operator algebra
10970:
10969:
10943:
10936:
10929:
10920:
10919:
10872:
10870:
10869:
10864:
10859:
10858:
10826:Triebel–Lizorkin
10796:
10794:
10793:
10788:
10786:
10782:
10781:
10776:
10745:
10743:
10742:
10737:
10735:
10734:
10706:
10704:
10703:
10698:
10696:
10695:
10669:
10667:
10666:
10661:
10650:
10649:
10579:
10577:
10576:
10571:
10566:
10557:
10552:
10513:
10511:
10510:
10505:
10366:
10344:
10326:Balanced/Circled
10124:Robinson-Ursescu
10042:Eberlein–Šmulian
9962:Spectral theorem
9758:Linear operators
9555:Uniformly smooth
9452:
9445:
9438:
9429:
9428:
9424:
9399:Wilansky, Albert
9394:
9369:Trèves, François
9364:
9337:
9303:
9291:
9284:(January 1991).
9277:
9250:
9217:Köthe, Gottfried
9212:
9185:
9165:
9151:
9124:
9103:Linear operators
9097:
9059:
9025:
8998:
8963:
8957:
8951:
8945:
8939:
8933:
8927:
8921:
8915:
8909:
8903:
8897:
8891:
8885:
8872:
8866:
8860:
8854:
8839:
8833:
8827:
8821:
8815:
8809:
8803:
8797:
8770:
8764:
8729:
8714:
8705:
8677:Compact operator
8673:
8657:
8655:
8654:
8649:
8644:
8643:
8605:
8603:
8602:
8597:
8589:
8569:
8553:
8551:
8550:
8545:
8537:
8520:
8504:
8502:
8501:
8496:
8494:
8493:
8456:
8454:
8453:
8448:
8436:
8434:
8433:
8428:
8417:
8409:
8401:
8384:
8383:
8364:
8362:
8361:
8356:
8338:
8336:
8335:
8330:
8328:
8311:
8303:
8295:
8288:
8282:
8281:
8261:
8246:
8244:
8243:
8238:
8226:
8224:
8223:
8218:
8213:
8196:
8190:
8173:
8167:
8151:
8134:
8121:
8115:
8114:
8097:
8082:
8080:
8079:
8074:
8071:
8054:
8038:
8036:
8035:
8030:
8027:
8010:
8004:
7981:
7979:
7978:
7973:
7953:
7936:
7925:
7908:
7902:
7899:
7844:
7842:
7841:
7836:
7818:
7816:
7815:
7810:
7808:
7782:
7780:
7779:
7774:
7753:
7751:
7750:
7745:
7737:
7729:
7717:
7715:
7714:
7709:
7694:
7692:
7691:
7686:
7674:
7672:
7671:
7666:
7654:
7652:
7651:
7646:
7630:
7628:
7627:
7622:
7607:
7605:
7604:
7599:
7563:
7561:
7560:
7555:
7540:
7538:
7537:
7532:
7516:
7514:
7513:
7508:
7489:
7487:
7486:
7481:
7462:
7460:
7459:
7454:
7433:
7431:
7430:
7425:
7413:
7411:
7410:
7405:
7393:
7391:
7390:
7385:
7359:
7358:
7328:
7327:
7279:
7277:
7276:
7271:
7255:
7253:
7252:
7247:
7235:
7233:
7232:
7227:
7215:
7213:
7212:
7207:
7188:
7186:
7185:
7180:
7161:
7159:
7158:
7153:
7141:
7139:
7138:
7133:
7088:
7086:
7085:
7080:
7062:
7060:
7059:
7054:
7034:
7032:
7031:
7026:
7011:
7009:
7008:
7003:
6987:
6985:
6984:
6979:
6934:
6932:
6931:
6926:
6908:
6906:
6905:
6900:
6855:
6853:
6852:
6847:
6830:
6828:
6827:
6822:
6801:
6799:
6798:
6793:
6781:
6779:
6778:
6773:
6757:
6755:
6754:
6749:
6733:
6731:
6730:
6725:
6709:
6707:
6706:
6701:
6683:
6681:
6680:
6675:
6657:
6655:
6654:
6649:
6630:
6628:
6627:
6622:
6607:(for example, a
6602:
6600:
6599:
6594:
6574:
6572:
6571:
6566:
6544:(for example, a
6539:
6537:
6536:
6531:
6511:
6509:
6508:
6503:
6487:sequential space
6484:
6482:
6481:
6476:
6460:
6458:
6457:
6452:
6440:
6438:
6437:
6432:
6410:
6408:
6407:
6402:
6390:
6388:
6387:
6382:
6367:
6365:
6364:
6359:
6340:
6338:
6337:
6332:
6314:
6312:
6311:
6306:
6286:
6284:
6283:
6278:
6262:is a continuous.
6261:
6259:
6258:
6253:
6251:
6243:
6225:
6223:
6222:
6217:
6203:
6201:
6200:
6195:
6184:
6176:
6164:
6162:
6161:
6156:
6144:
6142:
6141:
6136:
6122:
6120:
6119:
6114:
6095:
6093:
6092:
6087:
6075:
6073:
6072:
6067:
6047:
6045:
6044:
6039:
6024:
6022:
6021:
6016:
5997:
5995:
5994:
5989:
5969:
5967:
5966:
5961:
5956:
5955:
5933:
5931:
5930:
5925:
5917:
5900:
5894:
5872:
5870:
5869:
5864:
5848:(absolute) polar
5845:
5843:
5842:
5837:
5832:
5831:
5808:
5806:
5805:
5800:
5788:
5786:
5785:
5780:
5747:
5745:
5744:
5739:
5718:
5716:
5715:
5710:
5702:
5685:
5679:
5652:
5650:
5649:
5644:
5636:
5619:
5613:
5591:
5589:
5588:
5583:
5567:
5565:
5564:
5559:
5541:
5539:
5538:
5533:
5515:
5513:
5512:
5507:
5495:
5478:
5472:
5471:
5470:
5442:
5440:
5439:
5434:
5429:
5420:
5414:
5413:
5397:
5395:
5394:
5389:
5368:
5366:
5365:
5360:
5357:
5340:
5334:
5305:
5303:
5302:
5297:
5282:
5280:
5279:
5274:
5256:
5254:
5253:
5248:
5236:
5234:
5233:
5228:
5226:
5209:
5203:
5188:
5180:
5172:
5155:
5149:
5118:
5116:
5115:
5110:
5094:
5092:
5091:
5086:
5069:seminormed space
5040:
5038:
5037:
5032:
5014:
5012:
5011:
5006:
4994:
4992:
4991:
4986:
4974:
4972:
4971:
4966:
4954:
4937:
4931:
4908:
4906:
4905:
4900:
4895:
4879:
4877:
4876:
4871:
4850:
4848:
4847:
4842:
4824:
4822:
4821:
4816:
4797:
4795:
4794:
4789:
4773:
4771:
4770:
4765:
4747:
4745:
4744:
4739:
4725:
4723:
4722:
4717:
4699:
4697:
4696:
4691:
4658:
4656:
4655:
4650:
4633:
4615:
4613:
4612:
4607:
4605:
4604:
4569:
4567:
4566:
4561:
4553:
4536:
4530:
4508:
4506:
4505:
4500:
4498:
4497:
4475:
4473:
4472:
4467:
4465:
4447:
4445:
4444:
4439:
4421:
4419:
4418:
4413:
4411:
4410:
4375:
4373:
4372:
4367:
4359:
4342:
4336:
4314:
4312:
4311:
4306:
4304:
4303:
4283:
4281:
4280:
4275:
4273:
4272:
4254:
4252:
4251:
4246:
4235:
4218:
4212:
4190:
4188:
4187:
4182:
4180:
4179:
4148:
4146:
4145:
4140:
4138:
4137:
4119:
4117:
4116:
4111:
4106:
4105:
4074:
4072:
4071:
4066:
4054:
4052:
4051:
4046:
4031:
4029:
4028:
4023:
4018:
4007:in the codomain
4006:
4004:
4003:
3998:
3986:
3984:
3983:
3978:
3960:
3958:
3957:
3952:
3950:
3949:
3933:
3931:
3930:
3925:
3910:
3908:
3907:
3902:
3889:
3887:
3886:
3881:
3862:
3860:
3859:
3854:
3841:
3839:
3838:
3833:
3818:
3816:
3815:
3810:
3796:
3794:
3793:
3788:
3773:
3771:
3770:
3765:
3746:
3744:
3743:
3738:
3736:
3704:
3702:
3701:
3696:
3684:
3682:
3681:
3676:
3674:
3658:
3656:
3655:
3650:
3612:
3610:
3609:
3604:
3592:
3590:
3589:
3584:
3572:
3570:
3569:
3564:
3549:
3547:
3546:
3541:
3529:
3527:
3526:
3521:
3500:
3498:
3497:
3492:
3480:
3478:
3477:
3472:
3460:
3458:
3457:
3452:
3428:
3426:
3425:
3420:
3405:
3403:
3402:
3397:
3348:pseudometrizable
3343:be continuous.
3311:
3309:
3308:
3303:
3287:
3285:
3284:
3279:
3267:
3265:
3264:
3259:
3244:
3242:
3241:
3236:
3215:
3213:
3212:
3207:
3195:
3193:
3192:
3187:
3175:
3173:
3172:
3167:
3151:
3149:
3148:
3143:
3131:
3129:
3128:
3123:
3110:seminormed space
3035:
3033:
3032:
3027:
3011:
3009:
3008:
3003:
2991:is equivalent to
2988:
2986:
2985:
2980:
2965:
2963:
2962:
2957:
2933:
2931:
2930:
2925:
2850:) and thus also
2814:
2813:
2802:
2800:
2799:
2794:
2782:
2780:
2779:
2774:
2757:
2756:
2750:
2748:
2747:
2742:
2723:
2721:
2720:
2715:
2694:
2692:
2691:
2686:
2674:
2672:
2671:
2666:
2651:
2649:
2648:
2643:
2631:
2630:
2624:
2622:
2621:
2616:
2597:
2596:
2590:
2588:
2587:
2582:
2540:
2538:
2537:
2532:
2505:
2462:
2460:
2459:
2454:
2442:
2440:
2439:
2434:
2419:
2417:
2416:
2411:
2409:
2405:
2404:
2381:
2379:
2378:
2373:
2349:
2347:
2346:
2341:
2339:
2338:
2322:
2320:
2319:
2314:
2302:
2300:
2299:
2294:
2273:
2271:
2270:
2265:
2243:
2242:
2235:and is called a
2224:
2222:
2221:
2216:
2179:
2177:
2176:
2171:
2125:
2123:
2122:
2117:
2093:
2091:
2090:
2085:
2055:
2053:
2052:
2047:
2005:
2003:
2002:
1997:
1979:
1977:
1976:
1971:
1929:
1927:
1926:
1921:
1864:
1862:
1861:
1856:
1838:
1836:
1835:
1830:
1782:
1780:
1779:
1774:
1762:
1760:
1759:
1754:
1742:
1740:
1739:
1734:
1704:
1682:
1680:
1679:
1674:
1644:
1642:
1641:
1636:
1617:
1615:
1614:
1609:
1587:
1586:
1585:
1583:
1582:
1577:
1561:
1559:
1558:
1553:
1529:
1527:
1526:
1521:
1492:
1490:
1489:
1484:
1472:
1470:
1469:
1464:
1462:
1454:
1448:
1426:
1424:
1423:
1418:
1406:
1404:
1403:
1398:
1396:
1388:
1373:
1371:
1370:
1365:
1363:
1351:
1349:
1348:
1343:
1341:
1325:
1323:
1322:
1317:
1296:
1267:
1265:
1264:
1259:
1248:) then a subset
1246:seminormed space
1213:
1211:
1210:
1205:
1173:
1171:
1170:
1165:
1144:
1142:
1141:
1136:
1120:
1118:
1117:
1112:
1100:
1098:
1097:
1092:
1080:
1078:
1077:
1072:
1057:
1055:
1054:
1049:
1017:
1015: then
1014:
991:
988:
968:
965:
956:
954:
953:
948:
930:
928:
927:
922:
900:
898:
897:
892:
866:
864:
863:
858:
846:
844:
843:
838:
823:
821:
820:
815:
800:
798:
797:
792:
769:
767:
766:
761:
745:
743:
742:
737:
725:
723:
722:
717:
701:
699:
698:
693:
673:pseudometrizable
670:
668:
667:
662:
642:
640:
639:
634:
614:sequential space
611:
609:
608:
603:
588:
586:
585:
580:
575:
574:
558:
556:
555:
550:
548:
547:
527:
525:
524:
519:
517:
516:
504:
503:
488:
487:
482:
465:
463:
462:
457:
438:
436:
435:
430:
418:
416:
415:
410:
395:
393:
392:
387:
360:
358:
357:
352:
340:
338:
337:
332:
320:
318:
317:
312:
297:
295:
294:
289:
266:
264:
263:
258:
243:
241:
240:
235:
220:
218:
217:
212:
199:
197:
196:
191:
167:
165:
164:
159:
145:
143:
142:
137:
114:
112:
111:
106:
79:Bounded operator
12158:
12157:
12153:
12152:
12151:
12149:
12148:
12147:
12138:Operator theory
12118:
12117:
12116:
12111:
12093:
11855:B-complete/Ptak
11838:
11782:
11746:
11738:
11717:Bounding points
11680:
11622:Densely defined
11568:
11557:Bounded inverse
11503:
11437:
11431:
11401:
11396:
11378:
11342:Advanced topics
11337:
11261:
11240:
11199:
11165:Hilbert–Schmidt
11138:
11129:Gelfand–Naimark
11076:
11026:
10961:
10947:
10917:
10912:
10876:
10854:
10850:
10836:
10833:
10832:
10831:Wiener amalgam
10801:Segal–Bargmann
10777:
10772:
10771:
10767:
10762:
10759:
10758:
10730:
10726:
10724:
10721:
10720:
10691:
10687:
10685:
10682:
10681:
10639:
10635:
10633:
10630:
10629:
10584:Birnbaum–Orlicz
10562:
10553:
10542:
10536:
10533:
10532:
10487:
10484:
10483:
10461:
10417:Bounding points
10390:
10364:
10342:
10299:
10150:Banach manifold
10133:
10057:Gelfand–Naimark
9978:
9952:Spectral theory
9920:Banach algebras
9912:Operator theory
9906:
9867:Pseudo-monotone
9850:Hilbert–Schmidt
9830:Densely defined
9752:
9665:
9579:
9462:
9456:
9413:
9383:
9353:
9326:
9300:
9266:
9231:
9201:
9174:
9140:
9113:
9086:
9076:Springer-Verlag
9048:
9014:
8987:
8977:Springer-Verlag
8967:
8966:
8958:
8954:
8946:
8942:
8934:
8930:
8922:
8918:
8910:
8906:
8898:
8894:
8886:
8875:
8867:
8863:
8855:
8842:
8834:
8830:
8822:
8818:
8810:
8806:
8798:
8773:
8765:
8752:
8747:
8727:
8712:
8703:
8671:
8664:
8636:
8632:
8612:
8609:
8608:
8585:
8565:
8560:
8557:
8556:
8533:
8516:
8511:
8508:
8507:
8486:
8482:
8465:
8462:
8461:
8442:
8439:
8438:
8413:
8405:
8397:
8376:
8372:
8370:
8367:
8366:
8344:
8341:
8340:
8324:
8307:
8299:
8291:
8277:
8257:
8252:
8249:
8248:
8232:
8229:
8228:
8209:
8192:
8180:
8147:
8130:
8110:
8093:
8088:
8085:
8084:
8067:
8050:
8044:
8041:
8040:
8023:
8006:
7994:
7987:
7984:
7983:
7949:
7932:
7921:
7904:
7900: and
7898:
7850:
7847:
7846:
7824:
7821:
7820:
7804:
7790:
7787:
7786:
7759:
7756:
7755:
7754:if and only if
7733:
7725:
7723:
7720:
7719:
7700:
7697:
7696:
7680:
7677:
7676:
7660:
7657:
7656:
7640:
7637:
7636:
7616:
7613:
7612:
7569:
7566:
7565:
7546:
7543:
7542:
7526:
7523:
7522:
7502:
7499:
7498:
7495:
7472:
7469:
7468:
7439:
7436:
7435:
7419:
7416:
7415:
7399:
7396:
7395:
7394:for any subset
7351:
7347:
7320:
7316:
7314:
7311:
7310:
7286:
7265:
7262:
7261:
7241:
7238:
7237:
7221:
7218:
7217:
7201:
7198:
7197:
7174:
7171:
7170:
7147:
7144:
7143:
7121:
7118:
7117:
7107:
7093:(respectively,
7068:
7065:
7064:
7039:
7036:
7035:
7017:
7014:
7013:
6997:
6994:
6993:
6940:
6937:
6936:
6935:the half-space
6917:
6914:
6913:
6861:
6858:
6857:
6856:the half-space
6838:
6835:
6834:
6807:
6804:
6803:
6787:
6784:
6783:
6767:
6764:
6763:
6743:
6740:
6739:
6719:
6716:
6715:
6695:
6692:
6691:
6669:
6666:
6665:
6643:
6640:
6639:
6616:
6613:
6612:
6588:
6585:
6584:
6560:
6557:
6556:
6525:
6522:
6521:
6497:
6494:
6493:
6470:
6467:
6466:
6446:
6443:
6442:
6420:
6417:
6416:
6396:
6393:
6392:
6376:
6373:
6372:
6350:
6347:
6346:
6320:
6317:
6316:
6294:
6291:
6290:
6272:
6269:
6268:
6247:
6239:
6231:
6228:
6227:
6211:
6208:
6207:
6206:In particular,
6180:
6172:
6170:
6167:
6166:
6150:
6147:
6146:
6130:
6127:
6126:
6105:
6102:
6101:
6081:
6078:
6077:
6055:
6052:
6051:
6030:
6027:
6026:
6010:
6007:
6006:
5983:
5980:
5979:
5951:
5947:
5939:
5936:
5935:
5913:
5896:
5884:
5878:
5875:
5874:
5873:the inequality
5855:
5852:
5851:
5827:
5823:
5821:
5818:
5817:
5794:
5791:
5790:
5753:
5750:
5749:
5724:
5721:
5720:
5719:for every real
5698:
5681:
5666:
5660:
5657:
5656:
5632:
5615:
5603:
5597:
5594:
5593:
5577:
5574:
5573:
5547:
5544:
5543:
5521:
5518:
5517:
5491:
5474:
5466:
5462:
5455:
5448:
5445:
5444:
5418:
5409:
5405:
5403:
5400:
5399:
5374:
5371:
5370:
5353:
5336:
5324:
5311:
5308:
5307:
5288:
5285:
5284:
5262:
5259:
5258:
5242:
5239:
5238:
5222:
5205:
5193:
5184:
5176:
5168:
5151:
5136:
5130:
5127:
5126:
5104:
5101:
5100:
5080:
5077:
5076:
5020:
5017:
5016:
5015:if and only if
5000:
4997:
4996:
4980:
4977:
4976:
4950:
4933:
4921:
4914:
4911:
4910:
4891:
4889:
4886:
4885:
4856:
4853:
4852:
4830:
4827:
4826:
4810:
4807:
4806:
4802:of its domain.
4783:
4780:
4779:
4759:
4756:
4755:
4748:) inequalities.
4731:
4728:
4727:
4709:
4706:
4705:
4664:
4661:
4660:
4629:
4621:
4618:
4617:
4600:
4596:
4579:
4576:
4575:
4549:
4532:
4520:
4514:
4511:
4510:
4493:
4489:
4481:
4478:
4477:
4461:
4453:
4450:
4449:
4427:
4424:
4423:
4406:
4402:
4385:
4382:
4381:
4355:
4338:
4326:
4320:
4317:
4316:
4299:
4295:
4293:
4290:
4289:
4268:
4264:
4262:
4259:
4258:
4231:
4214:
4202:
4196:
4193:
4192:
4175:
4171:
4154:
4151:
4150:
4133:
4129:
4127:
4124:
4123:
4101:
4097:
4080:
4077:
4076:
4060:
4057:
4056:
4040:
4037:
4036:
4014:
4012:
4009:
4008:
3992:
3989:
3988:
3966:
3963:
3962:
3945:
3941:
3939:
3936:
3935:
3919:
3916:
3915:
3914:By definition,
3896:
3893:
3892:
3872:
3869:
3868:
3848:
3845:
3844:
3824:
3821:
3820:
3804:
3801:
3800:
3782:
3779:
3778:
3756:
3753:
3752:
3732:
3718:
3715:
3714:
3690:
3687:
3686:
3670:
3668:
3665:
3664:
3644:
3641:
3640:
3637:
3625:
3619:
3598:
3595:
3594:
3578:
3575:
3574:
3555:
3552:
3551:
3535:
3532:
3531:
3506:
3503:
3502:
3486:
3483:
3482:
3466:
3463:
3462:
3434:
3431:
3430:
3414:
3411:
3410:
3379:
3376:
3375:
3297:
3294:
3293:
3288:is necessarily
3273:
3270:
3269:
3253:
3250:
3249:
3224:
3221:
3220:
3201:
3198:
3197:
3181:
3178:
3177:
3161:
3158:
3157:
3137:
3134:
3133:
3117:
3114:
3113:
3098:locally bounded
3058:
3021:
3018:
3017:
2997:
2994:
2993:
2971:
2968:
2967:
2951:
2948:
2947:
2940:TVS-isomorphism
2907:
2904:
2903:
2879:
2836:
2812:locally bounded
2811:
2810:
2788:
2785:
2784:
2768:
2765:
2764:
2754:
2753:
2733:
2730:
2729:
2700:
2697:
2696:
2680:
2677:
2676:
2660:
2657:
2656:
2637:
2634:
2633:
2628:
2627:
2604:
2601:
2600:
2594:
2593:
2564:
2561:
2560:
2557:
2489:
2471:
2468:
2467:
2448:
2445:
2444:
2425:
2422:
2421:
2400:
2396:
2392:
2387:
2384:
2383:
2355:
2352:
2351:
2334:
2330:
2328:
2325:
2324:
2308:
2305:
2304:
2279:
2276:
2275:
2274:of its domain,
2253:
2250:
2249:
2238:
2237:
2198:
2195:
2194:
2191:
2132:
2129:
2128:
2099:
2096:
2095:
2061:
2058:
2057:
2011:
2008:
2007:
1985:
1982:
1981:
1935:
1932:
1931:
1870:
1867:
1866:
1844:
1841:
1840:
1788:
1785:
1784:
1768:
1765:
1764:
1748:
1745:
1744:
1694:
1688:
1685:
1684:
1650:
1647:
1646:
1627:
1624:
1623:
1594:
1591:
1590:
1571:
1568:
1567:
1565:
1564:
1535:
1532:
1531:
1509:
1506:
1505:
1478:
1475:
1474:
1458:
1450:
1438:
1432:
1429:
1428:
1412:
1409:
1408:
1392:
1384:
1382:
1379:
1378:
1359:
1357:
1354:
1353:
1337:
1335:
1332:
1331:
1286:
1280:
1277:
1276:
1275:, meaning that
1253:
1250:
1249:
1234:
1187:
1184:
1183:
1180:
1150:
1147:
1146:
1130:
1127:
1126:
1106:
1103:
1102:
1086:
1083:
1082:
1066:
1063:
1062:
1013:
987:
964:
962:
959:
958:
936:
933:
932:
910:
907:
906:
880:
877:
876:
852:
849:
848:
832:
829:
828:
806:
803:
802:
786:
783:
782:
755:
752:
751:
731:
728:
727:
711:
708:
707:
687:
684:
683:
656:
653:
652:
628:
625:
624:
597:
594:
593:
570:
566:
564:
561:
560:
543:
539:
537:
534:
533:
512:
508:
499:
495:
483:
481:
480:
478:
475:
474:
451:
448:
447:
424:
421:
420:
404:
401:
400:
366:
363:
362:
346:
343:
342:
326:
323:
322:
303:
300:
299:
283:
280:
279:
252:
249:
248:
226:
223:
222:
206:
203:
202:
176:
173:
172:
153:
150:
149:
131:
128:
127:
117:linear operator
88:
85:
84:
81:
75:
70:
60:
12:
11:
5:
12156:
12146:
12145:
12140:
12135:
12130:
12113:
12112:
12110:
12109:
12098:
12095:
12094:
12092:
12091:
12086:
12081:
12076:
12074:Ultrabarrelled
12066:
12060:
12055:
12049:
12044:
12039:
12034:
12029:
12024:
12015:
12009:
12004:
12002:Quasi-complete
11999:
11997:Quasibarrelled
11994:
11989:
11984:
11979:
11974:
11969:
11964:
11959:
11954:
11949:
11944:
11939:
11938:
11937:
11927:
11922:
11917:
11912:
11907:
11902:
11897:
11892:
11887:
11877:
11872:
11862:
11857:
11852:
11846:
11844:
11840:
11839:
11837:
11836:
11826:
11821:
11816:
11811:
11806:
11796:
11790:
11788:
11787:Set operations
11784:
11783:
11781:
11780:
11775:
11770:
11765:
11760:
11755:
11750:
11742:
11734:
11729:
11724:
11719:
11714:
11709:
11704:
11699:
11694:
11688:
11686:
11682:
11681:
11679:
11678:
11673:
11668:
11663:
11658:
11657:
11656:
11651:
11646:
11636:
11631:
11630:
11629:
11624:
11619:
11614:
11609:
11604:
11599:
11589:
11588:
11587:
11576:
11574:
11570:
11569:
11567:
11566:
11561:
11560:
11559:
11549:
11543:
11534:
11529:
11524:
11522:Banach–Alaoglu
11519:
11517:Anderson–Kadec
11513:
11511:
11505:
11504:
11502:
11501:
11496:
11491:
11486:
11481:
11476:
11471:
11466:
11461:
11456:
11451:
11445:
11443:
11442:Basic concepts
11439:
11438:
11430:
11429:
11422:
11415:
11407:
11398:
11397:
11395:
11394:
11383:
11380:
11379:
11377:
11376:
11371:
11366:
11361:
11359:Choquet theory
11356:
11351:
11345:
11343:
11339:
11338:
11336:
11335:
11325:
11320:
11315:
11310:
11305:
11300:
11295:
11290:
11285:
11280:
11275:
11269:
11267:
11263:
11262:
11260:
11259:
11254:
11248:
11246:
11242:
11241:
11239:
11238:
11233:
11228:
11223:
11218:
11213:
11211:Banach algebra
11207:
11205:
11201:
11200:
11198:
11197:
11192:
11187:
11182:
11177:
11172:
11167:
11162:
11157:
11152:
11146:
11144:
11140:
11139:
11137:
11136:
11134:Banach–Alaoglu
11131:
11126:
11121:
11116:
11111:
11106:
11101:
11096:
11090:
11088:
11082:
11081:
11078:
11077:
11075:
11074:
11069:
11064:
11062:Locally convex
11059:
11045:
11040:
11034:
11032:
11028:
11027:
11025:
11024:
11019:
11014:
11009:
11004:
10999:
10994:
10989:
10984:
10979:
10973:
10967:
10963:
10962:
10946:
10945:
10938:
10931:
10923:
10914:
10913:
10911:
10910:
10905:
10900:
10895:
10890:
10884:
10882:
10878:
10877:
10875:
10874:
10862:
10857:
10853:
10849:
10846:
10843:
10840:
10828:
10823:
10822:
10821:
10811:
10809:Sequence space
10806:
10798:
10785:
10780:
10775:
10770:
10766:
10754:
10753:
10752:
10747:
10733:
10729:
10710:
10709:
10708:
10694:
10690:
10671:
10659:
10656:
10653:
10648:
10645:
10642:
10638:
10625:
10617:
10612:
10599:
10594:
10586:
10581:
10569:
10565:
10561:
10556:
10551:
10548:
10545:
10541:
10528:
10520:
10515:
10503:
10500:
10497:
10494:
10491:
10480:
10471:
10469:
10463:
10462:
10460:
10459:
10449:
10444:
10439:
10434:
10429:
10424:
10419:
10414:
10404:
10398:
10396:
10392:
10391:
10389:
10388:
10383:
10378:
10373:
10368:
10360:
10346:
10338:
10333:
10328:
10323:
10318:
10313:
10307:
10305:
10301:
10300:
10298:
10297:
10287:
10286:
10285:
10280:
10275:
10265:
10264:
10263:
10258:
10253:
10243:
10242:
10241:
10236:
10231:
10226:
10224:Gelfand–Pettis
10221:
10216:
10206:
10205:
10204:
10199:
10194:
10189:
10184:
10174:
10169:
10164:
10159:
10158:
10157:
10147:
10141:
10139:
10135:
10134:
10132:
10131:
10126:
10121:
10116:
10111:
10106:
10101:
10096:
10091:
10086:
10081:
10076:
10075:
10074:
10064:
10059:
10054:
10049:
10044:
10039:
10034:
10029:
10024:
10019:
10014:
10009:
10004:
9999:
9997:Banach–Alaoglu
9994:
9992:Anderson–Kadec
9988:
9986:
9980:
9979:
9977:
9976:
9971:
9966:
9965:
9964:
9959:
9949:
9948:
9947:
9942:
9932:
9930:Operator space
9927:
9922:
9916:
9914:
9908:
9907:
9905:
9904:
9899:
9894:
9889:
9884:
9879:
9874:
9869:
9864:
9863:
9862:
9852:
9847:
9846:
9845:
9840:
9832:
9827:
9817:
9816:
9815:
9805:
9800:
9790:
9789:
9788:
9783:
9778:
9768:
9762:
9760:
9754:
9753:
9751:
9750:
9745:
9740:
9739:
9738:
9733:
9723:
9722:
9721:
9716:
9706:
9701:
9696:
9695:
9694:
9684:
9679:
9673:
9671:
9667:
9666:
9664:
9663:
9658:
9653:
9652:
9651:
9641:
9636:
9631:
9630:
9629:
9618:Locally convex
9615:
9614:
9613:
9603:
9598:
9593:
9587:
9585:
9581:
9580:
9578:
9577:
9570:Tensor product
9563:
9557:
9552:
9546:
9541:
9535:
9530:
9525:
9515:
9514:
9513:
9508:
9498:
9493:
9491:Banach lattice
9488:
9487:
9486:
9476:
9470:
9468:
9464:
9463:
9455:
9454:
9447:
9440:
9432:
9426:
9425:
9411:
9395:
9381:
9365:
9351:
9338:
9324:
9304:
9298:
9278:
9265:978-1584888666
9264:
9251:
9229:
9213:
9199:
9186:
9172:
9152:
9138:
9125:
9111:
9098:
9084:
9060:
9046:
9026:
9012:
8999:
8985:
8965:
8964:
8962:, p. 128.
8952:
8940:
8928:
8916:
8904:
8892:
8873:
8861:
8840:
8828:
8826:, p. 476.
8816:
8804:
8771:
8749:
8748:
8746:
8743:
8742:
8741:
8735:
8730:
8721:
8715:
8706:
8697:
8692:
8686:
8680:
8674:
8663:
8660:
8659:
8658:
8647:
8642:
8639:
8635:
8631:
8628:
8625:
8622:
8619:
8616:
8606:
8595:
8592:
8588:
8584:
8581:
8578:
8575:
8572:
8568:
8564:
8554:
8543:
8540:
8536:
8532:
8529:
8526:
8523:
8519:
8515:
8505:
8492:
8489:
8485:
8481:
8478:
8475:
8472:
8469:
8446:
8426:
8423:
8420:
8416:
8412:
8408:
8404:
8400:
8396:
8393:
8390:
8387:
8382:
8379:
8375:
8354:
8351:
8348:
8327:
8323:
8320:
8317:
8314:
8310:
8306:
8302:
8298:
8294:
8287:
8280:
8276:
8273:
8270:
8267:
8264:
8260:
8256:
8236:
8216:
8212:
8208:
8205:
8202:
8199:
8195:
8189:
8186:
8183:
8179:
8172:
8166:
8163:
8160:
8157:
8154:
8150:
8146:
8143:
8140:
8137:
8133:
8129:
8126:
8120:
8113:
8109:
8106:
8103:
8100:
8096:
8092:
8070:
8066:
8063:
8060:
8057:
8053:
8049:
8026:
8022:
8019:
8016:
8013:
8009:
8003:
8000:
7997:
7993:
7971:
7968:
7965:
7962:
7959:
7956:
7952:
7948:
7945:
7942:
7939:
7935:
7931:
7928:
7924:
7920:
7917:
7914:
7911:
7907:
7896:
7893:
7890:
7887:
7884:
7881:
7878:
7875:
7872:
7869:
7866:
7863:
7860:
7857:
7854:
7834:
7831:
7828:
7807:
7803:
7800:
7797:
7794:
7772:
7769:
7766:
7763:
7743:
7740:
7736:
7732:
7728:
7707:
7704:
7684:
7664:
7644:
7620:
7597:
7594:
7591:
7588:
7585:
7582:
7579:
7576:
7573:
7553:
7550:
7530:
7506:
7494:
7491:
7479:
7476:
7452:
7449:
7446:
7443:
7423:
7403:
7383:
7380:
7377:
7374:
7371:
7368:
7365:
7362:
7357:
7354:
7350:
7346:
7343:
7340:
7337:
7334:
7331:
7326:
7323:
7319:
7290:locally convex
7285:
7282:
7269:
7258:bounded subset
7245:
7225:
7205:
7192:
7178:
7151:
7131:
7128:
7125:
7106:
7103:
7078:
7075:
7072:
7052:
7049:
7046:
7043:
7024:
7021:
7001:
6990:
6989:
6977:
6974:
6971:
6968:
6965:
6962:
6959:
6956:
6953:
6950:
6947:
6944:
6924:
6921:
6910:
6898:
6895:
6892:
6889:
6886:
6883:
6880:
6877:
6874:
6871:
6868:
6865:
6845:
6842:
6833:For some real
6831:
6820:
6817:
6814:
6811:
6791:
6771:
6747:
6723:
6712:
6711:
6699:
6689:
6673:
6663:
6647:
6633:locally convex
6620:
6592:
6583:If the domain
6581:
6580:
6564:
6529:
6520:If the domain
6518:
6517:
6501:
6474:
6465:If the domain
6463:
6462:
6461:is continuous.
6450:
6430:
6427:
6424:
6400:
6380:
6369:
6368:
6357:
6354:
6330:
6327:
6324:
6304:
6301:
6298:
6288:
6276:
6265:
6264:
6263:
6250:
6246:
6242:
6238:
6235:
6215:
6193:
6190:
6187:
6183:
6179:
6175:
6154:
6134:
6123:
6112:
6109:
6099:
6085:
6065:
6062:
6059:
6048:
6037:
6034:
6014:
6005:The kernel of
6003:
6002:of its domain.
5987:
5977:
5976:
5975:
5972:duality theory
5959:
5954:
5950:
5946:
5943:
5923:
5920:
5916:
5912:
5909:
5906:
5903:
5899:
5893:
5890:
5887:
5883:
5862:
5859:
5835:
5830:
5826:
5814:
5798:
5778:
5775:
5772:
5769:
5766:
5763:
5760:
5757:
5737:
5734:
5731:
5728:
5708:
5705:
5701:
5697:
5694:
5691:
5688:
5684:
5678:
5675:
5672:
5669:
5665:
5642:
5639:
5635:
5631:
5628:
5625:
5622:
5618:
5612:
5609:
5606:
5602:
5581:
5570:
5569:
5568:
5557:
5554:
5551:
5531:
5528:
5525:
5504:
5501:
5498:
5494:
5490:
5487:
5484:
5481:
5477:
5469:
5465:
5461:
5458:
5454:
5432:
5426:
5423:
5417:
5412:
5408:
5387:
5384:
5381:
5378:
5356:
5352:
5349:
5346:
5343:
5339:
5333:
5330:
5327:
5323:
5318:
5315:
5295:
5292:
5272:
5269:
5266:
5246:
5225:
5221:
5218:
5215:
5212:
5208:
5202:
5199:
5196:
5192:
5187:
5183:
5179:
5175:
5171:
5167:
5164:
5161:
5158:
5154:
5148:
5145:
5142:
5139:
5135:
5108:
5084:
5074:
5073:
5072:
5058:
5046:
5041:
5030:
5027:
5024:
5004:
4984:
4963:
4960:
4957:
4953:
4949:
4946:
4943:
4940:
4936:
4930:
4927:
4924:
4920:
4898:
4894:
4882:bounded subset
4869:
4866:
4863:
4860:
4840:
4837:
4834:
4825:of some point
4814:
4787:
4763:
4753:
4752:
4751:
4750:
4749:
4736:
4714:
4689:
4686:
4683:
4680:
4677:
4674:
4671:
4668:
4648:
4645:
4642:
4639:
4636:
4632:
4628:
4625:
4603:
4599:
4595:
4592:
4589:
4586:
4583:
4574:condition for
4573:
4572:not sufficient
4559:
4556:
4552:
4548:
4545:
4542:
4539:
4535:
4529:
4526:
4523:
4519:
4496:
4492:
4488:
4485:
4464:
4460:
4457:
4437:
4434:
4431:
4409:
4405:
4401:
4398:
4395:
4392:
4389:
4380:condition for
4379:
4365:
4362:
4358:
4354:
4351:
4348:
4345:
4341:
4335:
4332:
4329:
4325:
4302:
4298:
4271:
4267:
4244:
4241:
4238:
4234:
4230:
4227:
4224:
4221:
4217:
4211:
4208:
4205:
4201:
4178:
4174:
4170:
4167:
4164:
4161:
4158:
4136:
4132:
4120:
4109:
4104:
4100:
4096:
4093:
4090:
4087:
4084:
4064:
4044:
4021:
4017:
3996:
3976:
3973:
3970:
3948:
3944:
3923:
3900:
3890:
3879:
3876:
3852:
3842:
3831:
3828:
3808:
3798:
3797:is continuous.
3786:
3763:
3760:
3735:
3731:
3728:
3725:
3722:
3711:locally convex
3694:
3673:
3648:
3636:
3633:
3618:
3615:
3602:
3582:
3562:
3559:
3539:
3519:
3516:
3513:
3510:
3490:
3470:
3450:
3447:
3444:
3441:
3438:
3418:
3395:
3392:
3389:
3386:
3383:
3342:
3318:
3301:
3277:
3257:
3234:
3231:
3228:
3205:
3185:
3165:
3141:
3121:
3099:
3071:
3057:
3054:
3043:
3038:normable space
3025:
3001:
2978:
2975:
2955:
2923:
2920:
2917:
2914:
2911:
2901:
2888:
2878:
2875:
2835:
2832:
2829:
2824:
2815:
2806:
2792:
2772:
2762:
2758:
2740:
2737:
2726:bounded subset
2713:
2710:
2707:
2704:
2684:
2664:
2641:
2632:
2614:
2611:
2608:
2598:
2591:is said to be
2580:
2577:
2574:
2571:
2568:
2530:
2527:
2524:
2521:
2518:
2515:
2512:
2509:
2504:
2501:
2498:
2495:
2492:
2488:
2484:
2481:
2478:
2475:
2452:
2432:
2429:
2408:
2403:
2399:
2395:
2391:
2371:
2368:
2365:
2362:
2359:
2337:
2333:
2312:
2292:
2289:
2286:
2283:
2263:
2260:
2257:
2244:
2234:
2229:is said to be
2214:
2211:
2208:
2205:
2202:
2169:
2166:
2163:
2160:
2157:
2154:
2151:
2148:
2145:
2142:
2139:
2136:
2115:
2112:
2109:
2106:
2103:
2083:
2080:
2077:
2074:
2071:
2068:
2065:
2045:
2042:
2039:
2036:
2033:
2030:
2027:
2024:
2021:
2018:
2015:
1995:
1992:
1989:
1969:
1966:
1963:
1960:
1957:
1954:
1951:
1948:
1945:
1942:
1939:
1919:
1916:
1913:
1910:
1907:
1904:
1901:
1898:
1895:
1892:
1889:
1886:
1883:
1880:
1877:
1874:
1854:
1851:
1848:
1828:
1825:
1822:
1819:
1816:
1813:
1810:
1807:
1804:
1801:
1798:
1795:
1792:
1772:
1752:
1732:
1729:
1726:
1723:
1720:
1717:
1714:
1711:
1708:
1703:
1700:
1697:
1693:
1672:
1669:
1666:
1663:
1660:
1657:
1654:
1634:
1631:
1620:bounded subset
1607:
1604:
1601:
1598:
1588:
1575:
1562:is said to be
1551:
1548:
1545:
1542:
1539:
1530:is a set then
1519:
1516:
1513:
1482:
1461:
1457:
1453:
1447:
1444:
1441:
1437:
1416:
1395:
1391:
1387:
1376:absolute value
1362:
1340:
1329:
1315:
1312:
1309:
1306:
1303:
1300:
1295:
1292:
1289:
1285:
1274:
1257:
1226:Bounded subset
1203:
1200:
1197:
1194:
1191:
1179:
1176:
1175:
1174:
1163:
1160:
1157:
1154:
1134:
1110:
1090:
1070:
1059:
1058:
1047:
1044:
1041:
1038:
1035:
1032:
1029:
1026:
1023:
1020:
1012:
1009:
1006:
1003:
1000:
997:
994:
989: if
986:
983:
980:
977:
974:
971:
946:
943:
940:
920:
917:
914:
890:
887:
884:
856:
836:
825:
824:
813:
810:
790:
759:
748:
747:
735:
715:
691:
660:
649:
648:
632:
601:
590:
589:
578:
573:
569:
546:
542:
530:equicontinuous
515:
511:
507:
502:
498:
494:
491:
486:
455:
428:
408:
397:
396:
385:
382:
379:
376:
373:
370:
350:
330:
310:
307:
287:
269:locally convex
256:
245:
244:
233:
230:
210:
200:
189:
186:
183:
180:
157:
147:
146:is continuous.
135:
104:
101:
98:
95:
92:
74:
71:
59:
56:
9:
6:
4:
3:
2:
12155:
12144:
12141:
12139:
12136:
12134:
12131:
12129:
12126:
12125:
12123:
12108:
12100:
12099:
12096:
12090:
12087:
12085:
12082:
12080:
12077:
12075:
12071:
12067:
12065:) convex
12064:
12061:
12059:
12056:
12054:
12050:
12048:
12045:
12043:
12040:
12038:
12037:Semi-complete
12035:
12033:
12030:
12028:
12025:
12023:
12019:
12016:
12014:
12010:
12008:
12005:
12003:
12000:
11998:
11995:
11993:
11990:
11988:
11985:
11983:
11980:
11978:
11975:
11973:
11970:
11968:
11965:
11963:
11960:
11958:
11955:
11953:
11952:Infrabarreled
11950:
11948:
11945:
11943:
11940:
11936:
11933:
11932:
11931:
11928:
11926:
11923:
11921:
11918:
11916:
11913:
11911:
11910:Distinguished
11908:
11906:
11903:
11901:
11898:
11896:
11893:
11891:
11888:
11886:
11882:
11878:
11876:
11873:
11871:
11867:
11863:
11861:
11858:
11856:
11853:
11851:
11848:
11847:
11845:
11843:Types of TVSs
11841:
11835:
11831:
11827:
11825:
11822:
11820:
11817:
11815:
11812:
11810:
11807:
11805:
11801:
11797:
11795:
11792:
11791:
11789:
11785:
11779:
11776:
11774:
11771:
11769:
11766:
11764:
11763:Prevalent/Shy
11761:
11759:
11756:
11754:
11753:Extreme point
11751:
11749:
11743:
11741:
11735:
11733:
11730:
11728:
11725:
11723:
11720:
11718:
11715:
11713:
11710:
11708:
11705:
11703:
11700:
11698:
11695:
11693:
11690:
11689:
11687:
11685:Types of sets
11683:
11677:
11674:
11672:
11669:
11667:
11664:
11662:
11659:
11655:
11652:
11650:
11647:
11645:
11642:
11641:
11640:
11637:
11635:
11632:
11628:
11627:Discontinuous
11625:
11623:
11620:
11618:
11615:
11613:
11610:
11608:
11605:
11603:
11600:
11598:
11595:
11594:
11593:
11590:
11586:
11583:
11582:
11581:
11578:
11577:
11575:
11571:
11565:
11562:
11558:
11555:
11554:
11553:
11550:
11547:
11544:
11542:
11538:
11535:
11533:
11530:
11528:
11525:
11523:
11520:
11518:
11515:
11514:
11512:
11510:
11506:
11500:
11497:
11495:
11492:
11490:
11487:
11485:
11484:Metrizability
11482:
11480:
11477:
11475:
11472:
11470:
11469:Fréchet space
11467:
11465:
11462:
11460:
11457:
11455:
11452:
11450:
11447:
11446:
11444:
11440:
11435:
11428:
11423:
11421:
11416:
11414:
11409:
11408:
11405:
11393:
11385:
11384:
11381:
11375:
11372:
11370:
11367:
11365:
11364:Weak topology
11362:
11360:
11357:
11355:
11352:
11350:
11347:
11346:
11344:
11340:
11333:
11329:
11326:
11324:
11321:
11319:
11316:
11314:
11311:
11309:
11306:
11304:
11301:
11299:
11296:
11294:
11291:
11289:
11288:Index theorem
11286:
11284:
11281:
11279:
11276:
11274:
11271:
11270:
11268:
11264:
11258:
11255:
11253:
11250:
11249:
11247:
11245:Open problems
11243:
11237:
11234:
11232:
11229:
11227:
11224:
11222:
11219:
11217:
11214:
11212:
11209:
11208:
11206:
11202:
11196:
11193:
11191:
11188:
11186:
11183:
11181:
11178:
11176:
11173:
11171:
11168:
11166:
11163:
11161:
11158:
11156:
11153:
11151:
11148:
11147:
11145:
11141:
11135:
11132:
11130:
11127:
11125:
11122:
11120:
11117:
11115:
11112:
11110:
11107:
11105:
11102:
11100:
11097:
11095:
11092:
11091:
11089:
11087:
11083:
11073:
11070:
11068:
11065:
11063:
11060:
11057:
11053:
11049:
11046:
11044:
11041:
11039:
11036:
11035:
11033:
11029:
11023:
11020:
11018:
11015:
11013:
11010:
11008:
11005:
11003:
11000:
10998:
10995:
10993:
10990:
10988:
10985:
10983:
10980:
10978:
10975:
10974:
10971:
10968:
10964:
10959:
10955:
10951:
10944:
10939:
10937:
10932:
10930:
10925:
10924:
10921:
10909:
10906:
10904:
10901:
10899:
10896:
10894:
10891:
10889:
10886:
10885:
10883:
10879:
10873:
10855:
10851:
10847:
10844:
10838:
10829:
10827:
10824:
10820:
10817:
10816:
10815:
10812:
10810:
10807:
10805:
10804:
10799:
10797:
10783:
10778:
10768:
10764:
10755:
10751:
10748:
10746:
10727:
10718:
10717:
10716:
10715:
10711:
10707:
10688:
10679:
10678:
10677:
10676:
10672:
10670:
10646:
10643:
10640:
10636:
10626:
10624:
10623:
10618:
10616:
10613:
10611:
10609:
10605:
10600:
10598:
10595:
10593:
10592:
10587:
10585:
10582:
10580:
10554:
10549:
10546:
10543:
10539:
10529:
10527:
10526:
10521:
10519:
10516:
10514:
10492:
10489:
10481:
10479:
10478:
10473:
10472:
10470:
10468:
10464:
10458:
10454:
10450:
10448:
10445:
10443:
10440:
10438:
10435:
10433:
10430:
10428:
10427:Extreme point
10425:
10423:
10420:
10418:
10415:
10413:
10409:
10405:
10403:
10400:
10399:
10397:
10393:
10387:
10384:
10382:
10379:
10377:
10374:
10372:
10369:
10367:
10361:
10358:
10354:
10350:
10347:
10345:
10339:
10337:
10334:
10332:
10329:
10327:
10324:
10322:
10319:
10317:
10314:
10312:
10309:
10308:
10306:
10304:Types of sets
10302:
10295:
10291:
10288:
10284:
10281:
10279:
10276:
10274:
10271:
10270:
10269:
10266:
10262:
10259:
10257:
10254:
10252:
10249:
10248:
10247:
10244:
10240:
10237:
10235:
10232:
10230:
10227:
10225:
10222:
10220:
10217:
10215:
10212:
10211:
10210:
10207:
10203:
10200:
10198:
10195:
10193:
10190:
10188:
10185:
10183:
10180:
10179:
10178:
10175:
10173:
10170:
10168:
10167:Convex series
10165:
10163:
10162:Bochner space
10160:
10156:
10153:
10152:
10151:
10148:
10146:
10143:
10142:
10140:
10136:
10130:
10127:
10125:
10122:
10120:
10117:
10115:
10114:Riesz's lemma
10112:
10110:
10107:
10105:
10102:
10100:
10099:Mazur's lemma
10097:
10095:
10092:
10090:
10087:
10085:
10082:
10080:
10077:
10073:
10070:
10069:
10068:
10065:
10063:
10060:
10058:
10055:
10053:
10052:Gelfand–Mazur
10050:
10048:
10045:
10043:
10040:
10038:
10035:
10033:
10030:
10028:
10025:
10023:
10020:
10018:
10015:
10013:
10010:
10008:
10005:
10003:
10000:
9998:
9995:
9993:
9990:
9989:
9987:
9985:
9981:
9975:
9972:
9970:
9967:
9963:
9960:
9958:
9955:
9954:
9953:
9950:
9946:
9943:
9941:
9938:
9937:
9936:
9933:
9931:
9928:
9926:
9923:
9921:
9918:
9917:
9915:
9913:
9909:
9903:
9900:
9898:
9895:
9893:
9890:
9888:
9885:
9883:
9880:
9878:
9875:
9873:
9870:
9868:
9865:
9861:
9858:
9857:
9856:
9853:
9851:
9848:
9844:
9841:
9839:
9836:
9835:
9833:
9831:
9828:
9826:
9822:
9818:
9814:
9811:
9810:
9809:
9806:
9804:
9801:
9799:
9795:
9791:
9787:
9784:
9782:
9779:
9777:
9774:
9773:
9772:
9769:
9767:
9764:
9763:
9761:
9759:
9755:
9749:
9746:
9744:
9741:
9737:
9734:
9732:
9729:
9728:
9727:
9724:
9720:
9717:
9715:
9712:
9711:
9710:
9707:
9705:
9702:
9700:
9697:
9693:
9690:
9689:
9688:
9685:
9683:
9680:
9678:
9675:
9674:
9672:
9668:
9662:
9659:
9657:
9654:
9650:
9647:
9646:
9645:
9642:
9640:
9637:
9635:
9632:
9628:
9624:
9621:
9620:
9619:
9616:
9612:
9609:
9608:
9607:
9604:
9602:
9599:
9597:
9594:
9592:
9589:
9588:
9586:
9582:
9575:
9571:
9567:
9564:
9562:
9558:
9556:
9553:
9551:) convex
9550:
9547:
9545:
9542:
9540:
9536:
9534:
9531:
9529:
9526:
9524:
9520:
9516:
9512:
9509:
9507:
9504:
9503:
9502:
9499:
9497:
9496:Grothendieck
9494:
9492:
9489:
9485:
9482:
9481:
9480:
9477:
9475:
9472:
9471:
9469:
9465:
9460:
9453:
9448:
9446:
9441:
9439:
9434:
9433:
9430:
9422:
9418:
9414:
9408:
9404:
9400:
9396:
9392:
9388:
9384:
9378:
9374:
9370:
9366:
9362:
9358:
9354:
9348:
9344:
9339:
9335:
9331:
9327:
9321:
9317:
9313:
9309:
9305:
9301:
9295:
9290:
9289:
9283:
9282:Rudin, Walter
9279:
9275:
9271:
9267:
9261:
9257:
9252:
9248:
9244:
9240:
9236:
9232:
9226:
9222:
9218:
9214:
9210:
9206:
9202:
9196:
9192:
9187:
9183:
9179:
9175:
9169:
9164:
9163:
9157:
9153:
9149:
9145:
9141:
9135:
9131:
9126:
9122:
9118:
9114:
9112:0-471-60848-3
9108:
9104:
9099:
9095:
9091:
9087:
9081:
9077:
9073:
9069:
9065:
9061:
9057:
9053:
9049:
9047:3-540-13627-4
9043:
9039:
9035:
9031:
9027:
9023:
9019:
9015:
9009:
9005:
9000:
8996:
8992:
8988:
8982:
8978:
8974:
8969:
8968:
8961:
8956:
8950:, p. 50.
8949:
8948:Wilansky 2013
8944:
8938:, p. 55.
8937:
8936:Wilansky 2013
8932:
8925:
8920:
8913:
8908:
8902:, p. 63.
8901:
8900:Wilansky 2013
8896:
8889:
8888:Wilansky 2013
8884:
8882:
8880:
8878:
8870:
8869:Wilansky 2013
8865:
8858:
8853:
8851:
8849:
8847:
8845:
8837:
8836:Wilansky 2013
8832:
8825:
8820:
8814:, p. 54.
8813:
8812:Wilansky 2013
8808:
8801:
8796:
8794:
8792:
8790:
8788:
8786:
8784:
8782:
8780:
8778:
8776:
8768:
8763:
8761:
8759:
8757:
8755:
8750:
8739:
8736:
8734:
8731:
8725:
8722:
8719:
8716:
8710:
8707:
8701:
8698:
8696:
8693:
8690:
8687:
8684:
8681:
8678:
8675:
8669:
8666:
8665:
8645:
8640:
8637:
8633:
8629:
8623:
8620:
8614:
8607:
8593:
8590:
8579:
8576:
8570:
8555:
8541:
8538:
8527:
8521:
8506:
8490:
8487:
8483:
8479:
8473:
8467:
8460:
8459:
8458:
8444:
8421:
8418:
8410:
8402:
8394:
8391:
8385:
8380:
8377:
8373:
8352:
8349:
8346:
8318:
8312:
8296:
8285:
8271:
8268:
8262:
8234:
8214:
8203:
8197:
8187:
8184:
8181:
8170:
8161:
8158:
8155:
8152:
8141:
8135:
8118:
8104:
8098:
8061:
8055:
8017:
8011:
8001:
7998:
7995:
7982:the supremum
7969:
7963:
7960:
7957:
7954:
7943:
7937:
7926:
7915:
7909:
7891:
7888:
7885:
7882:
7876:
7870:
7864:
7858:
7852:
7832:
7829:
7826:
7798:
7795:
7792:
7783:
7770:
7767:
7764:
7761:
7741:
7738:
7730:
7705:
7702:
7682:
7662:
7642:
7634:
7618:
7609:
7592:
7589:
7586:
7580:
7574:
7551:
7548:
7528:
7520:
7517:is a complex
7504:
7490:
7477:
7474:
7466:
7450:
7447:
7444:
7441:
7421:
7401:
7375:
7369:
7366:
7363:
7355:
7352:
7348:
7344:
7341:
7338:
7332:
7324:
7321:
7317:
7307:
7305:
7300:
7298:
7294:
7291:
7281:
7267:
7259:
7243:
7223:
7203:
7195:
7190:
7176:
7167:
7165:
7149:
7129:
7123:
7114:
7112:
7102:
7100:
7099:discontinuous
7096:
7092:
7076:
7073:
7070:
7050:
7047:
7044:
7041:
7022:
7019:
6999:
6972:
6969:
6963:
6957:
6954:
6951:
6948:
6945:
6922:
6919:
6912:For any real
6911:
6893:
6890:
6884:
6878:
6875:
6872:
6869:
6866:
6843:
6840:
6832:
6818:
6815:
6812:
6809:
6789:
6769:
6761:
6759:
6745:
6737:
6721:
6697:
6690:
6687:
6671:
6664:
6661:
6645:
6638:
6636:
6634:
6618:
6610:
6606:
6590:
6578:
6562:
6555:
6553:
6551:
6547:
6546:Fréchet space
6543:
6527:
6515:
6499:
6492:
6490:
6488:
6472:
6448:
6428:
6425:
6422:
6414:
6412:
6398:
6378:
6355:
6352:
6344:
6328:
6325:
6322:
6302:
6299:
6296:
6289:
6274:
6267:The graph of
6266:
6244:
6236:
6233:
6213:
6205:
6204:
6191:
6188:
6185:
6177:
6152:
6132:
6124:
6110:
6107:
6097:
6083:
6063:
6060:
6057:
6049:
6035:
6032:
6025:is closed in
6012:
6004:
6001:
5985:
5978:
5973:
5957:
5952:
5948:
5944:
5941:
5921:
5918:
5907:
5901:
5891:
5888:
5885:
5860:
5857:
5849:
5833:
5828:
5824:
5815:
5812:
5796:
5773:
5770:
5767:
5764:
5761:
5758:
5735:
5732:
5729:
5726:
5706:
5703:
5692:
5686:
5676:
5673:
5670:
5667:
5654:
5653:
5640:
5637:
5626:
5620:
5610:
5607:
5604:
5579:
5571:
5555:
5552:
5549:
5529:
5526:
5523:
5502:
5499:
5496:
5485:
5479:
5467:
5463:
5459:
5456:
5430:
5424:
5421:
5415:
5410:
5406:
5385:
5382:
5379:
5376:
5347:
5341:
5331:
5328:
5325:
5316:
5313:
5293:
5290:
5270:
5267:
5264:
5244:
5216:
5210:
5200:
5197:
5194:
5181:
5173:
5162:
5156:
5146:
5143:
5140:
5137:
5125:The equality
5124:
5123:
5122:
5106:
5098:
5082:
5075:
5070:
5066:
5062:
5056:
5054:
5050:
5044:
5042:
5028:
5025:
5022:
5002:
4982:
4961:
4955:
4944:
4938:
4928:
4925:
4922:
4896:
4883:
4864:
4858:
4838:
4835:
4832:
4812:
4804:
4803:
4801:
4785:
4777:
4761:
4754:
4734:
4712:
4703:
4684:
4681:
4678:
4675:
4669:
4666:
4646:
4643:
4640:
4637:
4634:
4626:
4623:
4601:
4597:
4593:
4587:
4581:
4571:
4557:
4554:
4543:
4537:
4527:
4524:
4521:
4494:
4490:
4486:
4483:
4458:
4455:
4435:
4432:
4429:
4407:
4403:
4399:
4393:
4387:
4378:not necessary
4377:
4363:
4360:
4349:
4343:
4333:
4330:
4327:
4300:
4296:
4287:
4269:
4265:
4256:
4255:
4242:
4239:
4236:
4225:
4219:
4209:
4206:
4203:
4176:
4172:
4168:
4162:
4156:
4134:
4130:
4121:
4107:
4102:
4098:
4094:
4088:
4082:
4062:
4042:
4035:
4019:
3994:
3974:
3971:
3968:
3946:
3942:
3921:
3913:
3912:
3898:
3891:
3877:
3874:
3866:
3850:
3843:
3829:
3826:
3806:
3799:
3784:
3777:
3776:
3775:
3761:
3758:
3750:
3726:
3723:
3720:
3712:
3708:
3692:
3662:
3646:
3632:
3630:
3624:
3614:
3600:
3580:
3560:
3557:
3537:
3514:
3508:
3488:
3468:
3448:
3442:
3439:
3436:
3416:
3409:
3393:
3387:
3384:
3381:
3372:
3371:
3367:
3365:
3361:
3357:
3353:
3350:(such as any
3349:
3344:
3340:
3338:
3334:
3329:
3328:
3324:
3322:
3316:
3313:
3299:
3291:
3275:
3255:
3246:
3232:
3226:
3219:
3203:
3183:
3163:
3155:
3139:
3119:
3111:
3107:
3103:
3097:
3094:
3093:
3089:
3087:
3083:
3079:
3075:
3069:
3067:
3063:
3053:
3051:
3047:
3041:
3039:
3023:
3015:
2999:
2992:
2976:
2973:
2953:
2945:
2941:
2937:
2921:
2915:
2912:
2909:
2899:
2896:
2894:
2892:
2886:
2884:
2874:
2872:
2868:
2864:
2859:
2857:
2853:
2849:
2845:
2841:
2831:
2827:
2822:
2820:
2816:
2809:
2804:
2790:
2770:
2760:
2752:
2738:
2735:
2727:
2708:
2702:
2682:
2662:
2655:
2639:
2626:
2612:
2609:
2606:
2592:
2578:
2572:
2569:
2566:
2556:
2551:
2550:
2546:
2544:
2525:
2516:
2510:
2502:
2499:
2493:
2482:
2476:
2466:
2465:operator norm
2450:
2430:
2427:
2406:
2401:
2397:
2393:
2389:
2369:
2363:
2360:
2357:
2335:
2331:
2310:
2287:
2281:
2261:
2258:
2255:
2248:
2245:if for every
2241:
2236:
2233:
2230:
2228:
2212:
2206:
2203:
2200:
2190:
2185:
2184:
2180:
2167:
2161:
2158:
2152:
2146:
2143:
2140:
2137:
2113:
2107:
2104:
2101:
2075:
2069:
2066:
2040:
2034:
2031:
2028:
2022:
2019:
2013:
1993:
1990:
1987:
1964:
1961:
1958:
1955:
1952:
1949:
1943:
1940:
1937:
1914:
1908:
1905:
1899:
1893:
1890:
1884:
1881:
1878:
1872:
1852:
1849:
1846:
1823:
1820:
1817:
1814:
1811:
1808:
1805:
1799:
1796:
1793:
1790:
1770:
1750:
1743:A linear map
1730:
1724:
1715:
1709:
1701:
1698:
1695:
1664:
1658:
1655:
1632:
1629:
1621:
1602:
1596:
1573:
1563:
1549:
1543:
1540:
1537:
1517:
1514:
1511:
1502:
1501:
1497:
1494:
1480:
1455:
1445:
1442:
1439:
1414:
1389:
1377:
1327:
1313:
1307:
1301:
1293:
1290:
1287:
1272:
1269:
1255:
1247:
1243:
1239:
1233:
1228:
1227:
1223:
1221:
1217:
1201:
1195:
1192:
1189:
1161:
1158:
1155:
1152:
1145:is closed in
1132:
1125:the graph of
1124:
1122:
1108:
1088:
1068:
1045:
1042:
1039:
1033:
1030:
1027:
1024:
1021:
1010:
1007:
1001:
998:
995:
984:
981:
978:
975:
972:
969:
944:
941:
938:
918:
915:
912:
904:
902:
885:
874:
870:
854:
834:
811:
808:
788:
781:
779:
777:
773:
757:
733:
713:
705:
689:
682:
680:
678:
674:
658:
646:
630:
623:
621:
619:
615:
599:
576:
567:
540:
531:
509:
496:
492:
489:
484:
473:
469:
453:
446:
444:
442:
426:
406:
383:
380:
377:
374:
371:
368:
348:
328:
308:
305:
285:
278:
274:
272:
270:
254:
231:
228:
208:
201:
187:
184:
181:
178:
171:
155:
148:
133:
126:
125:
124:
122:
118:
102:
96:
93:
90:
83:Suppose that
80:
69:
65:
55:
53:
49:
48:normed spaces
44:
42:
38:
35:
31:
27:
23:
19:
12013:Polynomially
11942:Grothendieck
11935:tame Fréchet
11885:Bornological
11745:Linear cone
11737:Convex cone
11712:Banach disks
11654:Sesquilinear
11606:
11509:Main results
11499:Vector space
11458:
11454:Completeness
11449:Banach space
11354:Balanced set
11328:Distribution
11266:Applications
11119:Krein–Milman
11104:Closed graph
10881:Applications
10802:
10713:
10674:
10621:
10607:
10603:
10590:
10524:
10476:
10363:Linear cone
10356:
10352:
10341:Convex cone
10234:Paley–Wiener
10094:Mackey–Arens
10084:Krein–Milman
10037:Closed range
10032:Closed graph
10002:Banach–Mazur
9882:Self-adjoint
9824:
9786:sesquilinear
9519:Polynomially
9459:Banach space
9402:
9372:
9342:
9311:
9287:
9255:
9220:
9190:
9161:
9129:
9102:
9067:
9064:Conway, John
9033:
9003:
8972:
8955:
8943:
8931:
8919:
8907:
8895:
8864:
8831:
8819:
8807:
7784:
7610:
7519:normed space
7496:
7308:
7304:bounded sets
7301:
7287:
7168:
7115:
7108:
6991:
6736:real numbers
6713:
6582:
6550:normed space
6519:
6464:
6370:
6341:denotes the
4995:is equal to
4034:neighborhood
3987:centered at
3705:need not be
3638:
3626:
3408:normed space
3373:
3369:
3368:
3352:normed space
3345:
3330:
3326:
3325:
3314:
3247:
3095:
3091:
3090:
3086:bornological
3076:valued in a
3059:
3050:automorphism
2897:
2895:
2880:
2871:normed space
2860:
2848:normed space
2837:
2654:neighborhood
2558:
2548:
2547:
2192:
2182:
2181:
1503:
1499:
1498:
1495:
1242:normed space
1235:
1225:
1224:
1182:Throughout,
1181:
1060:
826:
776:normed space
749:
677:Banach space
650:
591:
398:
246:
119:between two
82:
45:
29:
25:
15:
12007:Quasinormed
11920:FK-AK space
11814:Linear span
11809:Convex hull
11794:Affine hull
11597:Almost open
11537:Hahn–Banach
11283:Heat kernel
11273:Hardy space
11180:Trace class
11094:Hahn–Banach
11056:Topological
10602:Continuous
10437:Linear span
10422:Convex hull
10402:Affine hull
10261:holomorphic
10197:holomorphic
10177:Derivatives
10067:Hahn–Banach
10007:Banach–Saks
9925:C*-algebras
9892:Trace class
9855:Functionals
9743:Ultrastrong
9656:Quasinormed
8339:so that if
5061:boundedness
3102:bounded set
1566:bounded on
1374:) with the
774:(such as a
616:(such as a
532:subsets of
22:mathematics
12122:Categories
12047:Stereotype
11905:(DF)-space
11900:Convenient
11639:Functional
11607:Continuous
11592:Linear map
11532:F. Riesz's
11474:Linear map
11216:C*-algebra
11031:Properties
10355:), and (Hw
10256:continuous
10192:functional
9940:C*-algebra
9825:Continuous
9687:Dual space
9661:Stereotype
9639:Metrizable
9566:Projective
8745:References
7465:additivity
7284:Properties
7091:continuous
6988:is closed.
6909:is closed.
6802:such that
6287:is closed.
6165:such that
4851:such that
4702:polar sets
4075:such that
3961:of radius
3713:) and let
3621:See also:
3321:equivalent
3066:equivalent
3016:(which if
2883:continuous
2844:continuous
2828:at a point
2695:such that
2553:See also:
2187:See also:
1839:for every
1230:See also:
1216:linear map
957:such that
905:for every
361:such that
77:See also:
62:See also:
34:continuous
12063:Uniformly
12022:Reflexive
11870:Barrelled
11866:Countably
11778:Symmetric
11676:Transpose
11190:Unbounded
11185:Transpose
11143:Operators
11072:Separable
11067:Reflexive
11052:Algebraic
11038:Barrelled
10814:Sobolev W
10757:Schwartz
10732:∞
10693:∞
10689:ℓ
10655:Ω
10641:λ
10499:Σ
10381:Symmetric
10316:Absorbing
10229:regulated
10209:Integrals
10062:Goldstine
9897:Transpose
9834:Fredholm
9704:Ultraweak
9692:Dual norm
9623:Seminorms
9591:Barrelled
9561:Injective
9549:Uniformly
9523:Reflexive
9421:849801114
9391:853623322
9371:(2006) .
9334:840278135
9274:144216834
9247:840293704
9219:(1983) .
9032:(1987) .
9022:878109401
8995:297140003
8638:≤
8630:⊆
8591:≤
8539:≤
8488:≤
8480:⊆
8419:≤
8395:∈
8378:≤
8185:∈
8159:∈
7999:∈
7961:∈
7889:∈
7830:⊆
7802:→
7765:≤
7739:≤
7596:‖
7590:
7584:‖
7578:‖
7572:‖
7445:∈
7353:−
7322:−
7127:→
7074:
7045:
6970:≤
6949:∈
6891:≤
6870:∈
6813:≤
6426:
6343:real part
6326:
6300:
6186:≤
6100:dense in
5953:∘
5945:∈
5919:≤
5889:∈
5829:∘
5704:≤
5671:∈
5638:≤
5608:∈
5553:≠
5460:∈
5329:∈
5268:≠
5257:and when
5198:∈
5141:∈
4959:∞
4926:∈
4836:∈
4713:≤
4676:−
4594:⊆
4555:≤
4525:∈
4400:⊆
4331:∈
4237:≤
4207:∈
4169:⊆
4095:⊆
3730:→
3707:Hausdorff
3446:→
3391:→
3230:→
2919:→
2610:∈
2576:→
2529:∞
2523:‖
2508:‖
2500:≤
2497:‖
2491:‖
2480:‖
2474:‖
2367:→
2259:⊆
2210:→
2159:≤
2156:‖
2150:‖
2141:∈
2111:→
2079:‖
2076:⋅
2073:‖
2006:(because
1991:≠
1962:∈
1865:(because
1850:∈
1821:∈
1728:∞
1722:‖
1707:‖
1699:∈
1668:‖
1665:⋅
1662:‖
1645:which if
1547:→
1515:⊆
1443:∈
1390:⋅
1311:∞
1305:‖
1299:‖
1291:∈
1222:(TVSs).
1199:→
1156:×
1037:‖
1028:−
1019:‖
1011:δ
1005:‖
999:−
993:‖
979:∈
939:δ
889:‖
886:⋅
883:‖
867:are both
572:′
545:′
514:′
506:→
501:′
472:transpose
441:Hausdorff
439:are both
378:≤
372:∘
182:∈
100:→
12107:Category
12058:Strictly
12032:Schwartz
11972:LF-space
11967:LB-space
11925:FK-space
11895:Complete
11875:BK-space
11800:Relative
11747:(subset)
11739:(subset)
11666:Seminorm
11649:Bilinear
11392:Category
11204:Algebras
11086:Theorems
11043:Complete
11012:Schwartz
10958:glossary
10750:weighted
10620:Hilbert
10597:Bs space
10467:Examples
10432:Interior
10408:Relative
10386:Zonotope
10365:(subset)
10343:(subset)
10294:Strongly
10273:Lebesgue
10268:Measures
10138:Analysis
9984:Theorems
9935:Spectrum
9860:positive
9843:operator
9781:operator
9771:Bilinear
9736:operator
9719:operator
9699:Operator
9596:Complete
9544:Strictly
9401:(2013).
9361:24909067
9158:(1973).
9148:30593138
9121:18412261
9094:21195908
9066:(1990).
9056:17499190
8662:See also
8083:because
7633:open map
7434:and any
7297:normable
7169:Suppose
7105:Examples
5398:the set
4286:supremum
3573:so that
3012:being a
2599:a point
2225:between
1218:between
470:and its
277:seminorm
39:between
12072:)
12020:)
11962:K-space
11947:Hilbert
11930:Fréchet
11915:F-space
11890:Brauner
11883:)
11868:)
11850:Asplund
11832:)
11802:)
11722:Bounded
11617:Compact
11602:Bounded
11539: (
11195:Unitary
11175:Nuclear
11160:Compact
11155:Bounded
11150:Adjoint
11124:Min–max
11017:Sobolev
11002:Nuclear
10992:Hilbert
10987:Fréchet
10952: (
10615:Hardy H
10518:c space
10455:)
10410:)
10331:Bounded
10219:Dunford
10214:Bochner
10187:Gateaux
10182:Fréchet
9957:of ODEs
9902:Unitary
9877:Nuclear
9808:Compact
9798:Bounded
9766:Adjoint
9606:Fréchet
9601:F-space
9572: (
9568:)
9521:)
9501:Hilbert
9474:Asplund
9239:0248498
9209:8210342
7675:and if
7095:bounded
6050:Either
5053:bounded
3362:into a
3356:bounded
3339:but to
3337:bounded
3292:, then
3062:bounded
2944:bounded
2898:Example
2891:bounded
2867:bounded
2852:bounded
2751:It is "
2232:bounded
1328:bounded
1273:bounded
12084:Webbed
12070:Quasi-
11992:Montel
11982:Mackey
11881:Ultra-
11860:Banach
11768:Radial
11732:Convex
11702:Affine
11644:Linear
11612:Closed
11436:(TVSs)
11170:Normal
11007:Orlicz
10997:Hölder
10977:Banach
10966:Spaces
10954:topics
10531:Besov
10371:Radial
10336:Convex
10321:Affine
10290:Weakly
10283:Vector
10155:bundle
9945:radius
9872:Normal
9838:kernel
9803:Closed
9726:Strong
9644:Normed
9634:Mackey
9479:Banach
9461:topics
9419:
9409:
9389:
9379:
9359:
9349:
9332:
9322:
9296:
9272:
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8289:
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8174:
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8122:
8116:
7635:. If
7631:is an
6611:) and
5516:Using
5065:normed
3106:normed
2989:which
2946:, but
2942:) and
2763:point
1244:(or a
869:normed
12042:Smith
12027:Riesz
12018:Semi-
11830:Quasi
11824:Polar
10982:Besov
10606:with
10453:Quasi
10447:Polar
10251:Borel
10202:quasi
9731:polar
9714:polar
9528:Riesz
7718:then
7564:then
7191:every
6658:is a
6603:is a
6575:is a
6548:or a
6485:is a
5998:is a
5283:then
5119:is a
4880:is a
4798:is a
3747:be a
3659:be a
3354:) is
2902:: If
2830:").
2805:every
2724:is a
1618:is a
1214:is a
702:is a
612:is a
528:maps
115:is a
50:is a
32:is a
11661:Norm
11585:form
11573:Maps
11330:(or
11048:Dual
10604:C(K)
10239:weak
9776:form
9709:Weak
9682:Dual
9649:norm
9611:tame
9484:list
9417:OCLC
9407:ISBN
9387:OCLC
9377:ISBN
9357:OCLC
9347:ISBN
9330:OCLC
9320:ISBN
9294:ISBN
9270:OCLC
9260:ISBN
9243:OCLC
9225:ISBN
9205:OCLC
9195:ISBN
9178:OCLC
9168:ISBN
9144:OCLC
9134:ISBN
9117:OCLC
9107:ISBN
9090:OCLC
9080:ISBN
9052:OCLC
9042:ISBN
9018:OCLC
9008:ISBN
8991:OCLC
8981:ISBN
8350:>
7521:and
7089:are
7063:and
6391:and
5771:>
5730:>
5380:>
5055:but
4956:<
4735:<
4476:and
4361:<
3972:>
3639:Let
2893:".
2858:).
2761:some
2526:<
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1725:<
1308:<
1271:norm
1081:and
1040:<
1008:<
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916:>
847:and
419:and
66:and
24:, a
9821:Dis
9316:GTM
8563:sup
8514:sup
8305:sup
8255:sup
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8178:sup
8125:sup
8091:sup
8048:sup
7992:sup
7785:If
7497:If
7467:of
7414:of
7295:is
7260:of
7196:on
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6782:on
6684:is
6631:is
6540:is
6512:is
6441:of
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6345:of
6145:on
6098:not
6096:is
5882:sup
5850:of
5664:sup
5601:sup
5453:sup
5322:sup
5191:sup
5134:sup
5095:is
5067:or
5057:not
5045:not
4919:sup
4884:of
4774:is
4518:sup
4324:sup
4200:sup
4122:If
3867:of
3863:is
3751:on
3709:or
3366:.
3341:not
3108:or
3088:.
3084:or
3042:not
2887:not
2823:not
2728:of
2625:or
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2443:if
1692:sup
1622:of
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1436:sup
1352:or
1284:sup
1061:If
871:or
827:If
770:is
750:If
671:is
651:If
643:is
592:If
466:is
399:If
341:on
298:on
267:is
247:If
168:is
28:or
16:In
12124::
10956:–
10591:BV
10525:BK
10477:AC
10359:))
10292:/
9794:Un
9415:.
9385:.
9355:.
9328:.
9314:.
9268:.
9241:.
9235:MR
9233:.
9203:.
9176:.
9142:.
9115:.
9088:.
9078:.
9070:.
9050:.
9036:.
9016:.
8989:.
8979:.
8876:^
8843:^
8774:^
8753:^
8386::=
7927::=
7865::=
7587:Re
7288:A
7071:Im
7042:Re
6423:Im
6323:Re
6297:Re
6237::=
5556:0.
5527::=
5416::=
5317::=
5029:0.
4644:Id
4436:Id
3317:or
3070:or
2954:Id
2910:Id
2483::=
1944::=
1800::=
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12053:B
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11864:(
11828:(
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11419:t
11412:v
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10848:,
10845:X
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10839:W
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10779:n
10774:R
10769:(
10765:S
10728:L
10714:L
10675:â„“
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10540:B
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10490:b
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9819:(
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9625:/
9576:)
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9437:v
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7768:p
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7735:|
7731:f
7727:|
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7683:p
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7619:X
7593:f
7581:=
7575:f
7552:,
7549:X
7529:f
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7475:F
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7422:Y
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7367:+
7364:D
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7342:x
7339:+
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7333:D
7330:(
7325:1
7318:F
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7244:X
7224:X
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7077:f
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7048:f
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6973:r
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6923:,
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6879:f
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6844:,
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6698:f
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6662:.
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6591:X
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6449:f
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6399:Y
6379:X
6356:.
6353:f
6329:f
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