36:
8429:
7714:
93:
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1901:
All norms on a finite-dimensional vector space are equivalent from a topological viewpoint as they induce the same topology (although the resulting metric spaces need not be the same). And since any
Euclidean space is complete, we can thus conclude that all finite-dimensional normed vector spaces are
5345:
A straightforward argument involving elementary linear algebra shows that the only finite-dimensional seminormed spaces are those arising as the product space of a normed space and a space with trivial seminorm. Consequently, many of the more interesting examples and applications of seminormed
4787:
5320:
4897:
5047:
2142:
903:. Every normed vector space can be "uniquely extended" to a Banach space, which makes normed spaces intimately related to Banach spaces. Every Banach space is a normed space but converse is not true. For example, the set of the
5640:, Mathematics and its Applications (East European Series), vol. 29 (Translated from the Polish by Ewa Bednarczuk ed.), Dordrecht; Warsaw: D. Reidel Publishing Co.; PWN—Polish Scientific Publishers, pp. xvi+524,
4626:
571:
5185:
4422:
2004:. (In fact, a more general result is true: a topological vector space is locally compact if and only if it is finite-dimensional. The point here is that we don't assume the topology comes from a norm.)
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242:
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2864:
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2256:
2645:
2570:
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1289:
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628:
602:
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408:
346:
5117:
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is called a normed vector space. If it is clear from context which norm is intended, then it is common to denote the normed vector space simply by
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Even if a metrizable topological vector space has a topology that is defined by a family of norms, then it may nevertheless still fail to be
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57:
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The norm is a continuous function on its vector space. All linear maps between finite dimensional vector spaces are also continuous.
8220:
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5653:
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79:
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4782:{\displaystyle \left(x_{1},\ldots ,x_{n}\right)+\left(y_{1},\ldots ,y_{n}\right):=\left(x_{1}+y_{1},\ldots ,x_{n}+y_{n}\right)}
4291:
2837:
A product of a family of normable spaces is normable if and only if only finitely many of the spaces are non-trivial (that is,
1898:. On a finite-dimensional vector space, all norms are equivalent but this is not true for infinite dimensional vector spaces.
1046:
4572:
633:
6581:
6260:
6039:
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6352:
1610:
8287:
7889:
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6397:
6342:
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2147:
7490:
6175:
5315:{\displaystyle q\left(x_{1},\ldots ,x_{n}\right):=\left(\sum _{i=1}^{n}q_{i}\left(x_{i}\right)^{p}\right)^{\frac {1}{p}}}
8338:
7717:
7439:
7424:
7252:
6454:
6444:
6404:
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6299:
5844:
7454:
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4117:
to the base field (the complexes or the reals) — such linear maps are called "functionals". The norm of a functional
7847:
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7459:
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6322:
6317:
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50:
44:
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into a normed vector space. An important theorem about continuous linear functionals on normed vector spaces is the
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6615:
6429:
6414:
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6185:
6150:
5895:
5457:
7694:
6909:
6559:
6377:
4517:
2995:
8191:
8001:
7510:
6603:
6576:
6222:
5891:
7444:
7126:
6419:
6409:
6327:
61:
7964:
7959:
7952:
7947:
7819:
7759:
7546:
7347:
6860:
6265:
6192:
6146:
6061:
5886:
5747:
5400:
3141:
2613:
2370:
1797:
890:
7862:
7419:
6392:
6332:
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4275:
4144:
761:
6957:
4436:
zero. These functions form a subspace which we "quotient out", making them equivalent to the zero function.
351:
8225:
8206:
7882:
7643:
3365:
3333:
2493:
8453:
8414:
8404:
8388:
8088:
8037:
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7922:
7674:
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6434:
6347:
6128:
6044:
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5874:
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2700:
1563:
923:
715:
295:
5151:
4904:
2959:
1929:
413:
8383:
8083:
8070:
8052:
8017:
7218:
7213:
6688:
6636:
6593:
6517:
6470:
6207:
5869:
5836:
5809:
4892:{\displaystyle \alpha \left(x_{1},\ldots ,x_{n}\right):=\left(\alpha x_{1},\ldots ,\alpha x_{n}\right).}
4429:
3833:
2293:
1837:
sits as a dense subspace inside some Banach space; this Banach space is essentially uniquely defined by
1801:
1325:
1292:
918:
is a normed vector space whose norm is the square root of the inner product of a vector and itself. The
894:
7857:
6507:
5705:
4428:
on the right hand side is defined and finite. However, the seminorm is equal to zero for any function
2013:
267:
8399:
8343:
8322:
7657:
7156:
7009:
6362:
6357:
6068:
5952:
5858:
5565:. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY:
3599:
3454:
3253:
3198:
2222:
2008:
1895:
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820:
140:
is defined. A norm is a generalization of the intuitive notion of "length" in the physical world. If
5609:. Monografie Matematyczne (in French). Vol. 1. Warszawa: Subwencji Funduszu Kultury Narodowej.
3692:
3527:
2455:
1511:
1365:
8282:
8277:
8235:
7814:
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7561:
7275:
6800:
6757:
6571:
6294:
6024:
5831:
5466:
5394:
5373:
2929:
2803:
2218:
2206:
1793:
1601:
1155:
1040:
995:
915:
825:
575:
469:
97:
4185:
3924:
1726:
1704:
1655:
1450:
1428:
1242:
1216:
1194:
225:
203:
8267:
8210:
8159:
8155:
8144:
8129:
8125:
7996:
7986:
7648:
7515:
7233:
7144:
7048:
6811:
6782:
6778:
6767:
6737:
6733:
6554:
6512:
6119:
6029:
5974:
5821:
4250:
4049:
3397:
1785:
1021:
927:
137:
7160:
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3501:
2840:
2654:
2575:
2326:
2235:
5516:
5514:
2618:
2543:
2261:
1264:
1164:
7979:
7905:
7628:
6914:
6387:
6168:
6111:
6091:
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5683:
5667:
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5471:
5426:
4425:
4120:
3193:
3081:
2791:
1859:
105:
5125:
2809:
1697:
is jointly continuous. This follows from the triangle inequality and homogeneity of the norm.
8372:
8255:
7942:
7927:
7794:
7633:
7551:
7520:
7500:
7485:
7480:
7475:
7312:
6544:
6539:
6527:
6439:
6424:
6287:
6227:
6202:
6133:
6123:
5986:
5931:
5641:
5610:
5367:
4433:
4290:) involves a seminorm defined on a vector space and then the normed space is defined as the
3728:
3594:
3563:
3449:
2001:
1007:
607:
581:
495:
387:
325:
5663:
5095:
4298:
3642:
3431:
2750:
2376:
8347:
8195:
7495:
7449:
7397:
7392:
7363:
7244:
6564:
6549:
6475:
6449:
6277:
6270:
6237:
6197:
6163:
6155:
6083:
6051:
5916:
5848:
5659:
5614:
5042:{\displaystyle q\left(x_{1},\ldots ,x_{n}\right):=\sum _{i=1}^{n}q_{i}\left(x_{i}\right),}
4039:. Isometrically isomorphic normed vector spaces are identical for all practical purposes.
3837:
3422:
3087:
1923:
1890:
1701:
Similarly, for any seminormed vector space we can define the distance between two vectors
904:
7322:
3055:
2802:: A Hausdorff topological vector space is normable if and only if there exists a convex,
5358:, normed vector spaces which are complete with respect to the metric induced by the norm
5052:
4227:
3802:
2770:
2221:, generalizations of normed vector spaces with this property are studied under the name
1866:
1677:
1405:
1132:
8378:
8327:
8042:
7684:
7536:
7337:
7134:
7082:
6999:
6742:
6608:
6255:
6245:
5864:
5816:
5325:
5075:
4447:
4207:
4100:
4076:
4018:
3998:
3974:
3954:
3853:
3778:
3754:
3586:
3313:
3285:
3230:
3172:
3150:
3120:
2909:
2889:
2869:
2680:
2435:
2352:
1983:
1905:
1840:
1820:
1808:
1539:
1295:
1236:
1112:
919:
908:
799:
741:
695:
446:
247:
183:
163:
143:
133:
8447:
8362:
8272:
8215:
8175:
8103:
8078:
8022:
7974:
7910:
7689:
7613:
7342:
7327:
7317:
7139:
6945:
6752:
6706:
6674:
6641:
6492:
6487:
6480:
6101:
6034:
6007:
5826:
5799:
5592:
5560:
5446:
5377:
4566:
2210:
2137:{\displaystyle {\mathcal {N}}(x)=x+{\mathcal {N}}(0):=\{x+N:N\in {\mathcal {N}}(0)\}}
1977:
1600:
is jointly continuous with respect to this topology. This follows directly from the
8409:
8357:
8317:
8307:
8185:
8032:
8027:
7824:
7774:
7728:
7679:
7332:
7302:
6651:
6646:
6106:
6096:
5969:
5959:
5804:
5784:
5556:
5355:
4287:
1813:
1788:(notice this is weaker than a metric) and allows the definition of notions such as
1017:
899:
886:
125:
101:
92:
8367:
8352:
8245:
8139:
8134:
8119:
8098:
8062:
7969:
7789:
7608:
7598:
7505:
7307:
6940:
6856:
6762:
6747:
6727:
6701:
6666:
6217:
6180:
5853:
129:
113:
8180:
8093:
8057:
7917:
7799:
7541:
7381:
7377:
7373:
7045:
7006:
6696:
6661:
6502:
6250:
6012:
5645:
4043:
3948:
2214:
17:
5727:
5697:
8332:
8149:
6772:
6017:
5981:
5671:
5584:
3944:
2007:
The topology of a seminormed vector space has many nice properties. Given a
1425:
This metric is defined in the natural way: the distance between two vectors
5742:
8297:
8292:
8250:
8230:
8200:
7991:
7075:
7038:
6922:
6848:
6808:
6711:
6534:
4295:
4138:
2229:
1399:
1033:
96:
Hierarchy of mathematical spaces. Normed vector spaces are a superset of
8240:
6843:
6677: ((cs, lcs)-closed, (cs, bcs)-complete, (lower) ideally convex, (H
5926:
1239:, but other choices are possible. For example, for a vector space over
5753:
4294:
by the subspace of elements of seminorm zero. For instance, with the
994:
The study of normed spaces and Banach spaces is a fundamental part of
566:{\displaystyle \lVert \lambda x\rVert =|\lambda |\,\lVert x\rVert }
5370:, where the length of each tangent vector is determined by a norm
3829:
The most important maps between two normed vector spaces are the
4424:
is a seminorm on the vector space of all functions on which the
4417:{\displaystyle \|f\|_{p}=\left(\int |f(x)|^{p}\;dx\right)^{1/p}}
4042:
When speaking of normed vector spaces, we augment the notion of
1536:
continuous and which is compatible with the linear structure of
7732:
7248:
5757:
1796:. To put it more abstractly every seminormed vector space is a
29:
2043:
around 0 we can construct all other neighbourhood systems as
5456:, Cambridge Studies in Advanced Mathematics, vol. 125,
5391:– a vector space with a topology defined by convex open sets
4259:
4058:
3406:
3379:
3347:
3267:
3212:
2432:), which happens if and only if there exists some open ball
2117:
2077:
2052:
2019:
1508:
This topology is precisely the weakest topology which makes
3560:
such that the topology that this norm induces is equal to
5489:
5487:
3836:. Together with these maps, normed vector spaces form a
5523:, pp. 136–149, 195–201, 240–252, 335–390, 420–433.
4204:
ranges over all unit vectors (that is, vectors of norm
3663:
a normable space because there does not exist any norm
3498:
a normable space because there does not exist any norm
5638:
Functional analysis and control theory: Linear systems
3494:, is defined by a countable family of norms but it is
2998:
2608:
Metrizable topological vector space § Normability
7163:
7089:
7051:
7012:
6960:
6863:
6814:
5376:, normed vector spaces where the norm is given by an
5346:
spaces occur for infinite-dimensional vector spaces.
5328:
5188:
5154:
5128:
5098:
5078:
5055:
4941:
4907:
4795:
4629:
4575:
4520:
4470:
4450:
4330:
4301:
4286:
The definition of many normed spaces (in particular,
4282:
Normed spaces as quotient spaces of seminormed spaces
4253:
4230:
4210:
4188:
4147:
4123:
4103:
4079:
4052:
4021:
4001:
3977:
3957:
3927:
3876:
3856:
3805:
3781:
3757:
3731:
3725:
such that the topology this norm induces is equal to
3695:
3669:
3645:
3602:
3589:(meaning that its topology can not be defined by any
3566:
3530:
3504:
3457:
3434:
3400:
3368:
3336:
3316:
3288:
3256:
3233:
3201:
3175:
3153:
3123:
3090:
3058:
2962:
2932:
2912:
2892:
2872:
2843:
2812:
2773:
2753:
2703:
2683:
2657:
2621:
2578:
2546:
2496:
2458:
2438:
2399:
2379:
2355:
2329:
2296:
2264:
2238:
2150:
2049:
2016:
1986:
1932:
1908:
1869:
1843:
1823:
1751:
1729:
1707:
1680:
1658:
1645:{\displaystyle \,\cdot \,:\mathbb {K} \times V\to V,}
1613:
1566:
1542:
1514:
1475:
1453:
1431:
1408:
1368:
1328:
1298:
1267:
1245:
1219:
1197:
1167:
1135:
1115:
1049:
936:
835:
802:
764:
744:
718:
698:
636:
610:
584:
524:
498:
472:
449:
416:
390:
354:
328:
298:
270:
250:
228:
206:
186:
166:
146:
5710:
Topological Vector Spaces, Distributions and
Kernels
5682:. New York, NY: Springer New York Imprint Springer.
3914:{\displaystyle \|f(\mathbf {v} )\|=\|\mathbf {v} \|}
3659:
is defined by a countable family of norms but it is
1154:
This also shows that a vector norm is a (uniformly)
8168:
8112:
8010:
7898:
7833:
7767:
7667:
7591:
7570:
7529:
7468:
7410:
7356:
7291:
7206:
6791:
6720:
6629:
6463:
6308:
6236:
6082:
5995:
5909:
5792:
7604:Spectral theory of ordinary differential equations
7191:
7115:
7064:
7025:
6988:
6898:
6832:
5334:
5314:
5174:
5140:
5111:
5084:
5064:
5041:
4927:
4891:
4781:
4615:
4557:
4506:
4456:
4416:
4314:
4266:
4239:
4216:
4196:
4174:
4129:
4109:
4085:
4065:
4027:
4007:
3983:
3963:
3935:
3913:
3862:
3814:
3787:
3763:
3740:
3717:
3681:
3651:
3627:
3575:
3552:
3516:
3482:
3440:
3413:
3386:
3354:
3322:
3294:
3274:
3239:
3219:
3181:
3159:
3129:
3107:
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3044:
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2918:
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2532:
2482:
2444:
2424:
2385:
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2341:
2315:
2282:
2250:
2195:
2136:
2035:
1992:
1968:
1914:
1878:
1849:
1829:
1776:
1737:
1715:
1689:
1666:
1644:
1592:
1548:
1528:
1500:
1461:
1439:
1417:
1382:
1354:
1304:
1283:
1253:
1227:
1205:
1191:on the field of scalars. When the scalar field is
1183:
1144:
1121:
1101:
985:{\displaystyle d(A,B)=\|{\overrightarrow {AB}}\|.}
984:
877:
808:
788:
750:
730:
704:
681:
622:
596:
565:
510:
484:
455:
435:
402:
373:
340:
310:
284:
256:
236:
214:
192:
172:
152:
3850:between two normed vector spaces is a linear map
2866:). Furthermore, the quotient of a normable space
2425:{\displaystyle \tau _{\|\cdot \|}\subseteq \tau }
3635:whose definition can be found in the article on
3012:
2000:is finite-dimensional; this is a consequence of
712:is a real or complex vector space as above, and
7116:{\displaystyle S\left(\mathbb {R} ^{n}\right)}
5403: – Vector space with a notion of nearness
1888:Two norms on the same vector space are called
1777:{\displaystyle \|\mathbf {u} -\mathbf {v} \|.}
1501:{\displaystyle \|\mathbf {u} -\mathbf {v} \|.}
7744:
7260:
7224:Mathematical formulation of quantum mechanics
5769:
3045:{\textstyle x+C\mapsto \inf _{c\in C}\|x+c\|}
8:
5397:– mathematical set with some added structure
4558:{\displaystyle q_{i}:X_{i}\to \mathbb {R} ,}
4338:
4331:
3908:
3900:
3894:
3877:
3676:
3670:
3511:
3505:
3039:
3027:
2943:
2933:
2853:
2847:
2734:
2722:
2664:
2658:
2527:
2518:
2512:
2497:
2474:
2468:
2411:
2405:
2336:
2330:
2308:
2302:
2245:
2239:
2187:
2163:
2131:
2094:
1963:
1954:
1948:
1939:
1768:
1752:
1523:
1515:
1492:
1476:
1377:
1369:
1346:
1338:
1235:), this is usually taken to be the ordinary
1091:
1085:
1079:
1073:
1062:
1050:
976:
958:
869:
857:
780:
774:
725:
719:
673:
667:
661:
655:
649:
637:
560:
554:
534:
525:
424:
418:
362:
356:
305:
299:
5385: – Characterization of normable spaces
885:which makes any normed vector space into a
27:Vector space on which a distance is defined
7751:
7737:
7729:
7295:
7267:
7253:
7245:
5776:
5762:
5754:
5537:sfn error: no target: CITEREFJarchow1981 (
4388:
3951:isometry between the normed vector spaces
3637:spaces of test functions and distributions
3593:norm). An example of such a space is the
3492:spaces of test functions and distributions
2290:is continuous if and only if the topology
417:
355:
7180:
7162:
7103:
7099:
7098:
7088:
7056:
7050:
7017:
7011:
6965:
6959:
6899:{\displaystyle B_{p,q}^{s}(\mathbb {R} )}
6889:
6888:
6879:
6868:
6862:
6813:
5465:
5342:this defines the same topological space.
5327:
5301:
5290:
5280:
5265:
5255:
5244:
5220:
5201:
5187:
5168:
5167:
5153:
5127:
5103:
5097:
5077:
5054:
5026:
5012:
5002:
4991:
4973:
4954:
4940:
4921:
4920:
4906:
4875:
4853:
4827:
4808:
4794:
4768:
4755:
4736:
4723:
4700:
4681:
4658:
4639:
4628:
4607:
4597:
4586:
4574:
4548:
4547:
4538:
4525:
4519:
4493:
4480:
4469:
4449:
4404:
4400:
4382:
4377:
4359:
4341:
4329:
4306:
4300:
4258:
4252:
4229:
4209:
4189:
4187:
4167:
4159:
4148:
4146:
4122:
4102:
4078:
4057:
4051:
4020:
4000:
3976:
3956:
3928:
3926:
3903:
3886:
3875:
3855:
3804:
3780:
3756:
3730:
3700:
3694:
3668:
3644:
3607:
3601:
3565:
3535:
3529:
3503:
3462:
3456:
3433:
3405:
3399:
3378:
3373:
3367:
3346:
3341:
3335:
3315:
3287:
3266:
3261:
3255:
3232:
3211:
3206:
3200:
3189:has a bounded neighborhood of the origin.
3174:
3152:
3122:
3094:
3089:
3062:
3057:
3015:
2997:
2978:
2977:
2966:
2961:
2936:
2931:
2911:
2891:
2871:
2842:
2811:
2772:
2752:
2702:
2682:
2656:
2620:
2577:
2545:
2495:
2457:
2437:
2404:
2398:
2378:
2354:
2328:
2301:
2295:
2263:
2237:
2149:
2116:
2115:
2076:
2075:
2051:
2050:
2048:
2018:
2017:
2015:
1985:
1931:
1907:
1868:
1842:
1822:
1763:
1755:
1750:
1730:
1728:
1708:
1706:
1679:
1660:
1659:
1657:
1623:
1622:
1618:
1614:
1612:
1571:
1567:
1565:
1541:
1522:
1518:
1513:
1487:
1479:
1474:
1454:
1452:
1432:
1430:
1407:
1376:
1372:
1367:
1345:
1341:
1327:
1297:
1276:
1268:
1266:
1247:
1246:
1244:
1221:
1220:
1218:
1199:
1198:
1196:
1176:
1168:
1166:
1134:
1114:
1102:{\displaystyle \|x-y\|\geq |\|x\|-\|y\||}
1094:
1068:
1048:
961:
935:
911:, but it is not complete for this norm.
834:
801:
763:
743:
717:
697:
635:
609:
583:
553:
548:
540:
523:
497:
471:
448:
415:
389:
353:
327:
297:
278:
277:
269:
249:
230:
229:
227:
208:
207:
205:
185:
165:
145:
80:Learn how and when to remove this message
7557:Group algebra of a locally compact group
5505:
5493:
4616:{\displaystyle X:=\prod _{i=1}^{n}X_{i}}
4507:{\displaystyle \left(X_{i},q_{i}\right)}
4175:{\displaystyle |\varphi (\mathbf {v} )|}
4046:to take the norm into account. The dual
3943:). Isometries are always continuous and
789:{\displaystyle (V,\lVert \cdot \rVert )}
682:{\displaystyle \|x+y\|\leq \|x\|+\|y\|.}
318:, satisfying the following four axioms:
91:
43:This article includes a list of general
6989:{\displaystyle L^{\lambda ,p}(\Omega )}
5532:
5413:
5389:Locally convex topological vector space
1784:This turns the seminormed space into a
1161:Property 3 depends on a choice of norm
926:is a special case that allows defining
907:of real numbers can be normed with the
374:{\displaystyle \;\lVert x\rVert \geq 0}
7890:Uniform boundedness (Banach–Steinhaus)
7229:Ordinary Differential Equations (ODEs)
6343:Banach–Steinhaus (Uniform boundedness)
5520:
5364: – Concept in functional analysis
5712:. Mineola, N.Y.: Dover Publications.
4789:and scalar multiplication defined as
3387:{\displaystyle X_{\sigma }^{\prime }}
3355:{\displaystyle X_{\sigma }^{\prime }}
3330:is finite dimensional if and only if
2533:{\displaystyle \{x\in X:\|x\|<1\}}
2217:. As this property is very useful in
1902:Banach spaces. A normed vector space
7:
5567:McGraw-Hill Science/Engineering/Math
2740:{\displaystyle (x,y)\mapsto \|y-x\|}
2196:{\displaystyle x+N:=\{x+n:n\in N\}.}
1593:{\displaystyle \,+\,:V\times V\to V}
1041:variation of the triangle inequality
731:{\displaystyle \lVert \cdot \rVert }
311:{\displaystyle \lVert \cdot \rVert }
5175:{\displaystyle q:X\to \mathbb {R} }
4928:{\displaystyle q:X\to \mathbb {R} }
3144:then the following are equivalent:
2985:{\displaystyle X/C\to \mathbb {R} }
1980:, which is the case if and only if
1969:{\displaystyle B=\{x:\|x\|\leq 1\}}
1804:which is induced by the semi-norm.
1362:is a normed vector space, the norm
998:, a major subfield of mathematics.
436:{\displaystyle \;\lVert x\rVert =0}
7057:
7018:
6980:
6824:
5383:Kolmogorov's normability criterion
3870:which preserves the norm (meaning
3701:
3608:
3536:
3463:
2799:Kolmogorov's normability criterion
2316:{\displaystyle \tau _{\|\cdot \|}}
1811:normed spaces, which are known as
1674:is the underlying scalar field of
1355:{\displaystyle (V,\|\,\cdot \,\|)}
1032:is a vector space equipped with a
49:it lacks sufficient corresponding
25:
6721:Subsets / set operations
6498:Differentiation in Fréchet spaces
4623:where vector addition defined as
2036:{\displaystyle {\mathcal {N}}(0)}
384:Positive definiteness: for every
285:{\displaystyle V\to \mathbb {R} }
8428:
8427:
7713:
7712:
7639:Topological quantum field theory
5741:
5598:Théorie des Opérations Linéaires
4190:
4160:
3929:
3904:
3887:
3771:can be a defined by a family of
2906:is normable, and if in addition
2790:The following theorem is due to
1764:
1756:
1731:
1709:
1488:
1480:
1455:
1433:
466:Absolute homogeneity: for every
34:
8415:With the approximation property
7026:{\displaystyle \ell ^{\infty }}
3628:{\displaystyle C^{\infty }(K),}
3483:{\displaystyle C^{\infty }(K),}
3275:{\displaystyle X_{b}^{\prime }}
3220:{\displaystyle X_{b}^{\prime }}
2926:'s topology is given by a norm
2697:such that the canonical metric
1213:(or more generally a subset of
878:{\displaystyle d(x,y)=\|y-x\|.}
104:, which in turn is a subset of
7878:Open mapping (Banach–Schauder)
7186:
7167:
6983:
6977:
6893:
6885:
6827:
6821:
6415:Lomonosov's invariant subspace
6338:Banach–Schauder (open mapping)
5164:
5122:More generally, for each real
4917:
4544:
4378:
4373:
4367:
4360:
4168:
4164:
4156:
4149:
3891:
3883:
3718:{\displaystyle C^{\infty }(K)}
3712:
3706:
3619:
3613:
3553:{\displaystyle C^{\infty }(K)}
3547:
3541:
3474:
3468:
3008:
2974:
2719:
2716:
2704:
2634:
2622:
2559:
2547:
2483:{\displaystyle (X,\|\cdot \|)}
2477:
2459:
2277:
2265:
2258:on a topological vector space
2128:
2122:
2088:
2082:
2063:
2057:
2030:
2024:
1633:
1584:
1529:{\displaystyle \|\,\cdot \,\|}
1383:{\displaystyle \|\,\cdot \,\|}
1349:
1329:
1277:
1269:
1177:
1169:
1095:
1069:
952:
940:
851:
839:
783:
765:
549:
541:
274:
1:
7435:Uniform boundedness principle
5425:. New York: Springer-Verlag.
5092:is a norm if and only if all
3490:as defined in the article on
2949:{\displaystyle \|\,\cdot ,\|}
2540:for example) that is open in
2209:for the origin consisting of
1926:if and only if the unit ball
1817:. Every normed vector space
485:{\displaystyle \lambda \in K}
6300:Singular value decomposition
5454:-adic differential equations
4197:{\displaystyle \mathbf {v} }
3936:{\displaystyle \mathbf {v} }
3795:if and only if there exists
2886:by a closed vector subspace
1738:{\displaystyle \mathbf {v} }
1716:{\displaystyle \mathbf {u} }
1667:{\displaystyle \mathbb {K} }
1462:{\displaystyle \mathbf {v} }
1440:{\displaystyle \mathbf {u} }
1254:{\displaystyle \mathbb {Q} }
1228:{\displaystyle \mathbb {C} }
1206:{\displaystyle \mathbb {R} }
237:{\displaystyle \mathbb {C} }
215:{\displaystyle \mathbb {R} }
8099:Radially convex/Star-shaped
8084:Pre-compact/Totally bounded
7065:{\displaystyle L^{\infty }}
6833:{\displaystyle ba(\Sigma )}
6702:Radially convex/Star-shaped
5603:Theory of Linear Operations
4267:{\displaystyle V^{\prime }}
4066:{\displaystyle V^{\prime }}
3825:Linear maps and dual spaces
3748:In fact, the topology of a
3414:{\displaystyle X^{\prime }}
2572:(said different, such that
897:then the normed space is a
8470:
7785:Continuous linear operator
7578:Invariant subspace problem
7192:{\displaystyle W(X,L^{p})}
5458:Cambridge University Press
5421:Callier, Frank M. (1991).
5322:is a semi norm. For each
4324:, the function defined by
3682:{\displaystyle \|\cdot \|}
3517:{\displaystyle \|\cdot \|}
3052:is a well defined norm on
2859:{\displaystyle \neq \{0\}}
2670:{\displaystyle \|\cdot \|}
2605:
2591:{\displaystyle B\in \tau }
2342:{\displaystyle \|\cdot \|}
2251:{\displaystyle \|\cdot \|}
1607:The scalar multiplication
1005:
893:. If this metric space is
322:Non-negativity: for every
8423:
8130:Algebraic interior (core)
7872:Vector-valued Hahn–Banach
7760:Topological vector spaces
7708:
7298:
6738:Algebraic interior (core)
6353:Cauchy–Schwarz inequality
5996:Function space Topologies
5680:Topological Vector Spaces
5646:10.1007/978-94-015-7758-8
5636:Rolewicz, Stefan (1987),
4073:of a normed vector space
2640:{\displaystyle (X,\tau )}
2565:{\displaystyle (X,\tau )}
2283:{\displaystyle (X,\tau )}
2205:Moreover, there exists a
1284:{\displaystyle |\alpha |}
1184:{\displaystyle |\alpha |}
7960:Topological homomorphism
7820:Topological vector space
7547:Spectrum of a C*-algebra
5678:Schaefer, H. H. (1999).
5401:Topological vector space
4130:{\displaystyle \varphi }
4037:isometrically isomorphic
3142:topological vector space
2614:topological vector space
1894:if they define the same
1807:Of special interest are
1798:topological vector space
1556:in the following sense:
891:topological vector space
758:, then the ordered pair
7644:Noncommutative geometry
5141:{\displaystyle p\geq 1}
5049:which is a seminorm on
3639:, because its topology
2828:{\displaystyle 0\in X.}
2651:if there exists a norm
1028:seminormed vector space
292:, typically denoted by
160:is a vector space over
64:more precise citations.
8018:Absolutely convex/disk
7700:Tomita–Takesaki theory
7675:Approximation property
7619:Calculus of variations
7193:
7117:
7066:
7027:
6990:
6900:
6834:
6003:Banach–Mazur compactum
5793:Types of Banach spaces
5362:Banach–Mazur compactum
5336:
5316:
5260:
5176:
5142:
5113:
5086:
5066:
5043:
5007:
4929:
4901:Define a new function
4893:
4783:
4617:
4602:
4559:
4508:
4458:
4418:
4316:
4268:
4241:
4218:
4198:
4176:
4131:
4111:
4087:
4067:
4029:
4009:
3985:
3965:
3937:
3915:
3864:
3816:
3789:
3765:
3742:
3741:{\displaystyle \tau .}
3719:
3683:
3653:
3629:
3577:
3576:{\displaystyle \tau .}
3554:
3518:
3484:
3442:
3415:
3388:
3356:
3324:
3296:
3276:
3250:the strong dual space
3241:
3221:
3183:
3161:
3131:
3109:
3074:
3046:
2986:
2950:
2920:
2900:
2880:
2860:
2829:
2784:
2761:
2741:
2691:
2671:
2641:
2592:
2566:
2534:
2484:
2446:
2426:
2387:
2363:
2343:
2317:
2284:
2252:
2197:
2138:
2037:
1994:
1970:
1916:
1880:
1851:
1831:
1778:
1739:
1717:
1691:
1668:
1646:
1594:
1550:
1530:
1502:
1463:
1441:
1419:
1384:
1356:
1306:
1285:
1255:
1229:
1207:
1185:
1146:
1123:
1103:
986:
924:Euclidean vector space
879:
810:
790:
752:
732:
706:
683:
624:
623:{\displaystyle y\in V}
598:
597:{\displaystyle x\in V}
567:
512:
511:{\displaystyle x\in V}
486:
457:
437:
404:
403:{\displaystyle x\in V}
375:
342:
341:{\displaystyle x\in V}
312:
286:
258:
238:
216:
194:
174:
154:
109:
8053:Complemented subspace
7867:hyperplane separation
7695:Banach–Mazur distance
7658:Generalized functions
7219:Finite element method
7214:Differential operator
7194:
7118:
7067:
7028:
6991:
6901:
6835:
6675:Convex series related
6471:Abstract Wiener space
6398:hyperplane separation
5953:Minkowski functionals
5837:Polarization identity
5337:
5317:
5240:
5177:
5143:
5114:
5112:{\displaystyle q_{i}}
5087:
5067:
5044:
4987:
4930:
4894:
4784:
4618:
4582:
4560:
4509:
4459:
4440:Finite product spaces
4419:
4317:
4315:{\displaystyle L^{p}}
4269:
4242:
4219:
4199:
4177:
4132:
4112:
4088:
4068:
4030:
4010:
3993:isometric isomorphism
3986:
3966:
3938:
3916:
3865:
3817:
3790:
3766:
3743:
3720:
3684:
3654:
3652:{\displaystyle \tau }
3630:
3578:
3555:
3519:
3485:
3443:
3441:{\displaystyle \tau }
3416:
3389:
3357:
3325:
3297:
3277:
3242:
3222:
3184:
3162:
3132:
3110:
3075:
3047:
2987:
2951:
2921:
2901:
2881:
2861:
2830:
2785:
2762:
2760:{\displaystyle \tau }
2747:induces the topology
2742:
2692:
2672:
2642:
2593:
2567:
2535:
2485:
2447:
2427:
2388:
2386:{\displaystyle \tau }
2364:
2344:
2318:
2285:
2253:
2223:locally convex spaces
2198:
2139:
2038:
1995:
1971:
1917:
1881:
1852:
1832:
1802:topological structure
1779:
1740:
1718:
1692:
1669:
1647:
1595:
1551:
1531:
1503:
1464:
1442:
1420:
1385:
1357:
1318:Topological structure
1307:
1286:
1256:
1230:
1208:
1186:
1147:
1124:
1104:
987:
880:
826:(norm) induced metric
811:
791:
753:
733:
707:
684:
625:
599:
568:
513:
487:
458:
438:
405:
376:
343:
313:
287:
259:
239:
217:
195:
175:
155:
95:
8303:Locally convex space
7853:Closed graph theorem
7805:Locally convex space
7440:Kakutani fixed-point
7425:Riesz representation
7161:
7087:
7049:
7010:
6958:
6861:
6812:
6801:Absolute continuity
6455:Schauder fixed-point
6445:Riesz representation
6405:Kakutani fixed-point
6373:Freudenthal spectral
5859:L-semi-inner product
5750:at Wikimedia Commons
5423:Linear System Theory
5326:
5186:
5152:
5126:
5096:
5076:
5053:
4939:
4905:
4793:
4627:
4573:
4518:
4468:
4448:
4328:
4299:
4251:
4228:
4208:
4186:
4145:
4121:
4101:
4093:is the space of all
4077:
4050:
4019:
3999:
3975:
3955:
3925:
3874:
3854:
3803:
3779:
3755:
3750:locally convex space
3729:
3693:
3667:
3643:
3600:
3564:
3528:
3502:
3455:
3432:
3398:
3366:
3334:
3314:
3286:
3254:
3231:
3199:
3173:
3151:
3121:
3108:{\displaystyle X/C.}
3088:
3056:
2996:
2960:
2930:
2910:
2890:
2870:
2841:
2810:
2771:
2751:
2701:
2681:
2655:
2619:
2576:
2544:
2494:
2456:
2436:
2397:
2377:
2353:
2327:
2294:
2262:
2236:
2148:
2047:
2014:
2009:neighbourhood system
1984:
1930:
1906:
1867:
1841:
1821:
1749:
1727:
1705:
1678:
1656:
1611:
1564:
1560:The vector addition
1540:
1512:
1473:
1451:
1429:
1406:
1366:
1326:
1312:-adic absolute value
1296:
1265:
1243:
1217:
1195:
1165:
1133:
1113:
1047:
934:
833:
800:
762:
742:
716:
696:
634:
608:
582:
522:
496:
470:
447:
414:
388:
352:
326:
296:
268:
248:
226:
204:
200:is a field equal to
184:
164:
144:
98:inner product spaces
8283:Interpolation space
7815:Operator topologies
7624:Functional calculus
7583:Mahler's conjecture
7562:Von Neumann algebra
7276:Functional analysis
6884:
6622:measurable function
6572:Functional calculus
6435:Parseval's identity
6348:Bessel's inequality
6295:Polar decomposition
6074:Uniform convergence
5832:Inner product space
5562:Functional Analysis
5395:Space (mathematics)
5374:Inner product space
4276:Hahn–Banach theorem
3799:continuous norm on
3383:
3351:
3271:
3216:
3073:{\displaystyle X/C}
2804:von Neumann bounded
2219:functional analysis
2207:neighbourhood basis
1800:and thus carries a
1602:triangle inequality
1156:continuous function
1014:normed vector space
996:functional analysis
916:inner product space
576:Triangle inequality
463:is the zero vector.
136:numbers on which a
118:normed vector space
8313:(Pseudo)Metrizable
8145:Minkowski addition
7997:Sublinear function
7649:Riemann hypothesis
7348:Topological vector
7234:Validated numerics
7189:
7145:Sobolev inequality
7113:
7062:
7023:
6986:
6915:Bounded variation
6896:
6864:
6849:Banach coordinate
6830:
6768:Minkowski addition
6430:M. Riesz extension
5910:Banach spaces are:
5332:
5312:
5172:
5138:
5109:
5082:
5065:{\displaystyle X.}
5062:
5039:
4925:
4889:
4779:
4613:
4555:
4504:
4464:seminormed spaces
4454:
4414:
4312:
4264:
4240:{\displaystyle V.}
4237:
4214:
4194:
4172:
4137:is defined as the
4127:
4107:
4083:
4063:
4025:
4005:
3981:
3961:
3933:
3911:
3860:
3815:{\displaystyle X.}
3812:
3785:
3761:
3738:
3715:
3679:
3649:
3625:
3573:
3550:
3514:
3480:
3438:
3411:
3384:
3369:
3362:is normable (here
3352:
3337:
3320:
3292:
3272:
3257:
3237:
3217:
3202:
3179:
3157:
3127:
3105:
3070:
3042:
3026:
2982:
2946:
2916:
2896:
2876:
2856:
2825:
2783:{\displaystyle X.}
2780:
2757:
2737:
2687:
2667:
2637:
2588:
2562:
2530:
2480:
2442:
2422:
2383:
2359:
2339:
2313:
2280:
2248:
2193:
2134:
2033:
1990:
1966:
1912:
1879:{\displaystyle V.}
1876:
1857:and is called the
1847:
1827:
1786:pseudometric space
1774:
1735:
1713:
1690:{\displaystyle V,}
1687:
1664:
1642:
1590:
1546:
1526:
1498:
1459:
1437:
1418:{\displaystyle V.}
1415:
1398:) and therefore a
1380:
1352:
1302:
1281:
1251:
1225:
1203:
1181:
1145:{\displaystyle y.}
1142:
1119:
1099:
982:
928:Euclidean distance
875:
806:
786:
748:
728:
702:
679:
620:
594:
563:
508:
482:
453:
433:
400:
371:
338:
308:
282:
254:
234:
212:
190:
170:
150:
110:
106:topological spaces
8441:
8440:
8160:Relative interior
7906:Bilinear operator
7790:Linear functional
7726:
7725:
7629:Integral operator
7406:
7405:
7242:
7241:
6954:Morrey–Campanato
6936:compact Hausdorff
6783:Relative interior
6637:Absolutely convex
6604:Projection-valued
6213:Strictly singular
6139:on Hilbert spaces
5900:of Hilbert spaces
5746:Media related to
5719:978-0-486-45352-1
5689:978-1-4612-7155-0
5576:978-0-07-054236-5
5477:978-0-521-76879-5
5447:Kedlaya, Kiran S.
5335:{\displaystyle p}
5309:
5085:{\displaystyle q}
4457:{\displaystyle n}
4426:Lebesgue integral
4217:{\displaystyle 1}
4110:{\displaystyle V}
4097:linear maps from
4086:{\displaystyle V}
4028:{\displaystyle W}
4008:{\displaystyle V}
3984:{\displaystyle W}
3964:{\displaystyle V}
3863:{\displaystyle f}
3788:{\displaystyle X}
3764:{\displaystyle X}
3421:endowed with the
3323:{\displaystyle X}
3295:{\displaystyle X}
3240:{\displaystyle X}
3194:strong dual space
3182:{\displaystyle X}
3160:{\displaystyle X}
3130:{\displaystyle X}
3082:quotient topology
3080:that induces the
3011:
2919:{\displaystyle X}
2899:{\displaystyle C}
2879:{\displaystyle X}
2690:{\displaystyle X}
2445:{\displaystyle B}
2362:{\displaystyle X}
1993:{\displaystyle V}
1915:{\displaystyle V}
1850:{\displaystyle V}
1830:{\displaystyle V}
1549:{\displaystyle V}
1305:{\displaystyle p}
1122:{\displaystyle x}
974:
829:, by the formula
819:A norm induces a
809:{\displaystyle V}
751:{\displaystyle V}
705:{\displaystyle V}
456:{\displaystyle x}
257:{\displaystyle V}
244:, then a norm on
193:{\displaystyle K}
173:{\displaystyle K}
153:{\displaystyle V}
100:and a subset of
90:
89:
82:
16:(Redirected from
8461:
8431:
8430:
8405:Uniformly smooth
8074:
8066:
8033:Balanced/Circled
8023:Absorbing/Radial
7753:
7746:
7739:
7730:
7716:
7715:
7634:Jones polynomial
7552:Operator algebra
7296:
7269:
7262:
7255:
7246:
7198:
7196:
7195:
7190:
7185:
7184:
7152:Triebel–Lizorkin
7122:
7120:
7119:
7114:
7112:
7108:
7107:
7102:
7071:
7069:
7068:
7063:
7061:
7060:
7032:
7030:
7029:
7024:
7022:
7021:
6995:
6993:
6992:
6987:
6976:
6975:
6905:
6903:
6902:
6897:
6892:
6883:
6878:
6839:
6837:
6836:
6831:
6692:
6670:
6652:Balanced/Circled
6450:Robinson-Ursescu
6368:Eberlein–Šmulian
6288:Spectral theorem
6084:Linear operators
5881:Uniformly smooth
5778:
5771:
5764:
5755:
5745:
5731:
5706:Trèves, François
5701:
5674:
5632:
5630:
5629:
5623:
5617:. Archived from
5608:
5588:
5543:
5542:
5530:
5524:
5518:
5509:
5503:
5497:
5491:
5482:
5480:
5469:
5443:
5437:
5436:
5418:
5368:Finsler manifold
5341:
5339:
5338:
5333:
5321:
5319:
5318:
5313:
5311:
5310:
5302:
5300:
5296:
5295:
5294:
5289:
5285:
5284:
5270:
5269:
5259:
5254:
5230:
5226:
5225:
5224:
5206:
5205:
5181:
5179:
5178:
5173:
5171:
5147:
5145:
5144:
5139:
5118:
5116:
5115:
5110:
5108:
5107:
5091:
5089:
5088:
5083:
5071:
5069:
5068:
5063:
5048:
5046:
5045:
5040:
5035:
5031:
5030:
5017:
5016:
5006:
5001:
4983:
4979:
4978:
4977:
4959:
4958:
4934:
4932:
4931:
4926:
4924:
4898:
4896:
4895:
4890:
4885:
4881:
4880:
4879:
4858:
4857:
4837:
4833:
4832:
4831:
4813:
4812:
4788:
4786:
4785:
4780:
4778:
4774:
4773:
4772:
4760:
4759:
4741:
4740:
4728:
4727:
4710:
4706:
4705:
4704:
4686:
4685:
4668:
4664:
4663:
4662:
4644:
4643:
4622:
4620:
4619:
4614:
4612:
4611:
4601:
4596:
4564:
4562:
4561:
4556:
4551:
4543:
4542:
4530:
4529:
4513:
4511:
4510:
4505:
4503:
4499:
4498:
4497:
4485:
4484:
4463:
4461:
4460:
4455:
4434:Lebesgue measure
4423:
4421:
4420:
4415:
4413:
4412:
4408:
4399:
4395:
4387:
4386:
4381:
4363:
4346:
4345:
4321:
4319:
4318:
4313:
4311:
4310:
4273:
4271:
4270:
4265:
4263:
4262:
4246:
4244:
4243:
4238:
4223:
4221:
4220:
4215:
4203:
4201:
4200:
4195:
4193:
4181:
4179:
4178:
4173:
4171:
4163:
4152:
4136:
4134:
4133:
4128:
4116:
4114:
4113:
4108:
4092:
4090:
4089:
4084:
4072:
4070:
4069:
4064:
4062:
4061:
4034:
4032:
4031:
4026:
4014:
4012:
4011:
4006:
3990:
3988:
3987:
3982:
3970:
3968:
3967:
3962:
3942:
3940:
3939:
3934:
3932:
3921:for all vectors
3920:
3918:
3917:
3912:
3907:
3890:
3869:
3867:
3866:
3861:
3821:
3819:
3818:
3813:
3794:
3792:
3791:
3786:
3770:
3768:
3767:
3762:
3747:
3745:
3744:
3739:
3724:
3722:
3721:
3716:
3705:
3704:
3688:
3686:
3685:
3680:
3658:
3656:
3655:
3650:
3634:
3632:
3631:
3626:
3612:
3611:
3582:
3580:
3579:
3574:
3559:
3557:
3556:
3551:
3540:
3539:
3523:
3521:
3520:
3515:
3489:
3487:
3486:
3481:
3467:
3466:
3447:
3445:
3444:
3439:
3420:
3418:
3417:
3412:
3410:
3409:
3393:
3391:
3390:
3385:
3382:
3377:
3361:
3359:
3358:
3353:
3350:
3345:
3329:
3327:
3326:
3321:
3301:
3299:
3298:
3293:
3281:
3279:
3278:
3273:
3270:
3265:
3246:
3244:
3243:
3238:
3226:
3224:
3223:
3218:
3215:
3210:
3188:
3186:
3185:
3180:
3166:
3164:
3163:
3158:
3136:
3134:
3133:
3128:
3114:
3112:
3111:
3106:
3098:
3079:
3077:
3076:
3071:
3066:
3051:
3049:
3048:
3043:
3025:
2991:
2989:
2988:
2983:
2981:
2970:
2955:
2953:
2952:
2947:
2925:
2923:
2922:
2917:
2905:
2903:
2902:
2897:
2885:
2883:
2882:
2877:
2865:
2863:
2862:
2857:
2834:
2832:
2831:
2826:
2806:neighborhood of
2789:
2787:
2786:
2781:
2766:
2764:
2763:
2758:
2746:
2744:
2743:
2738:
2696:
2694:
2693:
2688:
2676:
2674:
2673:
2668:
2646:
2644:
2643:
2638:
2597:
2595:
2594:
2589:
2571:
2569:
2568:
2563:
2539:
2537:
2536:
2531:
2489:
2487:
2486:
2481:
2451:
2449:
2448:
2443:
2431:
2429:
2428:
2423:
2415:
2414:
2392:
2390:
2389:
2384:
2368:
2366:
2365:
2360:
2348:
2346:
2345:
2340:
2322:
2320:
2319:
2314:
2312:
2311:
2289:
2287:
2286:
2281:
2257:
2255:
2254:
2249:
2202:
2200:
2199:
2194:
2143:
2141:
2140:
2135:
2121:
2120:
2081:
2080:
2056:
2055:
2042:
2040:
2039:
2034:
2023:
2022:
1999:
1997:
1996:
1991:
1975:
1973:
1972:
1967:
1921:
1919:
1918:
1913:
1885:
1883:
1882:
1877:
1856:
1854:
1853:
1848:
1836:
1834:
1833:
1828:
1783:
1781:
1780:
1775:
1767:
1759:
1744:
1742:
1741:
1736:
1734:
1722:
1720:
1719:
1714:
1712:
1696:
1694:
1693:
1688:
1673:
1671:
1670:
1665:
1663:
1651:
1649:
1648:
1643:
1626:
1599:
1597:
1596:
1591:
1555:
1553:
1552:
1547:
1535:
1533:
1532:
1527:
1507:
1505:
1504:
1499:
1491:
1483:
1468:
1466:
1465:
1460:
1458:
1446:
1444:
1443:
1438:
1436:
1424:
1422:
1421:
1416:
1389:
1387:
1386:
1381:
1361:
1359:
1358:
1353:
1311:
1309:
1308:
1303:
1290:
1288:
1287:
1282:
1280:
1272:
1260:
1258:
1257:
1252:
1250:
1234:
1232:
1231:
1226:
1224:
1212:
1210:
1209:
1204:
1202:
1190:
1188:
1187:
1182:
1180:
1172:
1151:
1149:
1148:
1143:
1128:
1126:
1125:
1120:
1109:for any vectors
1108:
1106:
1105:
1100:
1098:
1072:
1030:
1029:
1020:equipped with a
1008:Seminormed space
991:
989:
988:
983:
975:
970:
962:
905:finite sequences
884:
882:
881:
876:
815:
813:
812:
807:
795:
793:
792:
787:
757:
755:
754:
749:
737:
735:
734:
729:
711:
709:
708:
703:
688:
686:
685:
680:
629:
627:
626:
621:
603:
601:
600:
595:
572:
570:
569:
564:
552:
544:
517:
515:
514:
509:
491:
489:
488:
483:
462:
460:
459:
454:
442:
440:
439:
434:
409:
407:
406:
401:
380:
378:
377:
372:
347:
345:
344:
339:
317:
315:
314:
309:
291:
289:
288:
283:
281:
263:
261:
260:
255:
243:
241:
240:
235:
233:
221:
219:
218:
213:
211:
199:
197:
196:
191:
179:
177:
176:
171:
159:
157:
156:
151:
85:
78:
74:
71:
65:
60:this article by
51:inline citations
38:
37:
30:
21:
8469:
8468:
8464:
8463:
8462:
8460:
8459:
8458:
8444:
8443:
8442:
8437:
8419:
8181:B-complete/Ptak
8164:
8108:
8072:
8064:
8043:Bounding points
8006:
7948:Densely defined
7894:
7883:Bounded inverse
7829:
7763:
7757:
7727:
7722:
7704:
7668:Advanced topics
7663:
7587:
7566:
7525:
7491:Hilbert–Schmidt
7464:
7455:Gelfand–Naimark
7402:
7352:
7287:
7273:
7243:
7238:
7202:
7176:
7159:
7158:
7157:Wiener amalgam
7127:Segal–Bargmann
7097:
7093:
7085:
7084:
7052:
7047:
7046:
7013:
7008:
7007:
6961:
6956:
6955:
6910:Birnbaum–Orlicz
6859:
6858:
6810:
6809:
6787:
6743:Bounding points
6716:
6690:
6668:
6625:
6476:Banach manifold
6459:
6383:Gelfand–Naimark
6304:
6278:Spectral theory
6246:Banach algebras
6238:Operator theory
6232:
6193:Pseudo-monotone
6176:Hilbert–Schmidt
6156:Densely defined
6078:
5991:
5905:
5788:
5782:
5738:
5720:
5704:
5690:
5677:
5656:
5635:
5627:
5625:
5621:
5606:
5591:
5577:
5555:
5552:
5547:
5546:
5536:
5531:
5527:
5519:
5512:
5504:
5500:
5492:
5485:
5481:, Theorem 1.3.6
5478:
5445:
5444:
5440:
5433:
5420:
5419:
5415:
5410:
5352:
5324:
5323:
5276:
5272:
5271:
5261:
5239:
5235:
5234:
5216:
5197:
5196:
5192:
5184:
5183:
5150:
5149:
5124:
5123:
5099:
5094:
5093:
5074:
5073:
5051:
5050:
5022:
5018:
5008:
4969:
4950:
4949:
4945:
4937:
4936:
4903:
4902:
4871:
4849:
4845:
4841:
4823:
4804:
4803:
4799:
4791:
4790:
4764:
4751:
4732:
4719:
4718:
4714:
4696:
4677:
4676:
4672:
4654:
4635:
4634:
4630:
4625:
4624:
4603:
4571:
4570:
4534:
4521:
4516:
4515:
4514:with seminorms
4489:
4476:
4475:
4471:
4466:
4465:
4446:
4445:
4442:
4376:
4355:
4351:
4350:
4337:
4326:
4325:
4302:
4297:
4296:
4284:
4254:
4249:
4248:
4226:
4225:
4206:
4205:
4184:
4183:
4143:
4142:
4119:
4118:
4099:
4098:
4075:
4074:
4053:
4048:
4047:
4017:
4016:
3997:
3996:
3973:
3972:
3953:
3952:
3923:
3922:
3872:
3871:
3852:
3851:
3827:
3801:
3800:
3777:
3776:
3753:
3752:
3727:
3726:
3696:
3691:
3690:
3665:
3664:
3641:
3640:
3603:
3598:
3597:
3562:
3561:
3531:
3526:
3525:
3500:
3499:
3458:
3453:
3452:
3430:
3429:
3423:weak-* topology
3401:
3396:
3395:
3364:
3363:
3332:
3331:
3312:
3311:
3284:
3283:
3252:
3251:
3229:
3228:
3197:
3196:
3171:
3170:
3149:
3148:
3137:is a Hausdorff
3119:
3118:
3086:
3085:
3054:
3053:
2994:
2993:
2958:
2957:
2928:
2927:
2908:
2907:
2888:
2887:
2868:
2867:
2839:
2838:
2808:
2807:
2769:
2768:
2749:
2748:
2699:
2698:
2679:
2678:
2653:
2652:
2617:
2616:
2610:
2604:
2602:Normable spaces
2574:
2573:
2542:
2541:
2492:
2491:
2490:(such as maybe
2454:
2453:
2434:
2433:
2400:
2395:
2394:
2375:
2374:
2351:
2350:
2325:
2324:
2297:
2292:
2291:
2260:
2259:
2234:
2233:
2146:
2145:
2045:
2044:
2012:
2011:
1982:
1981:
1928:
1927:
1924:locally compact
1904:
1903:
1865:
1864:
1839:
1838:
1819:
1818:
1747:
1746:
1725:
1724:
1703:
1702:
1676:
1675:
1654:
1653:
1609:
1608:
1562:
1561:
1538:
1537:
1510:
1509:
1471:
1470:
1449:
1448:
1427:
1426:
1404:
1403:
1364:
1363:
1324:
1323:
1320:
1294:
1293:
1263:
1262:
1261:one could take
1241:
1240:
1215:
1214:
1193:
1192:
1163:
1162:
1131:
1130:
1111:
1110:
1045:
1044:
1027:
1026:
1010:
1004:
963:
932:
931:
930:by the formula
831:
830:
798:
797:
760:
759:
740:
739:
714:
713:
694:
693:
632:
631:
606:
605:
580:
579:
520:
519:
494:
493:
468:
467:
445:
444:
443:if and only if
412:
411:
386:
385:
350:
349:
324:
323:
294:
293:
266:
265:
246:
245:
224:
223:
202:
201:
182:
181:
162:
161:
142:
141:
86:
75:
69:
66:
56:Please help to
55:
39:
35:
28:
23:
22:
15:
12:
11:
5:
8467:
8465:
8457:
8456:
8446:
8445:
8439:
8438:
8436:
8435:
8424:
8421:
8420:
8418:
8417:
8412:
8407:
8402:
8400:Ultrabarrelled
8392:
8386:
8381:
8375:
8370:
8365:
8360:
8355:
8350:
8341:
8335:
8330:
8328:Quasi-complete
8325:
8323:Quasibarrelled
8320:
8315:
8310:
8305:
8300:
8295:
8290:
8285:
8280:
8275:
8270:
8265:
8264:
8263:
8253:
8248:
8243:
8238:
8233:
8228:
8223:
8218:
8213:
8203:
8198:
8188:
8183:
8178:
8172:
8170:
8166:
8165:
8163:
8162:
8152:
8147:
8142:
8137:
8132:
8122:
8116:
8114:
8113:Set operations
8110:
8109:
8107:
8106:
8101:
8096:
8091:
8086:
8081:
8076:
8068:
8060:
8055:
8050:
8045:
8040:
8035:
8030:
8025:
8020:
8014:
8012:
8008:
8007:
8005:
8004:
7999:
7994:
7989:
7984:
7983:
7982:
7977:
7972:
7962:
7957:
7956:
7955:
7950:
7945:
7940:
7935:
7930:
7925:
7915:
7914:
7913:
7902:
7900:
7896:
7895:
7893:
7892:
7887:
7886:
7885:
7875:
7869:
7860:
7855:
7850:
7848:Banach–Alaoglu
7845:
7843:Anderson–Kadec
7839:
7837:
7831:
7830:
7828:
7827:
7822:
7817:
7812:
7807:
7802:
7797:
7792:
7787:
7782:
7777:
7771:
7769:
7768:Basic concepts
7765:
7764:
7758:
7756:
7755:
7748:
7741:
7733:
7724:
7723:
7721:
7720:
7709:
7706:
7705:
7703:
7702:
7697:
7692:
7687:
7685:Choquet theory
7682:
7677:
7671:
7669:
7665:
7664:
7662:
7661:
7651:
7646:
7641:
7636:
7631:
7626:
7621:
7616:
7611:
7606:
7601:
7595:
7593:
7589:
7588:
7586:
7585:
7580:
7574:
7572:
7568:
7567:
7565:
7564:
7559:
7554:
7549:
7544:
7539:
7537:Banach algebra
7533:
7531:
7527:
7526:
7524:
7523:
7518:
7513:
7508:
7503:
7498:
7493:
7488:
7483:
7478:
7472:
7470:
7466:
7465:
7463:
7462:
7460:Banach–Alaoglu
7457:
7452:
7447:
7442:
7437:
7432:
7427:
7422:
7416:
7414:
7408:
7407:
7404:
7403:
7401:
7400:
7395:
7390:
7388:Locally convex
7385:
7371:
7366:
7360:
7358:
7354:
7353:
7351:
7350:
7345:
7340:
7335:
7330:
7325:
7320:
7315:
7310:
7305:
7299:
7293:
7289:
7288:
7274:
7272:
7271:
7264:
7257:
7249:
7240:
7239:
7237:
7236:
7231:
7226:
7221:
7216:
7210:
7208:
7204:
7203:
7201:
7200:
7188:
7183:
7179:
7175:
7172:
7169:
7166:
7154:
7149:
7148:
7147:
7137:
7135:Sequence space
7132:
7124:
7111:
7106:
7101:
7096:
7092:
7080:
7079:
7078:
7073:
7059:
7055:
7036:
7035:
7034:
7020:
7016:
6997:
6985:
6982:
6979:
6974:
6971:
6968:
6964:
6951:
6943:
6938:
6925:
6920:
6912:
6907:
6895:
6891:
6887:
6882:
6877:
6874:
6871:
6867:
6854:
6846:
6841:
6829:
6826:
6823:
6820:
6817:
6806:
6797:
6795:
6789:
6788:
6786:
6785:
6775:
6770:
6765:
6760:
6755:
6750:
6745:
6740:
6730:
6724:
6722:
6718:
6717:
6715:
6714:
6709:
6704:
6699:
6694:
6686:
6672:
6664:
6659:
6654:
6649:
6644:
6639:
6633:
6631:
6627:
6626:
6624:
6623:
6613:
6612:
6611:
6606:
6601:
6591:
6590:
6589:
6584:
6579:
6569:
6568:
6567:
6562:
6557:
6552:
6550:Gelfand–Pettis
6547:
6542:
6532:
6531:
6530:
6525:
6520:
6515:
6510:
6500:
6495:
6490:
6485:
6484:
6483:
6473:
6467:
6465:
6461:
6460:
6458:
6457:
6452:
6447:
6442:
6437:
6432:
6427:
6422:
6417:
6412:
6407:
6402:
6401:
6400:
6390:
6385:
6380:
6375:
6370:
6365:
6360:
6355:
6350:
6345:
6340:
6335:
6330:
6325:
6323:Banach–Alaoglu
6320:
6318:Anderson–Kadec
6314:
6312:
6306:
6305:
6303:
6302:
6297:
6292:
6291:
6290:
6285:
6275:
6274:
6273:
6268:
6258:
6256:Operator space
6253:
6248:
6242:
6240:
6234:
6233:
6231:
6230:
6225:
6220:
6215:
6210:
6205:
6200:
6195:
6190:
6189:
6188:
6178:
6173:
6172:
6171:
6166:
6158:
6153:
6143:
6142:
6141:
6131:
6126:
6116:
6115:
6114:
6109:
6104:
6094:
6088:
6086:
6080:
6079:
6077:
6076:
6071:
6066:
6065:
6064:
6059:
6049:
6048:
6047:
6042:
6032:
6027:
6022:
6021:
6020:
6010:
6005:
5999:
5997:
5993:
5992:
5990:
5989:
5984:
5979:
5978:
5977:
5967:
5962:
5957:
5956:
5955:
5944:Locally convex
5941:
5940:
5939:
5929:
5924:
5919:
5913:
5911:
5907:
5906:
5904:
5903:
5896:Tensor product
5889:
5883:
5878:
5872:
5867:
5861:
5856:
5851:
5841:
5840:
5839:
5834:
5824:
5819:
5817:Banach lattice
5814:
5813:
5812:
5802:
5796:
5794:
5790:
5789:
5783:
5781:
5780:
5773:
5766:
5758:
5752:
5751:
5737:
5736:External links
5734:
5733:
5732:
5718:
5702:
5688:
5675:
5654:
5633:
5593:Banach, Stefan
5589:
5575:
5551:
5548:
5545:
5544:
5535:, p. 130.
5525:
5510:
5498:
5483:
5476:
5467:10.1.1.165.270
5438:
5431:
5412:
5411:
5409:
5406:
5405:
5404:
5398:
5392:
5386:
5380:
5371:
5365:
5359:
5351:
5348:
5331:
5308:
5305:
5299:
5293:
5288:
5283:
5279:
5275:
5268:
5264:
5258:
5253:
5250:
5247:
5243:
5238:
5233:
5229:
5223:
5219:
5215:
5212:
5209:
5204:
5200:
5195:
5191:
5170:
5166:
5163:
5160:
5157:
5137:
5134:
5131:
5106:
5102:
5081:
5061:
5058:
5038:
5034:
5029:
5025:
5021:
5015:
5011:
5005:
5000:
4997:
4994:
4990:
4986:
4982:
4976:
4972:
4968:
4965:
4962:
4957:
4953:
4948:
4944:
4923:
4919:
4916:
4913:
4910:
4888:
4884:
4878:
4874:
4870:
4867:
4864:
4861:
4856:
4852:
4848:
4844:
4840:
4836:
4830:
4826:
4822:
4819:
4816:
4811:
4807:
4802:
4798:
4777:
4771:
4767:
4763:
4758:
4754:
4750:
4747:
4744:
4739:
4735:
4731:
4726:
4722:
4717:
4713:
4709:
4703:
4699:
4695:
4692:
4689:
4684:
4680:
4675:
4671:
4667:
4661:
4657:
4653:
4650:
4647:
4642:
4638:
4633:
4610:
4606:
4600:
4595:
4592:
4589:
4585:
4581:
4578:
4554:
4550:
4546:
4541:
4537:
4533:
4528:
4524:
4502:
4496:
4492:
4488:
4483:
4479:
4474:
4453:
4441:
4438:
4411:
4407:
4403:
4398:
4394:
4391:
4385:
4380:
4375:
4372:
4369:
4366:
4362:
4358:
4354:
4349:
4344:
4340:
4336:
4333:
4309:
4305:
4292:quotient space
4283:
4280:
4261:
4257:
4236:
4233:
4213:
4192:
4170:
4166:
4162:
4158:
4155:
4151:
4126:
4106:
4082:
4060:
4056:
4024:
4004:
3980:
3960:
3931:
3910:
3906:
3902:
3899:
3896:
3893:
3889:
3885:
3882:
3879:
3859:
3826:
3823:
3811:
3808:
3798:
3784:
3774:
3760:
3737:
3734:
3714:
3711:
3708:
3703:
3699:
3678:
3675:
3672:
3662:
3648:
3624:
3621:
3618:
3615:
3610:
3606:
3592:
3587:normable space
3572:
3569:
3549:
3546:
3543:
3538:
3534:
3513:
3510:
3507:
3497:
3479:
3476:
3473:
3470:
3465:
3461:
3437:
3408:
3404:
3381:
3376:
3372:
3349:
3344:
3340:
3319:
3308:
3307:
3291:
3269:
3264:
3260:
3248:
3236:
3214:
3209:
3205:
3190:
3178:
3168:
3156:
3139:locally convex
3126:
3104:
3101:
3097:
3093:
3069:
3065:
3061:
3041:
3038:
3035:
3032:
3029:
3024:
3021:
3018:
3014:
3010:
3007:
3004:
3001:
2980:
2976:
2973:
2969:
2965:
2945:
2942:
2939:
2935:
2915:
2895:
2875:
2855:
2852:
2849:
2846:
2824:
2821:
2818:
2815:
2779:
2776:
2756:
2736:
2733:
2730:
2727:
2724:
2721:
2718:
2715:
2712:
2709:
2706:
2686:
2666:
2663:
2660:
2636:
2633:
2630:
2627:
2624:
2603:
2600:
2587:
2584:
2581:
2561:
2558:
2555:
2552:
2549:
2529:
2526:
2523:
2520:
2517:
2514:
2511:
2508:
2505:
2502:
2499:
2479:
2476:
2473:
2470:
2467:
2464:
2461:
2441:
2421:
2418:
2413:
2410:
2407:
2403:
2382:
2358:
2338:
2335:
2332:
2310:
2307:
2304:
2300:
2279:
2276:
2273:
2270:
2267:
2247:
2244:
2241:
2192:
2189:
2186:
2183:
2180:
2177:
2174:
2171:
2168:
2165:
2162:
2159:
2156:
2153:
2133:
2130:
2127:
2124:
2119:
2114:
2111:
2108:
2105:
2102:
2099:
2096:
2093:
2090:
2087:
2084:
2079:
2074:
2071:
2068:
2065:
2062:
2059:
2054:
2032:
2029:
2026:
2021:
1989:
1965:
1962:
1959:
1956:
1953:
1950:
1947:
1944:
1941:
1938:
1935:
1911:
1893:
1875:
1872:
1862:
1846:
1826:
1816:
1773:
1770:
1766:
1762:
1758:
1754:
1733:
1711:
1699:
1698:
1686:
1683:
1662:
1641:
1638:
1635:
1632:
1629:
1625:
1621:
1617:
1605:
1589:
1586:
1583:
1580:
1577:
1574:
1570:
1545:
1525:
1521:
1517:
1497:
1494:
1490:
1486:
1482:
1478:
1457:
1435:
1414:
1411:
1379:
1375:
1371:
1351:
1348:
1344:
1340:
1337:
1334:
1331:
1319:
1316:
1301:
1279:
1275:
1271:
1249:
1237:absolute value
1223:
1201:
1179:
1175:
1171:
1141:
1138:
1118:
1097:
1093:
1090:
1087:
1084:
1081:
1078:
1075:
1071:
1067:
1064:
1061:
1058:
1055:
1052:
1003:
1000:
981:
978:
973:
969:
966:
960:
957:
954:
951:
948:
945:
942:
939:
920:Euclidean norm
909:Euclidean norm
902:
874:
871:
868:
865:
862:
859:
856:
853:
850:
847:
844:
841:
838:
828:
805:
785:
782:
779:
776:
773:
770:
767:
747:
727:
724:
721:
701:
690:
689:
678:
675:
672:
669:
666:
663:
660:
657:
654:
651:
648:
645:
642:
639:
619:
616:
613:
593:
590:
587:
573:
562:
559:
556:
551:
547:
543:
539:
536:
533:
530:
527:
507:
504:
501:
481:
478:
475:
464:
452:
432:
429:
426:
423:
420:
399:
396:
393:
382:
370:
367:
364:
361:
358:
337:
334:
331:
307:
304:
301:
280:
276:
273:
253:
232:
210:
189:
169:
149:
88:
87:
42:
40:
33:
26:
24:
18:Normable space
14:
13:
10:
9:
6:
4:
3:
2:
8466:
8455:
8454:Normed spaces
8452:
8451:
8449:
8434:
8426:
8425:
8422:
8416:
8413:
8411:
8408:
8406:
8403:
8401:
8397:
8393:
8391:) convex
8390:
8387:
8385:
8382:
8380:
8376:
8374:
8371:
8369:
8366:
8364:
8363:Semi-complete
8361:
8359:
8356:
8354:
8351:
8349:
8345:
8342:
8340:
8336:
8334:
8331:
8329:
8326:
8324:
8321:
8319:
8316:
8314:
8311:
8309:
8306:
8304:
8301:
8299:
8296:
8294:
8291:
8289:
8286:
8284:
8281:
8279:
8278:Infrabarreled
8276:
8274:
8271:
8269:
8266:
8262:
8259:
8258:
8257:
8254:
8252:
8249:
8247:
8244:
8242:
8239:
8237:
8236:Distinguished
8234:
8232:
8229:
8227:
8224:
8222:
8219:
8217:
8214:
8212:
8208:
8204:
8202:
8199:
8197:
8193:
8189:
8187:
8184:
8182:
8179:
8177:
8174:
8173:
8171:
8169:Types of TVSs
8167:
8161:
8157:
8153:
8151:
8148:
8146:
8143:
8141:
8138:
8136:
8133:
8131:
8127:
8123:
8121:
8118:
8117:
8115:
8111:
8105:
8102:
8100:
8097:
8095:
8092:
8090:
8089:Prevalent/Shy
8087:
8085:
8082:
8080:
8079:Extreme point
8077:
8075:
8069:
8067:
8061:
8059:
8056:
8054:
8051:
8049:
8046:
8044:
8041:
8039:
8036:
8034:
8031:
8029:
8026:
8024:
8021:
8019:
8016:
8015:
8013:
8011:Types of sets
8009:
8003:
8000:
7998:
7995:
7993:
7990:
7988:
7985:
7981:
7978:
7976:
7973:
7971:
7968:
7967:
7966:
7963:
7961:
7958:
7954:
7953:Discontinuous
7951:
7949:
7946:
7944:
7941:
7939:
7936:
7934:
7931:
7929:
7926:
7924:
7921:
7920:
7919:
7916:
7912:
7909:
7908:
7907:
7904:
7903:
7901:
7897:
7891:
7888:
7884:
7881:
7880:
7879:
7876:
7873:
7870:
7868:
7864:
7861:
7859:
7856:
7854:
7851:
7849:
7846:
7844:
7841:
7840:
7838:
7836:
7832:
7826:
7823:
7821:
7818:
7816:
7813:
7811:
7810:Metrizability
7808:
7806:
7803:
7801:
7798:
7796:
7795:Fréchet space
7793:
7791:
7788:
7786:
7783:
7781:
7778:
7776:
7773:
7772:
7770:
7766:
7761:
7754:
7749:
7747:
7742:
7740:
7735:
7734:
7731:
7719:
7711:
7710:
7707:
7701:
7698:
7696:
7693:
7691:
7690:Weak topology
7688:
7686:
7683:
7681:
7678:
7676:
7673:
7672:
7670:
7666:
7659:
7655:
7652:
7650:
7647:
7645:
7642:
7640:
7637:
7635:
7632:
7630:
7627:
7625:
7622:
7620:
7617:
7615:
7614:Index theorem
7612:
7610:
7607:
7605:
7602:
7600:
7597:
7596:
7594:
7590:
7584:
7581:
7579:
7576:
7575:
7573:
7571:Open problems
7569:
7563:
7560:
7558:
7555:
7553:
7550:
7548:
7545:
7543:
7540:
7538:
7535:
7534:
7532:
7528:
7522:
7519:
7517:
7514:
7512:
7509:
7507:
7504:
7502:
7499:
7497:
7494:
7492:
7489:
7487:
7484:
7482:
7479:
7477:
7474:
7473:
7471:
7467:
7461:
7458:
7456:
7453:
7451:
7448:
7446:
7443:
7441:
7438:
7436:
7433:
7431:
7428:
7426:
7423:
7421:
7418:
7417:
7415:
7413:
7409:
7399:
7396:
7394:
7391:
7389:
7386:
7383:
7379:
7375:
7372:
7370:
7367:
7365:
7362:
7361:
7359:
7355:
7349:
7346:
7344:
7341:
7339:
7336:
7334:
7331:
7329:
7326:
7324:
7321:
7319:
7316:
7314:
7311:
7309:
7306:
7304:
7301:
7300:
7297:
7294:
7290:
7285:
7281:
7277:
7270:
7265:
7263:
7258:
7256:
7251:
7250:
7247:
7235:
7232:
7230:
7227:
7225:
7222:
7220:
7217:
7215:
7212:
7211:
7209:
7205:
7199:
7181:
7177:
7173:
7170:
7164:
7155:
7153:
7150:
7146:
7143:
7142:
7141:
7138:
7136:
7133:
7131:
7130:
7125:
7123:
7109:
7104:
7094:
7090:
7081:
7077:
7074:
7072:
7053:
7044:
7043:
7042:
7041:
7037:
7033:
7014:
7005:
7004:
7003:
7002:
6998:
6996:
6972:
6969:
6966:
6962:
6952:
6950:
6949:
6944:
6942:
6939:
6937:
6935:
6931:
6926:
6924:
6921:
6919:
6918:
6913:
6911:
6908:
6906:
6880:
6875:
6872:
6869:
6865:
6855:
6853:
6852:
6847:
6845:
6842:
6840:
6818:
6815:
6807:
6805:
6804:
6799:
6798:
6796:
6794:
6790:
6784:
6780:
6776:
6774:
6771:
6769:
6766:
6764:
6761:
6759:
6756:
6754:
6753:Extreme point
6751:
6749:
6746:
6744:
6741:
6739:
6735:
6731:
6729:
6726:
6725:
6723:
6719:
6713:
6710:
6708:
6705:
6703:
6700:
6698:
6695:
6693:
6687:
6684:
6680:
6676:
6673:
6671:
6665:
6663:
6660:
6658:
6655:
6653:
6650:
6648:
6645:
6643:
6640:
6638:
6635:
6634:
6632:
6630:Types of sets
6628:
6621:
6617:
6614:
6610:
6607:
6605:
6602:
6600:
6597:
6596:
6595:
6592:
6588:
6585:
6583:
6580:
6578:
6575:
6574:
6573:
6570:
6566:
6563:
6561:
6558:
6556:
6553:
6551:
6548:
6546:
6543:
6541:
6538:
6537:
6536:
6533:
6529:
6526:
6524:
6521:
6519:
6516:
6514:
6511:
6509:
6506:
6505:
6504:
6501:
6499:
6496:
6494:
6493:Convex series
6491:
6489:
6488:Bochner space
6486:
6482:
6479:
6478:
6477:
6474:
6472:
6469:
6468:
6466:
6462:
6456:
6453:
6451:
6448:
6446:
6443:
6441:
6440:Riesz's lemma
6438:
6436:
6433:
6431:
6428:
6426:
6425:Mazur's lemma
6423:
6421:
6418:
6416:
6413:
6411:
6408:
6406:
6403:
6399:
6396:
6395:
6394:
6391:
6389:
6386:
6384:
6381:
6379:
6378:Gelfand–Mazur
6376:
6374:
6371:
6369:
6366:
6364:
6361:
6359:
6356:
6354:
6351:
6349:
6346:
6344:
6341:
6339:
6336:
6334:
6331:
6329:
6326:
6324:
6321:
6319:
6316:
6315:
6313:
6311:
6307:
6301:
6298:
6296:
6293:
6289:
6286:
6284:
6281:
6280:
6279:
6276:
6272:
6269:
6267:
6264:
6263:
6262:
6259:
6257:
6254:
6252:
6249:
6247:
6244:
6243:
6241:
6239:
6235:
6229:
6226:
6224:
6221:
6219:
6216:
6214:
6211:
6209:
6206:
6204:
6201:
6199:
6196:
6194:
6191:
6187:
6184:
6183:
6182:
6179:
6177:
6174:
6170:
6167:
6165:
6162:
6161:
6159:
6157:
6154:
6152:
6148:
6144:
6140:
6137:
6136:
6135:
6132:
6130:
6127:
6125:
6121:
6117:
6113:
6110:
6108:
6105:
6103:
6100:
6099:
6098:
6095:
6093:
6090:
6089:
6087:
6085:
6081:
6075:
6072:
6070:
6067:
6063:
6060:
6058:
6055:
6054:
6053:
6050:
6046:
6043:
6041:
6038:
6037:
6036:
6033:
6031:
6028:
6026:
6023:
6019:
6016:
6015:
6014:
6011:
6009:
6006:
6004:
6001:
6000:
5998:
5994:
5988:
5985:
5983:
5980:
5976:
5973:
5972:
5971:
5968:
5966:
5963:
5961:
5958:
5954:
5950:
5947:
5946:
5945:
5942:
5938:
5935:
5934:
5933:
5930:
5928:
5925:
5923:
5920:
5918:
5915:
5914:
5912:
5908:
5901:
5897:
5893:
5890:
5888:
5884:
5882:
5879:
5877:) convex
5876:
5873:
5871:
5868:
5866:
5862:
5860:
5857:
5855:
5852:
5850:
5846:
5842:
5838:
5835:
5833:
5830:
5829:
5828:
5825:
5823:
5822:Grothendieck
5820:
5818:
5815:
5811:
5808:
5807:
5806:
5803:
5801:
5798:
5797:
5795:
5791:
5786:
5779:
5774:
5772:
5767:
5765:
5760:
5759:
5756:
5749:
5748:Normed spaces
5744:
5740:
5739:
5735:
5729:
5725:
5721:
5715:
5711:
5707:
5703:
5699:
5695:
5691:
5685:
5681:
5676:
5673:
5669:
5665:
5661:
5657:
5655:90-277-2186-6
5651:
5647:
5643:
5639:
5634:
5624:on 2014-01-11
5620:
5616:
5612:
5604:
5600:
5599:
5594:
5590:
5586:
5582:
5578:
5572:
5568:
5564:
5563:
5558:
5557:Rudin, Walter
5554:
5553:
5549:
5540:
5534:
5529:
5526:
5522:
5517:
5515:
5511:
5508:, p. 42.
5507:
5506:Schaefer 1999
5502:
5499:
5496:, p. 41.
5495:
5494:Schaefer 1999
5490:
5488:
5484:
5479:
5473:
5468:
5463:
5459:
5455:
5451:
5448:
5442:
5439:
5434:
5432:0-387-97573-X
5428:
5424:
5417:
5414:
5407:
5402:
5399:
5396:
5393:
5390:
5387:
5384:
5381:
5379:
5378:inner product
5375:
5372:
5369:
5366:
5363:
5360:
5357:
5354:
5353:
5349:
5347:
5343:
5329:
5306:
5303:
5297:
5291:
5286:
5281:
5277:
5273:
5266:
5262:
5256:
5251:
5248:
5245:
5241:
5236:
5231:
5227:
5221:
5217:
5213:
5210:
5207:
5202:
5198:
5193:
5189:
5161:
5158:
5155:
5135:
5132:
5129:
5120:
5104:
5100:
5079:
5072:The function
5059:
5056:
5036:
5032:
5027:
5023:
5019:
5013:
5009:
5003:
4998:
4995:
4992:
4988:
4984:
4980:
4974:
4970:
4966:
4963:
4960:
4955:
4951:
4946:
4942:
4914:
4911:
4908:
4899:
4886:
4882:
4876:
4872:
4868:
4865:
4862:
4859:
4854:
4850:
4846:
4842:
4838:
4834:
4828:
4824:
4820:
4817:
4814:
4809:
4805:
4800:
4796:
4775:
4769:
4765:
4761:
4756:
4752:
4748:
4745:
4742:
4737:
4733:
4729:
4724:
4720:
4715:
4711:
4707:
4701:
4697:
4693:
4690:
4687:
4682:
4678:
4673:
4669:
4665:
4659:
4655:
4651:
4648:
4645:
4640:
4636:
4631:
4608:
4604:
4598:
4593:
4590:
4587:
4583:
4579:
4576:
4568:
4567:product space
4552:
4539:
4535:
4531:
4526:
4522:
4500:
4494:
4490:
4486:
4481:
4477:
4472:
4451:
4439:
4437:
4435:
4431:
4427:
4409:
4405:
4401:
4396:
4392:
4389:
4383:
4370:
4364:
4356:
4352:
4347:
4342:
4334:
4323:
4307:
4303:
4293:
4289:
4288:Banach spaces
4281:
4279:
4277:
4255:
4234:
4231:
4211:
4153:
4140:
4124:
4104:
4096:
4080:
4054:
4045:
4040:
4038:
4022:
4002:
3994:
3991:is called an
3978:
3958:
3950:
3946:
3897:
3880:
3857:
3849:
3844:
3841:
3839:
3835:
3832:
3824:
3822:
3809:
3806:
3796:
3782:
3772:
3758:
3751:
3735:
3732:
3709:
3697:
3673:
3660:
3646:
3638:
3622:
3616:
3604:
3596:
3595:Fréchet space
3590:
3588:
3583:
3570:
3567:
3544:
3532:
3508:
3495:
3493:
3477:
3471:
3459:
3451:
3450:Fréchet space
3435:
3428:The topology
3426:
3424:
3402:
3374:
3370:
3342:
3338:
3317:
3310:Furthermore,
3305:
3289:
3262:
3258:
3249:
3234:
3207:
3203:
3195:
3191:
3176:
3169:
3154:
3147:
3146:
3145:
3143:
3140:
3124:
3115:
3102:
3099:
3095:
3091:
3083:
3067:
3063:
3059:
3036:
3033:
3030:
3022:
3019:
3016:
3005:
3002:
2999:
2971:
2967:
2963:
2956:then the map
2940:
2937:
2913:
2893:
2873:
2850:
2844:
2835:
2822:
2819:
2816:
2813:
2805:
2801:
2800:
2795:
2793:
2777:
2774:
2754:
2731:
2728:
2725:
2713:
2710:
2707:
2684:
2661:
2650:
2631:
2628:
2625:
2615:
2609:
2601:
2599:
2585:
2582:
2579:
2556:
2553:
2550:
2524:
2521:
2515:
2509:
2506:
2503:
2500:
2471:
2465:
2462:
2439:
2419:
2416:
2408:
2401:
2380:
2372:
2356:
2333:
2305:
2298:
2274:
2271:
2268:
2242:
2231:
2226:
2224:
2220:
2216:
2212:
2208:
2203:
2190:
2184:
2181:
2178:
2175:
2172:
2169:
2166:
2160:
2157:
2154:
2151:
2125:
2112:
2109:
2106:
2103:
2100:
2097:
2091:
2085:
2072:
2069:
2066:
2060:
2027:
2010:
2005:
2003:
2002:Riesz's lemma
1987:
1979:
1960:
1957:
1951:
1945:
1942:
1936:
1933:
1925:
1909:
1899:
1897:
1892:
1889:
1886:
1873:
1870:
1861:
1858:
1844:
1824:
1815:
1814:Banach spaces
1812:
1810:
1805:
1803:
1799:
1795:
1791:
1787:
1771:
1760:
1684:
1681:
1639:
1636:
1630:
1627:
1619:
1615:
1606:
1603:
1587:
1581:
1578:
1575:
1572:
1568:
1559:
1558:
1557:
1543:
1519:
1495:
1484:
1412:
1409:
1401:
1397:
1394:(a notion of
1393:
1373:
1342:
1335:
1332:
1317:
1315:
1313:
1299:
1273:
1238:
1173:
1159:
1157:
1152:
1139:
1136:
1116:
1088:
1082:
1076:
1065:
1059:
1056:
1053:
1042:
1037:
1035:
1031:
1023:
1019:
1015:
1009:
1001:
999:
997:
992:
979:
971:
967:
964:
955:
949:
946:
943:
937:
929:
925:
921:
917:
912:
910:
906:
901:
898:
896:
892:
888:
872:
866:
863:
860:
854:
848:
845:
842:
836:
827:
824:
823:, called its
822:
817:
803:
777:
771:
768:
745:
738:is a norm on
722:
699:
676:
670:
664:
658:
652:
646:
643:
640:
617:
614:
611:
591:
588:
585:
577:
574:
557:
545:
537:
531:
528:
505:
502:
499:
479:
476:
473:
465:
450:
430:
427:
421:
397:
394:
391:
383:
368:
365:
359:
335:
332:
329:
321:
320:
319:
302:
271:
251:
187:
167:
147:
139:
135:
131:
127:
123:
119:
115:
107:
103:
102:metric spaces
99:
94:
84:
81:
73:
70:December 2019
63:
59:
53:
52:
46:
41:
32:
31:
19:
8339:Polynomially
8268:Grothendieck
8261:tame Fréchet
8211:Bornological
8071:Linear cone
8063:Convex cone
8038:Banach disks
7980:Sesquilinear
7835:Main results
7825:Vector space
7780:Completeness
7775:Banach space
7680:Balanced set
7654:Distribution
7592:Applications
7445:Krein–Milman
7430:Closed graph
7207:Applications
7128:
7039:
7000:
6947:
6933:
6929:
6916:
6850:
6802:
6689:Linear cone
6682:
6678:
6667:Convex cone
6560:Paley–Wiener
6420:Mackey–Arens
6410:Krein–Milman
6363:Closed range
6358:Closed graph
6328:Banach–Mazur
6208:Self-adjoint
6112:sesquilinear
5845:Polynomially
5785:Banach space
5709:
5679:
5637:
5626:. Retrieved
5619:the original
5602:
5597:
5561:
5550:Bibliography
5533:Jarchow 1981
5528:
5501:
5453:
5450:
5441:
5422:
5416:
5356:Banach space
5344:
5182:defined by
5121:
4900:
4443:
4432:on a set of
4285:
4094:
4041:
4036:
3992:
3847:
3845:
3842:
3828:
3797:at least one
3584:
3427:
3309:
3247:is normable.
3167:is normable.
3116:
2836:
2797:
2796:
2648:
2611:
2227:
2204:
2006:
1900:
1887:
1806:
1700:
1469:is given by
1395:
1321:
1160:
1153:
1038:
1025:
1018:vector space
1013:
1011:
993:
913:
900:Banach space
887:metric space
818:
691:
578:: for every
126:vector space
122:normed space
121:
117:
111:
76:
67:
48:
8333:Quasinormed
8246:FK-AK space
8140:Linear span
8135:Convex hull
8120:Affine hull
7923:Almost open
7863:Hahn–Banach
7609:Heat kernel
7599:Hardy space
7506:Trace class
7420:Hahn–Banach
7382:Topological
6928:Continuous
6763:Linear span
6748:Convex hull
6728:Affine hull
6587:holomorphic
6523:holomorphic
6503:Derivatives
6393:Hahn–Banach
6333:Banach–Saks
6251:C*-algebras
6218:Trace class
6181:Functionals
6069:Ultrastrong
5982:Quasinormed
5521:Trèves 2006
5119:are norms.
4565:denote the
4247:This turns
4035:are called
3834:linear maps
2349:induces on
2228:A norm (or
2215:convex sets
1794:convergence
114:mathematics
62:introducing
8373:Stereotype
8231:(DF)-space
8226:Convenient
7965:Functional
7933:Continuous
7918:Linear map
7858:F. Riesz's
7800:Linear map
7542:C*-algebra
7357:Properties
6681:), and (Hw
6582:continuous
6518:functional
6266:C*-algebra
6151:Continuous
6013:Dual space
5987:Stereotype
5965:Metrizable
5892:Projective
5628:2020-07-11
5615:0005.20901
5408:References
4095:continuous
4044:dual space
3949:surjective
3831:continuous
3304:metrizable
2792:Kolmogorov
2647:is called
2606:See also:
2393:(meaning,
1891:equivalent
1860:completion
1790:continuity
1390:induces a
1291:to be the
1006:See also:
1002:Definition
45:references
8389:Uniformly
8348:Reflexive
8196:Barrelled
8192:Countably
8104:Symmetric
8002:Transpose
7516:Unbounded
7511:Transpose
7469:Operators
7398:Separable
7393:Reflexive
7378:Algebraic
7364:Barrelled
7140:Sobolev W
7083:Schwartz
7058:∞
7019:∞
7015:ℓ
6981:Ω
6967:λ
6825:Σ
6707:Symmetric
6642:Absorbing
6555:regulated
6535:Integrals
6388:Goldstine
6223:Transpose
6160:Fredholm
6030:Ultraweak
6018:Dual norm
5949:Seminorms
5917:Barrelled
5887:Injective
5875:Uniformly
5849:Reflexive
5728:853623322
5708:(2006) .
5698:840278135
5462:CiteSeerX
5242:∑
5211:…
5165:→
5133:≥
4989:∑
4964:…
4918:→
4869:α
4863:…
4847:α
4818:…
4797:α
4746:…
4691:…
4649:…
4584:∏
4545:→
4430:supported
4357:∫
4339:‖
4332:‖
4260:′
4154:φ
4125:φ
4059:′
3945:injective
3909:‖
3901:‖
3895:‖
3878:‖
3733:τ
3702:∞
3677:‖
3674:⋅
3671:‖
3647:τ
3609:∞
3568:τ
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3512:‖
3509:⋅
3506:‖
3464:∞
3436:τ
3407:′
3380:′
3375:σ
3348:′
3343:σ
3268:′
3213:′
3040:‖
3028:‖
3020:∈
3009:↦
2992:given by
2975:→
2944:‖
2938:⋅
2934:‖
2845:≠
2817:∈
2755:τ
2735:‖
2729:−
2723:‖
2720:↦
2665:‖
2662:⋅
2659:‖
2632:τ
2586:τ
2583:∈
2557:τ
2519:‖
2513:‖
2504:∈
2475:‖
2472:⋅
2469:‖
2420:τ
2417:⊆
2412:‖
2409:⋅
2406:‖
2402:τ
2381:τ
2337:‖
2334:⋅
2331:‖
2309:‖
2306:⋅
2303:‖
2299:τ
2275:τ
2246:‖
2243:⋅
2240:‖
2211:absorbing
2182:∈
2113:∈
1958:≤
1955:‖
1949:‖
1769:‖
1761:−
1753:‖
1634:→
1628:×
1616:⋅
1585:→
1579:×
1524:‖
1520:⋅
1516:‖
1493:‖
1485:−
1477:‖
1378:‖
1374:⋅
1370:‖
1347:‖
1343:⋅
1339:‖
1274:α
1174:α
1092:‖
1086:‖
1083:−
1080:‖
1074:‖
1066:≥
1063:‖
1057:−
1051:‖
1039:A useful
977:‖
972:→
959:‖
870:‖
864:−
858:‖
781:‖
778:⋅
775:‖
726:‖
723:⋅
720:‖
674:‖
668:‖
662:‖
656:‖
653:≤
650:‖
638:‖
615:∈
589:∈
561:‖
555:‖
546:λ
535:‖
529:λ
526:‖
503:∈
477:∈
474:λ
425:‖
419:‖
395:∈
366:≥
363:‖
357:‖
333:∈
306:‖
303:⋅
300:‖
275:→
264:is a map
128:over the
8448:Category
8433:Category
8384:Strictly
8358:Schwartz
8298:LF-space
8293:LB-space
8251:FK-space
8221:Complete
8201:BK-space
8126:Relative
8073:(subset)
8065:(subset)
7992:Seminorm
7975:Bilinear
7718:Category
7530:Algebras
7412:Theorems
7369:Complete
7338:Schwartz
7284:glossary
7076:weighted
6946:Hilbert
6923:Bs space
6793:Examples
6758:Interior
6734:Relative
6712:Zonotope
6691:(subset)
6669:(subset)
6620:Strongly
6599:Lebesgue
6594:Measures
6464:Analysis
6310:Theorems
6261:Spectrum
6186:positive
6169:operator
6107:operator
6097:Bilinear
6062:operator
6045:operator
6025:Operator
5922:Complete
5870:Strictly
5672:13064804
5595:(1932).
5585:21163277
5559:(1991).
5449:(2010),
5350:See also
5148:the map
4139:supremum
3848:isometry
3838:category
3394:denotes
2649:normable
2230:seminorm
1896:topology
1809:complete
1400:topology
1396:distance
1034:seminorm
895:complete
821:distance
180:, where
8398:)
8346:)
8288:K-space
8273:Hilbert
8256:Fréchet
8241:F-space
8216:Brauner
8209:)
8194:)
8176:Asplund
8158:)
8128:)
8048:Bounded
7943:Compact
7928:Bounded
7865: (
7521:Unitary
7501:Nuclear
7486:Compact
7481:Bounded
7476:Adjoint
7450:Min–max
7343:Sobolev
7328:Nuclear
7318:Hilbert
7313:Fréchet
7278: (
6941:Hardy H
6844:c space
6781:)
6736:)
6657:Bounded
6545:Dunford
6540:Bochner
6513:Gateaux
6508:Fréchet
6283:of ODEs
6228:Unitary
6203:Nuclear
6134:Compact
6124:Bounded
6092:Adjoint
5932:Fréchet
5927:F-space
5898: (
5894:)
5847:)
5827:Hilbert
5800:Asplund
5664:0920371
3448:of the
2371:coarser
1978:compact
134:complex
58:improve
8410:Webbed
8396:Quasi-
8318:Montel
8308:Mackey
8207:Ultra-
8186:Banach
8094:Radial
8058:Convex
8028:Affine
7970:Linear
7938:Closed
7762:(TVSs)
7496:Normal
7333:Orlicz
7323:Hölder
7303:Banach
7292:Spaces
7280:topics
6857:Besov
6697:Radial
6662:Convex
6647:Affine
6616:Weakly
6609:Vector
6481:bundle
6271:radius
6198:Normal
6164:kernel
6129:Closed
6052:Strong
5970:Normed
5960:Mackey
5805:Banach
5787:topics
5726:
5716:
5696:
5686:
5670:
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5613:
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5583:
5573:
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5464:
5429:
4444:Given
4322:spaces
4182:where
3995:, and
3591:single
1652:where
1392:metric
889:and a
222:or to
47:, but
8368:Smith
8353:Riesz
8344:Semi-
8156:Quasi
8150:Polar
7308:Besov
6932:with
6779:Quasi
6773:Polar
6577:Borel
6528:quasi
6057:polar
6040:polar
5854:Riesz
5622:(PDF)
5607:(PDF)
5601:[
4224:) in
3773:norms
2373:than
2323:that
2144:with
1016:is a
922:of a
124:is a
7987:Norm
7911:form
7899:Maps
7656:(or
7374:Dual
6930:C(K)
6565:weak
6102:form
6035:Weak
6008:Dual
5975:norm
5937:tame
5810:list
5724:OCLC
5714:ISBN
5694:OCLC
5684:ISBN
5668:OCLC
5650:ISBN
5581:OCLC
5571:ISBN
5539:help
5472:ISBN
5427:ISBN
4015:and
3971:and
3947:. A
3192:the
2522:<
2213:and
1792:and
1723:and
1447:and
1129:and
1024:. A
1022:norm
604:and
492:and
138:norm
130:real
116:, a
6147:Dis
5642:doi
5611:Zbl
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1976:is
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692:If
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120:or
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