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This formulation allows one to define bounded operators between general topological vector spaces as an operator which takes bounded sets to bounded sets. In this context, it is still true that every continuous map is bounded, however the converse fails; a bounded operator need not be continuous.
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is continuous. This fact is often summarized by saying that a linear operator that is bounded on some neighborhood of the origin is necessarily continuous. In particular, any linear functional that is bounded on some neighborhood of the origin is continuous (even if its domain is not a
1276:{\displaystyle \|Lx\|=\left\Vert {\|x\| \over \varepsilon }L\left(\varepsilon {x \over \|x\|}\right)\right\Vert ={\|x\| \over \varepsilon }\left\Vert L\left(\varepsilon {x \over \|x\|}\right)\right\Vert \leq {\|x\| \over \varepsilon }\cdot 1={1 \over \varepsilon }\|x\|.}
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linear operator between TVS is a bounded operator. This implies that every continuous linear operator between metrizable TVS is bounded. However, in general, a bounded linear operator between two TVSs need not be continuous.
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Bornological spaces are exactly those locally convex spaces for which every bounded linear operator into another locally convex space is necessarily continuous. That is, a locally convex TVS
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linear map between two TVSs is always bounded, but the converse requires additional assumptions to hold (such as the domain being bornological and the codomain being locally convex).
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norm, the linear operator to the real numbers which returns the sum of a sequence is bounded, with operator norm 1. If the same space is considered with the
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Consequently, in functional analysis, when a linear operator is called "bounded" then it is never meant in this abstract sense (of having a bounded image).
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Any linear operator between two finite-dimensional normed spaces is bounded, and such an operator may be viewed as multiplication by some fixed
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be a linear operator between topological vector spaces (not necessarily
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1513:) then a linear operators into any other locally convex spaces is bounded if and only if it is continuous. For
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The concept of a bounded linear operator has been extended from normed spaces to all topological vector spaces.
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which is a contradiction. Q.E.D. This proof readily generalizes to give even stronger characterizations of "
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Thus any linear map that is sequentially continuous at the origin is necessarily a bounded linear map.
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is a bounded subset of its codomain. A linear map has this property if and only if it is identically
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This also means that boundedness is no longer equivalent to
Lipschitz continuity in this context.
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4998:{\displaystyle F\left(z_{\bullet }\right)=\left(F\left(z_{i}\right)\right)_{i=1}^{\infty }}
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3415:{\displaystyle L(x_{0},x_{1},x_{2},\dots )=\left(0,x_{0},x_{1},x_{2},\ldots \right)}
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with a subsequence if necessary, it may be assumed without loss of generality that
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Conversely, it follows from the continuity at the zero vector that there exists a
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5565:. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press.
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3082:{\displaystyle \Delta :H^{2}(\mathbb {R} ^{n})\to L^{2}(\mathbb {R} ^{n})\,}
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Any linear operator defined on a finite-dimensional normed space is bounded.
2155:{\displaystyle r_{\bullet }=\left(r_{i}\right)_{i=1}^{\infty }\to \infty }
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A linear operator between normed spaces is bounded if and only if it is
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maps bounded subsets of its domain to bounded subsets of its codomain;
1517:, a weaker converse holds; any bounded linear map from an LF space is
3000:. The compact operators form an important class of bounded operators.
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2233:{\displaystyle r_{\bullet }=\left(r_{i}x_{i}\right)_{i=1}^{\infty }}
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is a bornological space if and only if for every locally convex TVS
3287:{\displaystyle x_{0}^{2}+x_{1}^{2}+x_{2}^{2}+\cdots <\infty ,\,}
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4785:{\displaystyle z_{\bullet }:=\left(x_{i}/i\right)_{i=1}^{\infty }}
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of eventually zero sequences of real numbers, considered with the
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maps every Mackey convergent null sequence to a bounded subset of
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is locally convex then the following may be added to this list:
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4367:{\displaystyle x_{\bullet }=\left(x_{i}\right)_{i=1}^{\infty }}
2057:{\displaystyle x_{\bullet }=\left(x_{i}\right)_{i=1}^{\infty }}
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5411:{\displaystyle \left(r_{i}x_{i}\right)_{i=1}^{\infty }\to 0}
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5232:{\displaystyle F\left(x_{i}\right)\in riV\subseteq i^{2}V,}
4291: – Linear operator defined on a dense linear subspace
1946:
is by definition a sequence that converges to the origin.
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A subset of a TVS is called bounded (or more precisely,
5317:{\displaystyle \left(r_{i}x_{i}\right)_{i=1}^{\infty }}
3587:{\displaystyle \|P\|=\int _{-\pi }^{\pi }\!|P(x)|\,dx.}
27:
Linear transformation between topological vector spaces
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2989:{\displaystyle (Lf)(y)=\int _{a}^{b}\!K(x,y)f(x)\,dx,}
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735:{\displaystyle \|L(x+h)-L(x)\|=\|L(h)\|\leq M\|h\|.}
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Properties of the space of bounded linear operators
3422:is bounded. Its operator norm is easily seen to be
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4308:Proof: Assume for the sake of contradiction that
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2492:(or equivalently, at every) point of its domain.
2689:are bounded linear operators. For instance, if
1939:maps every null sequence to a bounded sequence;
437:Outside of functional analysis, when a function
3881:The space of all bounded linear operators from
6980:{\displaystyle S\left(\mathbb {R} ^{n}\right)}
5595:. Mineola, New York: Dover Publications, Inc.
5561:Narici, Lawrence; Beckenstein, Edward (2011).
5259:is bounded." For example, the word "such that
3817:{\displaystyle \|L(v_{n})\|=2\pi n\to \infty }
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7088:Mathematical formulation of quantum mechanics
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1769:Characterizations of bounded linear operators
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347:{\displaystyle \|Lx\|_{Y}\leq M\|x\|_{X}.}
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2666:
2660:
2639:
2633:
2612:
2606:
2559:
2531:
2507:
2473:
2450:
2426:
2400:
2380:
2352:
2346:
2320:
2293:
2273:
2245:
2224:
2213:
2202:
2192:
2173:
2167:
2140:
2129:
2119:
2101:
2095:
2074:
2048:
2037:
2027:
2009:
2003:
1978:
1958:
1924:
1881:
1858:
1836:
1813:
1778:
1735:
1712:
1692:
1653:
1630:
1601:
1581:
1561:
1529:
1447:
1418:
1398:
1368:
1320:
1288:
1251:
1224:
1193:
1159:
1128:
1099:
1077:
1048:
1013:
987:
910:
884:
853:
830:
810:
787:
767:
747:
652:
632:
600:
580:
560:Equivalence of boundedness and continuity
542:
510:
481:
442:
403:
383:
359:
335:
313:
298:
270:
244:
224:
200:
180:
157:
137:
113:
93:
57:
7421:Group algebra of a locally compact group
5459:
6853:{\displaystyle L^{\lambda ,p}(\Omega )}
5449:
4301:
2996:is bounded. This operator is in fact a
2682:norm, the same operator is not bounded.
1036:{\displaystyle \|h\|\leq \varepsilon .}
7093:Ordinary Differential Equations (ODEs)
6207:Banach–Steinhaus (Uniform boundedness)
4229:Bounded set (topological vector space)
1473:it. In a normed space (and even in a
1469:) if every neighborhood of the origin
4264: – Branch of functional analysis
4231: – Generalization of boundedness
2576:sequentially continuous at the origin
2090:if there exists a divergent sequence
7:
1765:Every normed space is bornological.
4276: – Mathematical field of study
473:" then this usually means that its
6921:
6882:
6844:
6688:
5397:
5309:
4990:
4874:
4835:
4777:
4469:
4359:
3837:
3811:
3277:
3097:and it takes values in a space of
3013:
2667:
2225:
2162:of positive real number such that
2149:
2141:
2049:
595:is bounded. Then, for all vectors
25:
6585:Subsets / set operations
6362:Differentiation in Fréchet spaces
898:{\displaystyle \varepsilon >0}
7577:
7576:
7503:Topological quantum field theory
4258: – Length in a vector space
3757:{\displaystyle \|v_{n}\|=2\pi ,}
2550:if and only if it is continuous.
6890:{\displaystyle \ell ^{\infty }}
4120:is a closed linear subspace of
2675:{\displaystyle \ell ^{\infty }}
2490:sequentially continuous at some
2067:Mackey convergent to the origin
7796:Theory of continuous functions
7050:
7031:
6847:
6841:
6757:
6749:
6691:
6685:
6279:Lomonosov's invariant subspace
6202:Banach–Schauder (open mapping)
5402:
4879:
4165:
4153:
4087:
4075:
4039:
4027:
3981:
3969:
3943:
3931:
3834:
3808:
3790:
3777:
3712:{\displaystyle n=1,2,\ldots ,}
3628:that maps a polynomial to its
3612:
3570:
3566:
3560:
3553:
3496:
3481:
3348:
3303:
3075:
3060:
3047:
3044:
3029:
2973:
2967:
2961:
2949:
2924:
2918:
2915:
2906:
2866:
2854:
2827:
2815:
2795:
2783:
2735:
2732:
2720:
2714:
2702:
2146:
1904:
1898:
1789:
1746:
1612:
1606:
1540:
1429:
1423:
1331:
1217:
1178:
1152:
1095:
960:
954:
945:
939:
924:
918:
708:
702:
687:
681:
672:
660:
492:
486:
453:
219:(a special type of TVS), then
68:
1:
7688:Uniform boundedness principle
7299:Uniform boundedness principle
5520:Narici & Beckenstein 2011
5508:Narici & Beckenstein 2011
5493:Narici & Beckenstein 2011
5472:Narici & Beckenstein 2011
4571:does not absorb the sequence
4235:Contraction (operator theory)
3846:{\displaystyle n\to \infty ,}
3671:{\displaystyle v_{n}=e^{inx}}
2844:and with values in the space
805:Moreover, since the constant
6164:Singular value decomposition
4632:{\displaystyle x_{\bullet }}
3632:is not bounded. Indeed, for
1382:{\displaystyle B\subseteq X}
1311:In topological vector spaces
6929:{\displaystyle L^{\infty }}
6697:{\displaystyle ba(\Sigma )}
6566:Radially convex/Star-shaped
5542:Encyclopedia of Mathematics
4693:for every positive integer
4093:{\displaystyle A\in B(X,Y)}
3099:square-integrable functions
2808:of continuous functions on
7812:
7442:Invariant subspace problem
7056:{\displaystyle W(X,L^{p})}
4246:Continuous linear operator
3446:Unbounded linear operators
2395:into bornivorous disks in
1680:
1481:Continuity and boundedness
533:Every bounded operator is
29:
7572:
7162:
6602:Algebraic interior (core)
6217:Cauchy–Schwarz inequality
5860:Function space Topologies
5563:Topological Vector Spaces
5150:is an integer then since
3994:is a normed vector space.
3472:trigonometric polynomials
3131:{\displaystyle \ell ^{2}}
2648:{\displaystyle \ell ^{1}}
1349:topological vector spaces
86:topological vector spaces
7411:Spectrum of a C*-algebra
4241:Discontinuous linear map
3621:{\displaystyle L:X\to X}
1798:{\displaystyle F:X\to Y}
1755:{\displaystyle F:X\to Y}
1549:{\displaystyle F:X\to Y}
1340:{\displaystyle F:X\to Y}
848:this shows that in fact
620:{\displaystyle x,h\in X}
462:{\displaystyle f:X\to Y}
77:{\displaystyle L:X\to Y}
30:Not to be confused with
7765:Ultrabornological space
7508:Noncommutative geometry
5324:is a bounded subset of
2548:sequentially continuous
2497:sequentially continuous
2467:
2314:
2240:is a bounded subset of
1625:is a bounded subset of
1519:sequentially continuous
1487:sequentially continuous
1354:bounded linear operator
1065:{\displaystyle x\in X,}
1043:Thus, for all non-zero
529:In normed vector spaces
287:{\displaystyle x\in X,}
47:bounded linear operator
7564:Tomita–Takesaki theory
7539:Approximation property
7483:Calculus of variations
7057:
6981:
6930:
6891:
6854:
6764:
6698:
5867:Banach–Mazur compactum
5657:Types of Banach spaces
5435:
5412:
5341:
5318:
5253:
5233:
5164:
5144:
5143:{\displaystyle i>r}
5118:
5095:
5048:
5047:{\displaystyle r>1}
5022:
4999:
4909:
4889:
4786:
4710:
4687:
4633:
4606:
4565:
4545:
4525:
4501:
4478:
4388:
4368:
4212:
4192:
4172:
4171:{\displaystyle B(X,Y)}
4134:
4114:
4094:
4046:
4045:{\displaystyle B(X,Y)}
4017:is Banach, then so is
4011:
3988:
3987:{\displaystyle B(X,Y)}
3950:
3949:{\displaystyle B(X,Y)}
3915:
3895:
3867:
3847:
3818:
3758:
3713:
3672:
3622:
3588:
3506:
3464:
3436:
3416:
3288:
3204:
3132:
3083:
2990:
2893:
2873:
2834:
2802:
2767:
2747:
2676:
2649:
2622:
2621:{\displaystyle c_{00}}
2568:
2540:
2516:
2482:
2459:
2435:
2412:
2389:
2365:
2364:{\displaystyle F^{-1}}
2329:
2302:
2282:
2257:
2234:
2156:
2083:
2058:
1990:
1967:
1933:
1911:
1867:
1845:
1822:
1799:
1756:
1724:
1701:
1662:
1642:
1619:
1590:
1570:
1550:
1459:
1436:
1407:
1383:
1341:
1297:
1277:
1066:
1037:
1002:
1001:{\displaystyle h\in X}
976:
899:
862:
842:
819:
799:
776:
762:go to zero shows that
756:
736:
641:
621:
589:
551:
519:
499:
463:
421:
420:{\displaystyle \|L\|.}
392:
368:
348:
288:
259:
258:{\displaystyle M>0}
233:
209:
189:
169:
152:to bounded subsets of
146:
122:
102:
78:
7745:Quasi-barrelled space
7559:Banach–Mazur distance
7522:Generalized functions
7083:Finite element method
7078:Differential operator
7058:
6982:
6931:
6892:
6855:
6765:
6699:
6539:Convex series related
6335:Abstract Wiener space
6262:hyperplane separation
5817:Minkowski functionals
5701:Polarization identity
5436:
5413:
5342:
5319:
5254:
5234:
5165:
5145:
5119:
5096:
5049:
5023:
5000:
4910:
4890:
4787:
4711:
4688:
4634:
4607:
4566:
4546:
4526:
4502:
4479:
4389:
4369:
4213:
4193:
4173:
4135:
4115:
4095:
4047:
4012:
3989:
3951:
3916:
3896:
3868:
3848:
3819:
3759:
3714:
3673:
3623:
3589:
3507:
3465:
3437:
3417:
3289:
3210:of real numbers with
3205:
3133:
3084:
2991:
2899:given by the formula
2894:
2874:
2835:
2803:
2773:defined on the space
2768:
2748:
2677:
2650:
2623:
2569:
2541:
2517:
2483:
2460:
2436:
2413:
2390:
2366:
2330:
2303:
2283:
2258:
2235:
2157:
2084:
2059:
1991:
1968:
1934:
1912:
1868:
1846:
1828:is (locally) bounded;
1823:
1800:
1757:
1725:
1702:
1663:
1643:
1620:
1591:
1571:
1551:
1460:
1437:
1408:
1384:
1342:
1298:
1278:
1067:
1038:
1003:
977:
900:
863:
843:
820:
800:
777:
757:
737:
642:
622:
590:
552:
520:
500:
464:
422:
393:
369:
349:
289:
260:
234:
210:
190:
170:
147:
123:
103:
79:
51:linear transformation
7760:Ultrabarrelled space
7750:Infrabarrelled space
7304:Kakutani fixed-point
7289:Riesz representation
7025:
6951:
6913:
6874:
6822:
6725:
6676:
6665:Absolute continuity
6319:Schauder fixed-point
6309:Riesz representation
6269:Kakutani fixed-point
6237:Freudenthal spectral
5723:L-semi-inner product
5422:
5351:
5328:
5263:
5243:
5174:
5154:
5128:
5105:
5058:
5032:
5009:
4919:
4915:) so by assumption,
4899:
4796:
4720:
4697:
4643:
4616:
4575:
4555:
4535:
4515:
4488:
4398:
4378:
4312:
4202:
4198:is nontrivial, then
4182:
4147:
4124:
4104:
4063:
4021:
4001:
3963:
3925:
3905:
3885:
3857:
3828:
3768:
3723:
3682:
3636:
3600:
3518:
3478:
3470:be the space of all
3454:
3426:
3297:
3214:
3145:
3115:
3010:
2903:
2883:
2848:
2812:
2777:
2757:
2693:
2659:
2632:
2605:
2558:
2530:
2506:
2472:
2449:
2425:
2399:
2379:
2345:
2319:
2292:
2272:
2244:
2166:
2094:
2073:
2002:
1977:
1957:
1923:
1880:
1857:
1835:
1812:
1777:
1734:
1711:
1691:
1652:
1629:
1618:{\displaystyle F(U)}
1600:
1580:
1560:
1528:
1503:pseudometrizable TVS
1446:
1435:{\displaystyle F(B)}
1417:
1397:
1367:
1319:
1287:
1076:
1047:
1012:
986:
909:
883:
874:Lipschitz continuous
870:uniformly continuous
852:
829:
809:
786:
766:
746:
651:
631:
599:
579:
541:
535:Lipschitz continuous
509:
498:{\displaystyle f(X)}
480:
441:
402:
382:
358:
297:
269:
243:
223:
217:normed vector spaces
199:
179:
156:
136:
112:
92:
56:
7488:Functional calculus
7447:Mahler's conjecture
7426:Von Neumann algebra
7140:Functional analysis
6748:
6486:measurable function
6436:Functional calculus
6299:Parseval's identity
6212:Bessel's inequality
6159:Polar decomposition
5938:Uniform convergence
5696:Inner product space
5522:, pp. 451–457.
5495:, pp. 441–457.
5474:, pp. 156–175.
5401:
5313:
4994:
4878:
4839:
4781:
4473:
4363:
3550:
3267:
3249:
3231:
2944:
2687:integral transforms
2339:into bounded disks.
2229:
2145:
2053:
1677:Bornological spaces
1497:If the domain is a
1467:von Neumann bounded
1351:(TVSs) is called a
825:does not depend on
39:functional analysis
7652:Bornological space
7513:Riemann hypothesis
7212:Topological vector
7098:Validated numerics
7053:
7009:Sobolev inequality
6977:
6926:
6887:
6850:
6779:Bounded variation
6760:
6728:
6713:Banach coordinate
6694:
6632:Minkowski addition
6294:M. Riesz extension
5774:Banach spaces are:
5537:"Bounded operator"
5434:{\displaystyle X.}
5431:
5408:
5354:
5340:{\displaystyle X.}
5337:
5314:
5266:
5249:
5229:
5160:
5140:
5117:{\displaystyle i.}
5114:
5101:for every integer
5091:
5044:
5021:{\displaystyle Y.}
5018:
4995:
4946:
4905:
4885:
4843:
4799:
4782:
4736:
4709:{\displaystyle i.}
4706:
4683:
4629:
4602:
4561:
4541:
4521:
4500:{\displaystyle Y.}
4497:
4484:is not bounded in
4474:
4425:
4384:
4364:
4328:
4289:Unbounded operator
4256:Norm (mathematics)
4208:
4188:
4168:
4130:
4110:
4090:
4042:
4007:
3984:
3946:
3911:
3891:
3863:
3843:
3814:
3754:
3709:
3668:
3618:
3584:
3533:
3502:
3460:
3435:{\displaystyle 1.}
3432:
3412:
3284:
3253:
3235:
3217:
3200:
3128:
3079:
2986:
2930:
2889:
2869:
2830:
2798:
2763:
2743:
2672:
2645:
2618:
2564:
2536:
2512:
2478:
2455:
2443:bornological space
2431:
2411:{\displaystyle X.}
2408:
2385:
2361:
2325:
2298:
2278:
2256:{\displaystyle X.}
2253:
2230:
2182:
2152:
2110:
2079:
2054:
2018:
1989:{\displaystyle Y.}
1986:
1963:
1929:
1907:
1863:
1841:
1818:
1795:
1752:
1730:a linear operator
1723:{\displaystyle Y,}
1720:
1697:
1683:Bornological space
1658:
1641:{\displaystyle Y,}
1638:
1615:
1586:
1566:
1546:
1499:bornological space
1458:{\displaystyle Y.}
1455:
1432:
1403:
1379:
1337:
1315:A linear operator
1293:
1273:
1062:
1033:
998:
972:
895:
858:
841:{\displaystyle x,}
838:
815:
798:{\displaystyle x.}
795:
772:
752:
732:
637:
617:
585:
573:
550:{\displaystyle 0.}
547:
518:{\displaystyle 0.}
515:
495:
459:
417:
388:
364:
354:The smallest such
344:
284:
265:such that for all
255:
229:
205:
185:
168:{\displaystyle Y.}
165:
142:
118:
98:
74:
18:Bounded linear map
7773:
7772:
7590:
7589:
7493:Integral operator
7270:
7269:
7106:
7105:
6818:Morrey–Campanato
6800:compact Hausdorff
6647:Relative interior
6501:Absolutely convex
6468:Projection-valued
6077:Strictly singular
6003:on Hilbert spaces
5764:of Hilbert spaces
5602:978-0-486-49353-4
5553:Kreyszig, Erwin:
5462:, pp. 47–50.
5252:{\displaystyle F}
5163:{\displaystyle V}
4908:{\displaystyle X}
4564:{\displaystyle V}
4544:{\displaystyle Y}
4531:of the origin in
4524:{\displaystyle V}
4387:{\displaystyle 0}
4251:Local boundedness
4211:{\displaystyle Y}
4191:{\displaystyle X}
4133:{\displaystyle X}
4113:{\displaystyle A}
4052:; in particular,
4010:{\displaystyle Y}
3914:{\displaystyle Y}
3894:{\displaystyle X}
3866:{\displaystyle L}
3505:{\displaystyle ,}
3463:{\displaystyle X}
2892:{\displaystyle L}
2872:{\displaystyle C}
2840:endowed with the
2801:{\displaystyle C}
2766:{\displaystyle L}
2567:{\displaystyle F}
2539:{\displaystyle F}
2515:{\displaystyle X}
2481:{\displaystyle F}
2458:{\displaystyle Y}
2434:{\displaystyle X}
2388:{\displaystyle Y}
2328:{\displaystyle F}
2301:{\displaystyle Y}
2281:{\displaystyle X}
2082:{\displaystyle X}
1966:{\displaystyle F}
1932:{\displaystyle F}
1866:{\displaystyle F}
1844:{\displaystyle F}
1821:{\displaystyle F}
1700:{\displaystyle X}
1661:{\displaystyle F}
1589:{\displaystyle X}
1576:of the origin in
1569:{\displaystyle U}
1406:{\displaystyle X}
1296:{\displaystyle L}
1283:This proves that
1259:
1240:
1209:
1175:
1144:
1115:
861:{\displaystyle L}
818:{\displaystyle M}
782:is continuous at
775:{\displaystyle L}
755:{\displaystyle h}
640:{\displaystyle h}
588:{\displaystyle L}
571:
391:{\displaystyle L}
367:{\displaystyle M}
232:{\displaystyle L}
208:{\displaystyle Y}
188:{\displaystyle X}
145:{\displaystyle X}
121:{\displaystyle Y}
101:{\displaystyle X}
16:(Redirected from
7803:
7786:Linear operators
7714:Saturated family
7683:Bounded operator
7617:
7610:
7603:
7594:
7580:
7579:
7498:Jones polynomial
7416:Operator algebra
7160:
7133:
7126:
7119:
7110:
7062:
7060:
7059:
7054:
7049:
7048:
7016:Triebel–Lizorkin
6986:
6984:
6983:
6978:
6976:
6972:
6971:
6966:
6935:
6933:
6932:
6927:
6925:
6924:
6896:
6894:
6893:
6888:
6886:
6885:
6859:
6857:
6856:
6851:
6840:
6839:
6769:
6767:
6766:
6761:
6756:
6747:
6742:
6703:
6701:
6700:
6695:
6556:
6534:
6516:Balanced/Circled
6314:Robinson-Ursescu
6232:Eberlein–Šmulian
6152:Spectral theorem
5948:Linear operators
5745:Uniformly smooth
5642:
5635:
5628:
5619:
5614:
5589:Wilansky, Albert
5584:
5550:
5523:
5517:
5511:
5505:
5496:
5490:
5475:
5469:
5463:
5457:
5442:
5440:
5438:
5437:
5432:
5417:
5415:
5414:
5409:
5400:
5395:
5384:
5380:
5379:
5378:
5369:
5368:
5346:
5344:
5343:
5338:
5323:
5321:
5320:
5315:
5312:
5307:
5296:
5292:
5291:
5290:
5281:
5280:
5258:
5256:
5255:
5250:
5238:
5236:
5235:
5230:
5222:
5221:
5197:
5193:
5192:
5169:
5167:
5166:
5161:
5149:
5147:
5146:
5141:
5123:
5121:
5120:
5115:
5100:
5098:
5097:
5092:
5081:
5077:
5076:
5053:
5051:
5050:
5045:
5027:
5025:
5024:
5019:
5004:
5002:
5001:
4996:
4993:
4988:
4977:
4973:
4972:
4968:
4967:
4942:
4938:
4937:
4914:
4912:
4911:
4906:
4894:
4892:
4891:
4886:
4877:
4872:
4861:
4857:
4856:
4838:
4833:
4822:
4818:
4817:
4816:
4791:
4789:
4788:
4783:
4780:
4775:
4764:
4760:
4756:
4751:
4750:
4732:
4731:
4715:
4713:
4712:
4707:
4692:
4690:
4689:
4684:
4679:
4678:
4666:
4662:
4661:
4638:
4636:
4635:
4630:
4628:
4627:
4611:
4609:
4608:
4603:
4598:
4594:
4593:
4570:
4568:
4567:
4562:
4550:
4548:
4547:
4542:
4530:
4528:
4527:
4522:
4506:
4504:
4503:
4498:
4483:
4481:
4480:
4475:
4472:
4467:
4456:
4452:
4451:
4447:
4446:
4421:
4417:
4416:
4393:
4391:
4390:
4385:
4373:
4371:
4370:
4365:
4362:
4357:
4346:
4342:
4341:
4324:
4323:
4306:
4285:
4262:Operator algebra
4217:
4215:
4214:
4209:
4197:
4195:
4194:
4189:
4177:
4175:
4174:
4169:
4139:
4137:
4136:
4131:
4119:
4117:
4116:
4111:
4099:
4097:
4096:
4091:
4051:
4049:
4048:
4043:
4016:
4014:
4013:
4008:
3993:
3991:
3990:
3985:
3955:
3953:
3952:
3947:
3920:
3918:
3917:
3912:
3900:
3898:
3897:
3892:
3873:is not bounded.
3872:
3870:
3869:
3864:
3852:
3850:
3849:
3844:
3823:
3821:
3820:
3815:
3789:
3788:
3763:
3761:
3760:
3755:
3738:
3737:
3718:
3716:
3715:
3710:
3677:
3675:
3674:
3669:
3667:
3666:
3648:
3647:
3627:
3625:
3624:
3619:
3593:
3591:
3590:
3585:
3573:
3556:
3549:
3544:
3511:
3509:
3508:
3503:
3469:
3467:
3466:
3461:
3441:
3439:
3438:
3433:
3421:
3419:
3418:
3413:
3411:
3407:
3400:
3399:
3387:
3386:
3374:
3373:
3341:
3340:
3328:
3327:
3315:
3314:
3293:
3291:
3290:
3285:
3266:
3261:
3248:
3243:
3230:
3225:
3209:
3207:
3206:
3201:
3199:
3195:
3188:
3187:
3175:
3174:
3162:
3161:
3137:
3135:
3134:
3129:
3127:
3126:
3088:
3086:
3085:
3080:
3074:
3073:
3068:
3059:
3058:
3043:
3042:
3037:
3028:
3027:
3005:Laplace operator
2998:compact operator
2995:
2993:
2992:
2987:
2943:
2938:
2898:
2896:
2895:
2890:
2878:
2876:
2875:
2870:
2839:
2837:
2836:
2833:{\displaystyle }
2831:
2807:
2805:
2804:
2799:
2772:
2770:
2769:
2764:
2752:
2750:
2749:
2744:
2742:
2681:
2679:
2678:
2673:
2671:
2670:
2654:
2652:
2651:
2646:
2644:
2643:
2627:
2625:
2624:
2619:
2617:
2616:
2573:
2571:
2570:
2565:
2545:
2543:
2542:
2537:
2524:sequential space
2521:
2519:
2518:
2513:
2487:
2485:
2484:
2479:
2464:
2462:
2461:
2456:
2440:
2438:
2437:
2432:
2417:
2415:
2414:
2409:
2394:
2392:
2391:
2386:
2370:
2368:
2367:
2362:
2360:
2359:
2334:
2332:
2331:
2326:
2307:
2305:
2304:
2299:
2287:
2285:
2284:
2279:
2262:
2260:
2259:
2254:
2239:
2237:
2236:
2231:
2228:
2223:
2212:
2208:
2207:
2206:
2197:
2196:
2178:
2177:
2161:
2159:
2158:
2153:
2144:
2139:
2128:
2124:
2123:
2106:
2105:
2088:
2086:
2085:
2080:
2063:
2061:
2060:
2055:
2052:
2047:
2036:
2032:
2031:
2014:
2013:
1995:
1993:
1992:
1987:
1972:
1970:
1969:
1964:
1938:
1936:
1935:
1930:
1916:
1914:
1913:
1908:
1872:
1870:
1869:
1864:
1850:
1848:
1847:
1842:
1827:
1825:
1824:
1819:
1804:
1802:
1801:
1796:
1761:
1759:
1758:
1753:
1729:
1727:
1726:
1721:
1706:
1704:
1703:
1698:
1667:
1665:
1664:
1659:
1647:
1645:
1644:
1639:
1624:
1622:
1621:
1616:
1595:
1593:
1592:
1587:
1575:
1573:
1572:
1567:
1555:
1553:
1552:
1547:
1501:(for example, a
1475:seminormed space
1464:
1462:
1461:
1456:
1441:
1439:
1438:
1433:
1412:
1410:
1409:
1404:
1388:
1386:
1385:
1380:
1346:
1344:
1343:
1338:
1302:
1300:
1299:
1294:
1282:
1280:
1279:
1274:
1260:
1252:
1241:
1236:
1225:
1220:
1216:
1215:
1211:
1210:
1208:
1194:
1176:
1171:
1160:
1155:
1151:
1150:
1146:
1145:
1143:
1129:
1116:
1111:
1100:
1071:
1069:
1068:
1063:
1042:
1040:
1039:
1034:
1007:
1005:
1004:
999:
982:for all vectors
981:
979:
978:
973:
904:
902:
901:
896:
867:
865:
864:
859:
847:
845:
844:
839:
824:
822:
821:
816:
804:
802:
801:
796:
781:
779:
778:
773:
761:
759:
758:
753:
741:
739:
738:
733:
647:nonzero we have
646:
644:
643:
638:
626:
624:
623:
618:
594:
592:
591:
586:
556:
554:
553:
548:
524:
522:
521:
516:
504:
502:
501:
496:
468:
466:
465:
460:
431:and vice versa.
426:
424:
423:
418:
397:
395:
394:
389:
373:
371:
370:
365:
353:
351:
350:
345:
340:
339:
318:
317:
293:
291:
290:
285:
264:
262:
261:
256:
238:
236:
235:
230:
214:
212:
211:
206:
194:
192:
191:
186:
174:
172:
171:
166:
151:
149:
148:
143:
127:
125:
124:
119:
107:
105:
104:
99:
83:
81:
80:
75:
32:bounded function
21:
7811:
7810:
7806:
7805:
7804:
7802:
7801:
7800:
7791:Operator theory
7776:
7775:
7774:
7769:
7735:Barrelled space
7718:
7709:Bornivorous set
7692:
7666:
7642:Barrelled space
7630:
7621:
7591:
7586:
7568:
7532:Advanced topics
7527:
7451:
7430:
7389:
7355:Hilbert–Schmidt
7328:
7319:Gelfand–Naimark
7266:
7216:
7151:
7137:
7107:
7102:
7066:
7040:
7023:
7022:
7021:Wiener amalgam
6991:Segal–Bargmann
6961:
6957:
6949:
6948:
6916:
6911:
6910:
6877:
6872:
6871:
6825:
6820:
6819:
6774:Birnbaum–Orlicz
6723:
6722:
6674:
6673:
6651:
6607:Bounding points
6580:
6554:
6532:
6489:
6340:Banach manifold
6323:
6247:Gelfand–Naimark
6168:
6142:Spectral theory
6110:Banach algebras
6102:Operator theory
6096:
6057:Pseudo-monotone
6040:Hilbert–Schmidt
6020:Densely defined
5942:
5855:
5769:
5652:
5646:
5603:
5587:
5573:
5560:
5535:
5532:
5527:
5526:
5518:
5514:
5506:
5499:
5491:
5478:
5470:
5466:
5458:
5451:
5446:
5445:
5420:
5419:
5370:
5360:
5359:
5355:
5349:
5348:
5326:
5325:
5282:
5272:
5271:
5267:
5261:
5260:
5241:
5240:
5213:
5184:
5180:
5172:
5171:
5152:
5151:
5126:
5125:
5103:
5102:
5068:
5064:
5056:
5055:
5030:
5029:
5028:So pick a real
5007:
5006:
4959:
4955:
4951:
4947:
4929:
4925:
4917:
4916:
4897:
4896:
4848:
4844:
4808:
4804:
4800:
4794:
4793:
4742:
4741:
4737:
4723:
4718:
4717:
4695:
4694:
4670:
4653:
4649:
4641:
4640:
4619:
4614:
4613:
4585:
4581:
4573:
4572:
4553:
4552:
4533:
4532:
4513:
4512:
4486:
4485:
4438:
4434:
4430:
4426:
4408:
4404:
4396:
4395:
4376:
4375:
4333:
4329:
4315:
4310:
4309:
4307:
4303:
4298:
4283:
4274:Operator theory
4225:
4200:
4199:
4180:
4179:
4145:
4144:
4122:
4121:
4102:
4101:
4061:
4060:
4019:
4018:
3999:
3998:
3961:
3960:
3923:
3922:
3903:
3902:
3883:
3882:
3879:
3855:
3854:
3826:
3825:
3780:
3766:
3765:
3729:
3721:
3720:
3680:
3679:
3652:
3639:
3634:
3633:
3598:
3597:
3516:
3515:
3512:with the norm
3476:
3475:
3452:
3451:
3448:
3424:
3423:
3391:
3378:
3365:
3358:
3354:
3332:
3319:
3306:
3295:
3294:
3212:
3211:
3179:
3166:
3153:
3152:
3148:
3143:
3142:
3118:
3113:
3112:
3063:
3050:
3032:
3019:
3008:
3007:
2901:
2900:
2881:
2880:
2846:
2845:
2810:
2809:
2775:
2774:
2755:
2754:
2691:
2690:
2662:
2657:
2656:
2635:
2630:
2629:
2608:
2603:
2602:
2585:
2556:
2555:
2528:
2527:
2504:
2503:
2470:
2469:
2447:
2446:
2423:
2422:
2397:
2396:
2377:
2376:
2348:
2343:
2342:
2317:
2316:
2290:
2289:
2270:
2269:
2242:
2241:
2198:
2188:
2187:
2183:
2169:
2164:
2163:
2115:
2111:
2097:
2092:
2091:
2071:
2070:
2023:
2019:
2005:
2000:
1999:
1975:
1974:
1955:
1954:
1921:
1920:
1878:
1877:
1855:
1854:
1833:
1832:
1810:
1809:
1775:
1774:
1771:
1732:
1731:
1709:
1708:
1689:
1688:
1685:
1679:
1650:
1649:
1627:
1626:
1598:
1597:
1578:
1577:
1558:
1557:
1526:
1525:
1483:
1444:
1443:
1415:
1414:
1395:
1394:
1365:
1364:
1317:
1316:
1313:
1308:
1285:
1284:
1226:
1198:
1189:
1185:
1181:
1177:
1161:
1133:
1124:
1120:
1101:
1098:
1094:
1074:
1073:
1045:
1044:
1010:
1009:
984:
983:
907:
906:
881:
880:
850:
849:
827:
826:
807:
806:
784:
783:
764:
763:
744:
743:
649:
648:
629:
628:
597:
596:
577:
576:
562:
539:
538:
531:
507:
506:
478:
477:
439:
438:
400:
399:
398:and denoted by
380:
379:
356:
355:
331:
309:
295:
294:
267:
266:
241:
240:
221:
220:
197:
196:
177:
176:
154:
153:
134:
133:
110:
109:
90:
89:
54:
53:
43:operator theory
35:
28:
23:
22:
15:
12:
11:
5:
7809:
7807:
7799:
7798:
7793:
7788:
7778:
7777:
7771:
7770:
7768:
7767:
7762:
7752:
7747:
7737:
7726:
7724:
7723:Related spaces
7720:
7719:
7717:
7716:
7711:
7706:
7700:
7698:
7694:
7693:
7691:
7690:
7685:
7674:
7672:
7668:
7667:
7665:
7664:
7654:
7649:
7644:
7638:
7636:
7635:Basic concepts
7632:
7631:
7622:
7620:
7619:
7612:
7605:
7597:
7588:
7587:
7585:
7584:
7573:
7570:
7569:
7567:
7566:
7561:
7556:
7551:
7549:Choquet theory
7546:
7541:
7535:
7533:
7529:
7528:
7526:
7525:
7515:
7510:
7505:
7500:
7495:
7490:
7485:
7480:
7475:
7470:
7465:
7459:
7457:
7453:
7452:
7450:
7449:
7444:
7438:
7436:
7432:
7431:
7429:
7428:
7423:
7418:
7413:
7408:
7403:
7401:Banach algebra
7397:
7395:
7391:
7390:
7388:
7387:
7382:
7377:
7372:
7367:
7362:
7357:
7352:
7347:
7342:
7336:
7334:
7330:
7329:
7327:
7326:
7324:Banach–Alaoglu
7321:
7316:
7311:
7306:
7301:
7296:
7291:
7286:
7280:
7278:
7272:
7271:
7268:
7267:
7265:
7264:
7259:
7254:
7252:Locally convex
7249:
7235:
7230:
7224:
7222:
7218:
7217:
7215:
7214:
7209:
7204:
7199:
7194:
7189:
7184:
7179:
7174:
7169:
7163:
7157:
7153:
7152:
7138:
7136:
7135:
7128:
7121:
7113:
7104:
7103:
7101:
7100:
7095:
7090:
7085:
7080:
7074:
7072:
7068:
7067:
7065:
7064:
7052:
7047:
7043:
7039:
7036:
7033:
7030:
7018:
7013:
7012:
7011:
7001:
6999:Sequence space
6996:
6988:
6975:
6970:
6965:
6960:
6956:
6944:
6943:
6942:
6937:
6923:
6919:
6900:
6899:
6898:
6884:
6880:
6861:
6849:
6846:
6843:
6838:
6835:
6832:
6828:
6815:
6807:
6802:
6789:
6784:
6776:
6771:
6759:
6755:
6751:
6746:
6741:
6738:
6735:
6731:
6718:
6710:
6705:
6693:
6690:
6687:
6684:
6681:
6670:
6661:
6659:
6653:
6652:
6650:
6649:
6639:
6634:
6629:
6624:
6619:
6614:
6609:
6604:
6594:
6588:
6586:
6582:
6581:
6579:
6578:
6573:
6568:
6563:
6558:
6550:
6536:
6528:
6523:
6518:
6513:
6508:
6503:
6497:
6495:
6491:
6490:
6488:
6487:
6477:
6476:
6475:
6470:
6465:
6455:
6454:
6453:
6448:
6443:
6433:
6432:
6431:
6426:
6421:
6416:
6414:Gelfand–Pettis
6411:
6406:
6396:
6395:
6394:
6389:
6384:
6379:
6374:
6364:
6359:
6354:
6349:
6348:
6347:
6337:
6331:
6329:
6325:
6324:
6322:
6321:
6316:
6311:
6306:
6301:
6296:
6291:
6286:
6281:
6276:
6271:
6266:
6265:
6264:
6254:
6249:
6244:
6239:
6234:
6229:
6224:
6219:
6214:
6209:
6204:
6199:
6194:
6189:
6187:Banach–Alaoglu
6184:
6182:Anderson–Kadec
6178:
6176:
6170:
6169:
6167:
6166:
6161:
6156:
6155:
6154:
6149:
6139:
6138:
6137:
6132:
6122:
6120:Operator space
6117:
6112:
6106:
6104:
6098:
6097:
6095:
6094:
6089:
6084:
6079:
6074:
6069:
6064:
6059:
6054:
6053:
6052:
6042:
6037:
6036:
6035:
6030:
6022:
6017:
6007:
6006:
6005:
5995:
5990:
5980:
5979:
5978:
5973:
5968:
5958:
5952:
5950:
5944:
5943:
5941:
5940:
5935:
5930:
5929:
5928:
5923:
5913:
5912:
5911:
5906:
5896:
5891:
5886:
5885:
5884:
5874:
5869:
5863:
5861:
5857:
5856:
5854:
5853:
5848:
5843:
5842:
5841:
5831:
5826:
5821:
5820:
5819:
5808:Locally convex
5805:
5804:
5803:
5793:
5788:
5783:
5777:
5775:
5771:
5770:
5768:
5767:
5760:Tensor product
5753:
5747:
5742:
5736:
5731:
5725:
5720:
5715:
5705:
5704:
5703:
5698:
5688:
5683:
5681:Banach lattice
5678:
5677:
5676:
5666:
5660:
5658:
5654:
5653:
5647:
5645:
5644:
5637:
5630:
5622:
5616:
5615:
5601:
5585:
5572:978-1584888666
5571:
5558:
5551:
5531:
5528:
5525:
5524:
5512:
5510:, p. 444.
5497:
5476:
5464:
5448:
5447:
5444:
5443:
5430:
5427:
5407:
5404:
5399:
5394:
5391:
5388:
5383:
5377:
5373:
5367:
5363:
5358:
5336:
5333:
5311:
5306:
5303:
5300:
5295:
5289:
5285:
5279:
5275:
5270:
5248:
5228:
5225:
5220:
5216:
5212:
5209:
5206:
5203:
5200:
5196:
5191:
5187:
5183:
5179:
5159:
5139:
5136:
5133:
5113:
5110:
5090:
5087:
5084:
5080:
5075:
5071:
5067:
5063:
5043:
5040:
5037:
5017:
5014:
5005:is bounded in
4992:
4987:
4984:
4981:
4976:
4971:
4966:
4962:
4958:
4954:
4950:
4945:
4941:
4936:
4932:
4928:
4924:
4904:
4895:is bounded in
4884:
4881:
4876:
4871:
4868:
4865:
4860:
4855:
4851:
4847:
4842:
4837:
4832:
4829:
4826:
4821:
4815:
4811:
4807:
4803:
4779:
4774:
4771:
4768:
4763:
4759:
4755:
4749:
4745:
4740:
4735:
4730:
4726:
4705:
4702:
4682:
4677:
4673:
4669:
4665:
4660:
4656:
4652:
4648:
4626:
4622:
4601:
4597:
4592:
4588:
4584:
4580:
4560:
4540:
4520:
4496:
4493:
4471:
4466:
4463:
4460:
4455:
4450:
4445:
4441:
4437:
4433:
4429:
4424:
4420:
4415:
4411:
4407:
4403:
4383:
4361:
4356:
4353:
4350:
4345:
4340:
4336:
4332:
4327:
4322:
4318:
4300:
4299:
4297:
4294:
4293:
4292:
4286:
4277:
4271:
4265:
4259:
4253:
4248:
4243:
4238:
4232:
4224:
4221:
4220:
4219:
4207:
4187:
4178:is Banach and
4167:
4164:
4161:
4158:
4155:
4152:
4141:
4129:
4109:
4100:the kernel of
4089:
4086:
4083:
4080:
4077:
4074:
4071:
4068:
4057:
4041:
4038:
4035:
4032:
4029:
4026:
4006:
3995:
3983:
3980:
3977:
3974:
3971:
3968:
3945:
3942:
3939:
3936:
3933:
3930:
3921:is denoted by
3910:
3890:
3878:
3875:
3862:
3842:
3839:
3836:
3833:
3813:
3810:
3807:
3804:
3801:
3798:
3795:
3792:
3787:
3783:
3779:
3776:
3773:
3753:
3750:
3747:
3744:
3741:
3736:
3732:
3728:
3708:
3705:
3702:
3699:
3696:
3693:
3690:
3687:
3665:
3662:
3659:
3655:
3651:
3646:
3642:
3617:
3614:
3611:
3608:
3605:
3583:
3580:
3577:
3572:
3568:
3565:
3562:
3559:
3555:
3548:
3543:
3540:
3536:
3532:
3529:
3526:
3523:
3501:
3498:
3495:
3492:
3489:
3486:
3483:
3459:
3447:
3444:
3443:
3442:
3431:
3410:
3406:
3403:
3398:
3394:
3390:
3385:
3381:
3377:
3372:
3368:
3364:
3361:
3357:
3353:
3350:
3347:
3344:
3339:
3335:
3331:
3326:
3322:
3318:
3313:
3309:
3305:
3302:
3282:
3279:
3276:
3273:
3270:
3265:
3260:
3256:
3252:
3247:
3242:
3238:
3234:
3229:
3224:
3220:
3198:
3194:
3191:
3186:
3182:
3178:
3173:
3169:
3165:
3160:
3156:
3151:
3125:
3121:
3106:shift operator
3102:
3077:
3072:
3067:
3062:
3057:
3053:
3049:
3046:
3041:
3036:
3031:
3026:
3022:
3018:
3015:
3001:
2985:
2982:
2979:
2975:
2972:
2969:
2966:
2963:
2960:
2957:
2954:
2951:
2948:
2942:
2937:
2933:
2929:
2926:
2923:
2920:
2917:
2914:
2911:
2908:
2888:
2868:
2865:
2862:
2859:
2856:
2853:
2829:
2826:
2823:
2820:
2817:
2797:
2794:
2791:
2788:
2785:
2782:
2762:
2741:
2737:
2734:
2731:
2728:
2725:
2722:
2719:
2716:
2713:
2710:
2707:
2704:
2701:
2698:
2683:
2669:
2665:
2642:
2638:
2615:
2611:
2600:sequence space
2596:
2593:
2584:
2581:
2580:
2579:
2563:
2553:
2552:
2551:
2535:
2511:
2502:If the domain
2500:
2477:
2454:
2430:
2419:
2418:
2407:
2404:
2384:
2358:
2355:
2351:
2340:
2324:
2310:locally convex
2297:
2277:
2266:
2265:
2264:
2263:
2252:
2249:
2227:
2222:
2219:
2216:
2211:
2205:
2201:
2195:
2191:
2186:
2181:
2176:
2172:
2151:
2148:
2143:
2138:
2135:
2132:
2127:
2122:
2118:
2114:
2109:
2104:
2100:
2078:
2064:is said to be
2051:
2046:
2043:
2040:
2035:
2030:
2026:
2022:
2017:
2012:
2008:
1985:
1982:
1962:
1952:
1951:
1950:
1947:
1928:
1918:
1906:
1903:
1900:
1897:
1894:
1891:
1888:
1885:
1862:
1852:
1840:
1831:(Definition):
1829:
1817:
1794:
1791:
1788:
1785:
1782:
1770:
1767:
1751:
1748:
1745:
1742:
1739:
1719:
1716:
1696:
1681:Main article:
1678:
1675:
1657:
1637:
1634:
1614:
1611:
1608:
1605:
1585:
1565:
1545:
1542:
1539:
1536:
1533:
1482:
1479:
1454:
1451:
1442:is bounded in
1431:
1428:
1425:
1422:
1402:
1378:
1375:
1372:
1361:
1355:
1336:
1333:
1330:
1327:
1324:
1312:
1309:
1292:
1272:
1269:
1266:
1263:
1258:
1255:
1250:
1247:
1244:
1239:
1235:
1232:
1229:
1223:
1219:
1214:
1207:
1204:
1201:
1197:
1192:
1188:
1184:
1180:
1174:
1170:
1167:
1164:
1158:
1154:
1149:
1142:
1139:
1136:
1132:
1127:
1123:
1119:
1114:
1110:
1107:
1104:
1097:
1093:
1090:
1087:
1084:
1081:
1061:
1058:
1055:
1052:
1032:
1029:
1026:
1023:
1020:
1017:
997:
994:
991:
971:
968:
965:
962:
959:
956:
953:
950:
947:
944:
941:
938:
935:
932:
929:
926:
923:
920:
917:
914:
894:
891:
888:
857:
837:
834:
814:
794:
791:
771:
751:
731:
728:
725:
722:
719:
716:
713:
710:
707:
704:
701:
698:
695:
692:
689:
686:
683:
680:
677:
674:
671:
668:
665:
662:
659:
656:
636:
616:
613:
610:
607:
604:
584:
570:
561:
558:
546:
530:
527:
514:
494:
491:
488:
485:
458:
455:
452:
449:
446:
416:
413:
410:
407:
387:
374:is called the
363:
343:
338:
334:
330:
327:
324:
321:
316:
312:
308:
305:
302:
283:
280:
277:
274:
254:
251:
248:
228:
204:
184:
164:
161:
141:
117:
97:
73:
70:
67:
64:
61:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
7808:
7797:
7794:
7792:
7789:
7787:
7784:
7783:
7781:
7766:
7763:
7761:
7757:
7753:
7751:
7748:
7746:
7742:
7738:
7736:
7732:
7728:
7727:
7725:
7721:
7715:
7712:
7710:
7707:
7705:
7704:Barrelled set
7702:
7701:
7699:
7695:
7689:
7686:
7684:
7680:
7676:
7675:
7673:
7669:
7663:
7659:
7655:
7653:
7650:
7648:
7645:
7643:
7640:
7639:
7637:
7633:
7629:
7625:
7618:
7613:
7611:
7606:
7604:
7599:
7598:
7595:
7583:
7575:
7574:
7571:
7565:
7562:
7560:
7557:
7555:
7554:Weak topology
7552:
7550:
7547:
7545:
7542:
7540:
7537:
7536:
7534:
7530:
7523:
7519:
7516:
7514:
7511:
7509:
7506:
7504:
7501:
7499:
7496:
7494:
7491:
7489:
7486:
7484:
7481:
7479:
7478:Index theorem
7476:
7474:
7471:
7469:
7466:
7464:
7461:
7460:
7458:
7454:
7448:
7445:
7443:
7440:
7439:
7437:
7435:Open problems
7433:
7427:
7424:
7422:
7419:
7417:
7414:
7412:
7409:
7407:
7404:
7402:
7399:
7398:
7396:
7392:
7386:
7383:
7381:
7378:
7376:
7373:
7371:
7368:
7366:
7363:
7361:
7358:
7356:
7353:
7351:
7348:
7346:
7343:
7341:
7338:
7337:
7335:
7331:
7325:
7322:
7320:
7317:
7315:
7312:
7310:
7307:
7305:
7302:
7300:
7297:
7295:
7292:
7290:
7287:
7285:
7282:
7281:
7279:
7277:
7273:
7263:
7260:
7258:
7255:
7253:
7250:
7247:
7243:
7239:
7236:
7234:
7231:
7229:
7226:
7225:
7223:
7219:
7213:
7210:
7208:
7205:
7203:
7200:
7198:
7195:
7193:
7190:
7188:
7185:
7183:
7180:
7178:
7175:
7173:
7170:
7168:
7165:
7164:
7161:
7158:
7154:
7149:
7145:
7141:
7134:
7129:
7127:
7122:
7120:
7115:
7114:
7111:
7099:
7096:
7094:
7091:
7089:
7086:
7084:
7081:
7079:
7076:
7075:
7073:
7069:
7063:
7045:
7041:
7037:
7034:
7028:
7019:
7017:
7014:
7010:
7007:
7006:
7005:
7002:
7000:
6997:
6995:
6994:
6989:
6987:
6973:
6968:
6958:
6954:
6945:
6941:
6938:
6936:
6917:
6908:
6907:
6906:
6905:
6901:
6897:
6878:
6869:
6868:
6867:
6866:
6862:
6860:
6836:
6833:
6830:
6826:
6816:
6814:
6813:
6808:
6806:
6803:
6801:
6799:
6795:
6790:
6788:
6785:
6783:
6782:
6777:
6775:
6772:
6770:
6744:
6739:
6736:
6733:
6729:
6719:
6717:
6716:
6711:
6709:
6706:
6704:
6682:
6679:
6671:
6669:
6668:
6663:
6662:
6660:
6658:
6654:
6648:
6644:
6640:
6638:
6635:
6633:
6630:
6628:
6625:
6623:
6620:
6618:
6617:Extreme point
6615:
6613:
6610:
6608:
6605:
6603:
6599:
6595:
6593:
6590:
6589:
6587:
6583:
6577:
6574:
6572:
6569:
6567:
6564:
6562:
6559:
6557:
6551:
6548:
6544:
6540:
6537:
6535:
6529:
6527:
6524:
6522:
6519:
6517:
6514:
6512:
6509:
6507:
6504:
6502:
6499:
6498:
6496:
6494:Types of sets
6492:
6485:
6481:
6478:
6474:
6471:
6469:
6466:
6464:
6461:
6460:
6459:
6456:
6452:
6449:
6447:
6444:
6442:
6439:
6438:
6437:
6434:
6430:
6427:
6425:
6422:
6420:
6417:
6415:
6412:
6410:
6407:
6405:
6402:
6401:
6400:
6397:
6393:
6390:
6388:
6385:
6383:
6380:
6378:
6375:
6373:
6370:
6369:
6368:
6365:
6363:
6360:
6358:
6357:Convex series
6355:
6353:
6352:Bochner space
6350:
6346:
6343:
6342:
6341:
6338:
6336:
6333:
6332:
6330:
6326:
6320:
6317:
6315:
6312:
6310:
6307:
6305:
6304:Riesz's lemma
6302:
6300:
6297:
6295:
6292:
6290:
6289:Mazur's lemma
6287:
6285:
6282:
6280:
6277:
6275:
6272:
6270:
6267:
6263:
6260:
6259:
6258:
6255:
6253:
6250:
6248:
6245:
6243:
6242:Gelfand–Mazur
6240:
6238:
6235:
6233:
6230:
6228:
6225:
6223:
6220:
6218:
6215:
6213:
6210:
6208:
6205:
6203:
6200:
6198:
6195:
6193:
6190:
6188:
6185:
6183:
6180:
6179:
6177:
6175:
6171:
6165:
6162:
6160:
6157:
6153:
6150:
6148:
6145:
6144:
6143:
6140:
6136:
6133:
6131:
6128:
6127:
6126:
6123:
6121:
6118:
6116:
6113:
6111:
6108:
6107:
6105:
6103:
6099:
6093:
6090:
6088:
6085:
6083:
6080:
6078:
6075:
6073:
6070:
6068:
6065:
6063:
6060:
6058:
6055:
6051:
6048:
6047:
6046:
6043:
6041:
6038:
6034:
6031:
6029:
6026:
6025:
6023:
6021:
6018:
6016:
6012:
6008:
6004:
6001:
6000:
5999:
5996:
5994:
5991:
5989:
5985:
5981:
5977:
5974:
5972:
5969:
5967:
5964:
5963:
5962:
5959:
5957:
5954:
5953:
5951:
5949:
5945:
5939:
5936:
5934:
5931:
5927:
5924:
5922:
5919:
5918:
5917:
5914:
5910:
5907:
5905:
5902:
5901:
5900:
5897:
5895:
5892:
5890:
5887:
5883:
5880:
5879:
5878:
5875:
5873:
5870:
5868:
5865:
5864:
5862:
5858:
5852:
5849:
5847:
5844:
5840:
5837:
5836:
5835:
5832:
5830:
5827:
5825:
5822:
5818:
5814:
5811:
5810:
5809:
5806:
5802:
5799:
5798:
5797:
5794:
5792:
5789:
5787:
5784:
5782:
5779:
5778:
5776:
5772:
5765:
5761:
5757:
5754:
5752:
5748:
5746:
5743:
5741:) convex
5740:
5737:
5735:
5732:
5730:
5726:
5724:
5721:
5719:
5716:
5714:
5710:
5706:
5702:
5699:
5697:
5694:
5693:
5692:
5689:
5687:
5686:Grothendieck
5684:
5682:
5679:
5675:
5672:
5671:
5670:
5667:
5665:
5662:
5661:
5659:
5655:
5650:
5643:
5638:
5636:
5631:
5629:
5624:
5623:
5620:
5612:
5608:
5604:
5598:
5594:
5590:
5586:
5582:
5578:
5574:
5568:
5564:
5559:
5557:, Wiley, 1989
5556:
5552:
5548:
5544:
5543:
5538:
5534:
5533:
5529:
5521:
5516:
5513:
5509:
5504:
5502:
5498:
5494:
5489:
5487:
5485:
5483:
5481:
5477:
5473:
5468:
5465:
5461:
5460:Wilansky 2013
5456:
5454:
5450:
5428:
5425:
5405:
5392:
5389:
5386:
5381:
5375:
5371:
5365:
5361:
5356:
5334:
5331:
5304:
5301:
5298:
5293:
5287:
5283:
5277:
5273:
5268:
5246:
5226:
5223:
5218:
5214:
5210:
5207:
5204:
5201:
5198:
5194:
5189:
5185:
5181:
5177:
5170:is balanced,
5157:
5137:
5134:
5131:
5111:
5108:
5088:
5085:
5082:
5078:
5073:
5069:
5065:
5061:
5041:
5038:
5035:
5015:
5012:
4985:
4982:
4979:
4974:
4969:
4964:
4960:
4956:
4952:
4948:
4943:
4939:
4934:
4930:
4926:
4922:
4902:
4882:
4869:
4866:
4863:
4858:
4853:
4849:
4845:
4840:
4830:
4827:
4824:
4819:
4813:
4809:
4805:
4801:
4772:
4769:
4766:
4761:
4757:
4753:
4747:
4743:
4738:
4733:
4728:
4724:
4716:The sequence
4703:
4700:
4680:
4675:
4671:
4667:
4663:
4658:
4654:
4650:
4646:
4624:
4620:
4599:
4595:
4590:
4586:
4582:
4578:
4558:
4538:
4518:
4511:neighborhood
4510:
4507:Pick an open
4494:
4491:
4464:
4461:
4458:
4453:
4448:
4443:
4439:
4435:
4431:
4427:
4422:
4418:
4413:
4409:
4405:
4401:
4381:
4374:converges to
4354:
4351:
4348:
4343:
4338:
4334:
4330:
4325:
4320:
4316:
4305:
4302:
4295:
4290:
4287:
4281:
4278:
4275:
4272:
4269:
4268:Operator norm
4266:
4263:
4260:
4257:
4254:
4252:
4249:
4247:
4244:
4242:
4239:
4236:
4233:
4230:
4227:
4226:
4222:
4205:
4185:
4162:
4159:
4156:
4150:
4142:
4127:
4107:
4084:
4081:
4078:
4072:
4069:
4066:
4058:
4055:
4036:
4033:
4030:
4024:
4004:
3996:
3978:
3975:
3972:
3966:
3959:
3958:
3957:
3940:
3937:
3934:
3928:
3908:
3888:
3876:
3874:
3860:
3840:
3831:
3805:
3802:
3799:
3796:
3785:
3781:
3774:
3751:
3748:
3745:
3742:
3734:
3730:
3706:
3703:
3700:
3697:
3694:
3691:
3688:
3685:
3663:
3660:
3657:
3653:
3649:
3644:
3640:
3631:
3615:
3609:
3606:
3603:
3596:The operator
3594:
3581:
3578:
3575:
3563:
3557:
3546:
3541:
3538:
3534:
3530:
3524:
3513:
3499:
3493:
3490:
3487:
3484:
3473:
3457:
3445:
3429:
3408:
3404:
3401:
3396:
3392:
3388:
3383:
3379:
3375:
3370:
3366:
3362:
3359:
3355:
3351:
3345:
3342:
3337:
3333:
3329:
3324:
3320:
3316:
3311:
3307:
3300:
3280:
3274:
3271:
3268:
3263:
3258:
3254:
3250:
3245:
3240:
3236:
3232:
3227:
3222:
3218:
3196:
3192:
3189:
3184:
3180:
3176:
3171:
3167:
3163:
3158:
3154:
3149:
3141:
3123:
3119:
3111:
3107:
3103:
3101:) is bounded.
3100:
3096:
3095:Sobolev space
3092:
3070:
3055:
3051:
3039:
3024:
3020:
3016:
3006:
3002:
2999:
2983:
2980:
2977:
2970:
2964:
2958:
2955:
2952:
2946:
2940:
2935:
2931:
2927:
2921:
2912:
2909:
2886:
2863:
2860:
2857:
2851:
2843:
2824:
2821:
2818:
2792:
2789:
2786:
2780:
2760:
2729:
2726:
2723:
2717:
2711:
2708:
2705:
2699:
2696:
2688:
2684:
2663:
2640:
2636:
2613:
2609:
2601:
2597:
2594:
2591:
2587:
2586:
2582:
2577:
2561:
2554:
2549:
2533:
2525:
2509:
2501:
2498:
2494:
2493:
2491:
2475:
2468:
2466:
2452:
2444:
2428:
2405:
2402:
2382:
2374:
2356:
2353:
2349:
2341:
2338:
2335:maps bounded
2322:
2315:
2313:
2311:
2295:
2275:
2250:
2247:
2220:
2217:
2214:
2209:
2203:
2199:
2193:
2189:
2184:
2179:
2174:
2170:
2136:
2133:
2130:
2125:
2120:
2116:
2112:
2107:
2102:
2098:
2089:
2076:
2068:
2044:
2041:
2038:
2033:
2028:
2024:
2020:
2015:
2010:
2006:
1997:
1996:
1983:
1980:
1960:
1953:
1948:
1945:
1944:null sequence
1941:
1940:
1926:
1919:
1901:
1895:
1892:
1889:
1886:
1883:
1876:
1860:
1853:
1838:
1830:
1815:
1808:
1807:
1806:
1792:
1786:
1783:
1780:
1768:
1766:
1763:
1749:
1743:
1740:
1737:
1717:
1714:
1694:
1684:
1676:
1674:
1672:
1655:
1635:
1632:
1609:
1603:
1583:
1563:
1543:
1537:
1534:
1531:
1522:
1520:
1516:
1512:
1508:
1507:Fréchet space
1504:
1500:
1495:
1491:
1488:
1480:
1478:
1476:
1472:
1468:
1452:
1449:
1426:
1420:
1400:
1392:
1376:
1373:
1370:
1362:
1359:
1356:
1353:
1350:
1334:
1328:
1325:
1322:
1310:
1307:
1306:
1290:
1270:
1264:
1256:
1253:
1248:
1245:
1242:
1237:
1230:
1221:
1212:
1202:
1195:
1190:
1186:
1182:
1172:
1165:
1156:
1147:
1137:
1130:
1125:
1121:
1117:
1112:
1105:
1091:
1085:
1082:
1059:
1056:
1053:
1050:
1030:
1027:
1024:
1018:
995:
992:
989:
969:
966:
957:
951:
948:
942:
936:
930:
921:
915:
892:
889:
886:
877:
875:
871:
855:
835:
832:
812:
792:
789:
769:
749:
729:
723:
717:
714:
705:
699:
693:
684:
678:
675:
669:
666:
663:
657:
634:
614:
611:
608:
605:
602:
582:
575:Suppose that
569:
567:
559:
557:
544:
536:
528:
526:
512:
489:
483:
476:
472:
456:
450:
447:
444:
435:
432:
430:
414:
408:
385:
377:
376:operator norm
361:
341:
336:
328:
322:
319:
314:
306:
303:
281:
278:
275:
272:
252:
249:
246:
226:
218:
202:
182:
162:
159:
139:
131:
115:
95:
87:
71:
65:
62:
59:
52:
48:
44:
40:
34:(set theory).
33:
19:
7682:
7544:Balanced set
7518:Distribution
7456:Applications
7344:
7309:Krein–Milman
7294:Closed graph
7071:Applications
6992:
6903:
6864:
6811:
6797:
6793:
6780:
6714:
6666:
6553:Linear cone
6546:
6542:
6531:Convex cone
6424:Paley–Wiener
6284:Mackey–Arens
6274:Krein–Milman
6227:Closed range
6222:Closed graph
6192:Banach–Mazur
6072:Self-adjoint
5987:
5976:sesquilinear
5709:Polynomially
5649:Banach space
5592:
5562:
5554:
5540:
5530:Bibliography
5515:
5467:
4304:
3880:
3595:
3514:
3449:
2842:uniform norm
2420:
2267:
2065:
1943:
1772:
1764:
1686:
1671:normed space
1523:
1511:normed space
1496:
1492:
1484:
1363:if whenever
1358:
1352:
1347:between two
1314:
1303:is bounded.
878:
574:
563:
532:
436:
433:
46:
36:
7647:Bounded set
7624:Boundedness
7473:Heat kernel
7463:Hardy space
7370:Trace class
7284:Hahn–Banach
7246:Topological
6792:Continuous
6627:Linear span
6612:Convex hull
6592:Affine hull
6451:holomorphic
6387:holomorphic
6367:Derivatives
6257:Hahn–Banach
6197:Banach–Saks
6115:C*-algebras
6082:Trace class
6045:Functionals
5933:Ultrastrong
5846:Quasinormed
4056:are Banach.
4054:dual spaces
2373:bornivorous
1998:A sequence
872:, and even
469:is called "
132:subsets of
7780:Categories
7406:C*-algebra
7221:Properties
6545:), and (Hw
6446:continuous
6382:functional
6130:C*-algebra
6015:Continuous
5877:Dual space
5851:Stereotype
5829:Metrizable
5756:Projective
5054:such that
4612:Replacing
4551:such that
4296:References
4218:is Banach.
3630:derivative
2522:is also a
1596:such that
905:such that
566:continuous
429:continuous
128:that maps
7741:Countably
7731:Countably
7671:Operators
7662:Bornology
7628:bornology
7380:Unbounded
7375:Transpose
7333:Operators
7262:Separable
7257:Reflexive
7242:Algebraic
7228:Barrelled
7004:Sobolev W
6947:Schwartz
6922:∞
6883:∞
6879:ℓ
6845:Ω
6831:λ
6689:Σ
6571:Symmetric
6506:Absorbing
6419:regulated
6399:Integrals
6252:Goldstine
6087:Transpose
6024:Fredholm
5894:Ultraweak
5882:Dual norm
5813:Seminorms
5781:Barrelled
5751:Injective
5739:Uniformly
5713:Reflexive
5611:849801114
5581:144216834
5547:EMS Press
5403:→
5398:∞
5310:∞
5211:⊆
5199:∈
5083:∈
4991:∞
4935:∙
4880:→
4875:∞
4836:∞
4778:∞
4729:∙
4625:∙
4591:∙
4470:∞
4414:∙
4360:∞
4321:∙
4070:∈
3838:∞
3835:→
3812:∞
3809:→
3803:π
3794:‖
3772:‖
3749:π
3740:‖
3727:‖
3704:…
3613:→
3547:π
3542:π
3539:−
3535:∫
3528:‖
3522:‖
3494:π
3488:π
3485:−
3405:…
3346:…
3278:∞
3272:⋯
3193:…
3140:sequences
3120:ℓ
3048:→
3014:Δ
2932:∫
2736:→
2718:×
2668:∞
2664:ℓ
2637:ℓ
2375:disks in
2354:−
2226:∞
2175:∙
2150:∞
2147:→
2142:∞
2103:∙
2050:∞
2011:∙
1887:
1790:→
1747:→
1541:→
1515:LF spaces
1374:⊆
1332:→
1268:‖
1262:‖
1257:ε
1243:⋅
1238:ε
1234:‖
1228:‖
1222:≤
1206:‖
1200:‖
1191:ε
1173:ε
1169:‖
1163:‖
1141:‖
1135:‖
1126:ε
1113:ε
1109:‖
1103:‖
1089:‖
1080:‖
1054:∈
1028:ε
1025:≤
1022:‖
1016:‖
993:∈
967:≤
964:‖
949:−
934:‖
928:‖
913:‖
887:ε
727:‖
721:‖
715:≤
712:‖
697:‖
691:‖
676:−
655:‖
612:∈
454:→
412:‖
406:‖
333:‖
326:‖
320:≤
311:‖
301:‖
276:∈
69:→
7582:Category
7394:Algebras
7276:Theorems
7233:Complete
7202:Schwartz
7148:glossary
6940:weighted
6810:Hilbert
6787:Bs space
6657:Examples
6622:Interior
6598:Relative
6576:Zonotope
6555:(subset)
6533:(subset)
6484:Strongly
6463:Lebesgue
6458:Measures
6328:Analysis
6174:Theorems
6125:Spectrum
6050:positive
6033:operator
5971:operator
5961:Bilinear
5926:operator
5909:operator
5889:Operator
5786:Complete
5734:Strictly
5591:(2013).
4668:∉
4509:balanced
4280:Seminorm
4223:See also
4059:For any
3719:we have
3110:Lp space
2583:Examples
1357:or just
1218:‖
1179:‖
1153:‖
1096:‖
1072:one has
742:Letting
84:between
7697:Subsets
7385:Unitary
7365:Nuclear
7350:Compact
7345:Bounded
7340:Adjoint
7314:Min–max
7207:Sobolev
7192:Nuclear
7182:Hilbert
7177:Fréchet
7142: (
6805:Hardy H
6708:c space
6645:)
6600:)
6521:Bounded
6409:Dunford
6404:Bochner
6377:Gateaux
6372:Fréchet
6147:of ODEs
6092:Unitary
6067:Nuclear
5998:Compact
5988:Bounded
5956:Adjoint
5796:Fréchet
5791:F-space
5762: (
5758:)
5711:)
5691:Hilbert
5664:Asplund
5549:, 2001
3138:of all
3108:on the
2598:On the
2526:, then
1471:absorbs
1391:bounded
1360:bounded
471:bounded
130:bounded
88:(TVSs)
7756:Quasi-
7658:Vector
7360:Normal
7197:Orlicz
7187:Hölder
7167:Banach
7156:Spaces
7144:topics
6721:Besov
6561:Radial
6526:Convex
6511:Affine
6480:Weakly
6473:Vector
6345:bundle
6135:radius
6062:Normal
6028:kernel
5993:Closed
5916:Strong
5834:Normed
5824:Mackey
5669:Banach
5651:topics
5609:
5599:
5579:
5569:
3764:while
3091:domain
2590:matrix
1485:Every
1305:Q.E.D.
7172:Besov
6796:with
6643:Quasi
6637:Polar
6441:Borel
6392:quasi
5921:polar
5904:polar
5718:Riesz
3678:with
3093:is a
3089:(its
2879:with
2685:Many
2441:is a
2371:maps
2337:disks
1875:image
1648:then
1413:then
1008:with
627:with
572:Proof
475:image
49:is a
7626:and
7520:(or
7238:Dual
6794:C(K)
6429:weak
5966:form
5899:Weak
5872:Dual
5839:norm
5801:tame
5674:list
5607:OCLC
5597:ISBN
5577:OCLC
5567:ISBN
5135:>
5039:>
4394:but
3450:Let
3275:<
3104:The
3003:The
2445:and
2308:are
2288:and
1773:Let
1673:).
1509:, a
1505:, a
890:>
250:>
215:are
195:and
108:and
45:, a
41:and
6011:Dis
5418:in
5124:If
4143:If
3997:If
3901:to
3853:so
3824:as
3474:on
2574:is
2546:is
2488:is
2421:if
2268:if
2069:in
1524:If
1393:in
1389:is
868:is
537:at
378:of
175:If
37:In
7782::
7758:)
7743:)
7733:)
7679:Un
7660:)
7146:–
6781:BV
6715:BK
6667:AC
6549:))
6482:/
5984:Un
5605:.
5575:.
5545:,
5539:,
5500:^
5479:^
5452:^
4734::=
3956:.
3430:1.
2614:00
2495:A
1942:A
1893::=
1884:Im
1521:.
876:.
568:.
545:0.
513:0.
7754:(
7739:(
7729:(
7681:)
7677:(
7656:(
7616:e
7609:t
7602:v
7524:)
7248:)
7244:/
7240:(
7150:)
7132:e
7125:t
7118:v
7051:)
7046:p
7042:L
7038:,
7035:X
7032:(
7029:W
6993:F
6974:)
6969:n
6964:R
6959:(
6955:S
6918:L
6904:L
6865:â„“
6848:)
6842:(
6837:p
6834:,
6827:L
6812:H
6798:K
6758:)
6754:R
6750:(
6745:s
6740:q
6737:,
6734:p
6730:B
6692:)
6686:(
6683:a
6680:b
6641:(
6596:(
6547:x
6543:x
6013:)
6009:(
5986:)
5982:(
5815:/
5766:)
5749:(
5729:B
5727:(
5707:(
5641:e
5634:t
5627:v
5613:.
5583:.
5441:"
5429:.
5426:X
5406:0
5393:1
5390:=
5387:i
5382:)
5376:i
5372:x
5366:i
5362:r
5357:(
5335:.
5332:X
5305:1
5302:=
5299:i
5294:)
5288:i
5284:x
5278:i
5274:r
5269:(
5247:F
5227:,
5224:V
5219:2
5215:i
5208:V
5205:i
5202:r
5195:)
5190:i
5186:x
5182:(
5178:F
5158:V
5138:r
5132:i
5112:.
5109:i
5089:V
5086:r
5079:)
5074:i
5070:z
5066:(
5062:F
5042:1
5036:r
5016:.
5013:Y
4986:1
4983:=
4980:i
4975:)
4970:)
4965:i
4961:z
4957:(
4953:F
4949:(
4944:=
4940:)
4931:z
4927:(
4923:F
4903:X
4883:0
4870:1
4867:=
4864:i
4859:)
4854:i
4850:x
4846:(
4841:=
4831:1
4828:=
4825:i
4820:)
4814:i
4810:z
4806:i
4802:(
4773:1
4770:=
4767:i
4762:)
4758:i
4754:/
4748:i
4744:x
4739:(
4725:z
4704:.
4701:i
4681:V
4676:2
4672:i
4664:)
4659:i
4655:x
4651:(
4647:F
4621:x
4600:.
4596:)
4587:x
4583:(
4579:F
4559:V
4539:Y
4519:V
4495:.
4492:Y
4465:1
4462:=
4459:i
4454:)
4449:)
4444:i
4440:x
4436:(
4432:F
4428:(
4423:=
4419:)
4410:x
4406:(
4402:F
4382:0
4355:1
4352:=
4349:i
4344:)
4339:i
4335:x
4331:(
4326:=
4317:x
4206:Y
4186:X
4166:)
4163:Y
4160:,
4157:X
4154:(
4151:B
4140:.
4128:X
4108:A
4088:)
4085:Y
4082:,
4079:X
4076:(
4073:B
4067:A
4040:)
4037:Y
4034:,
4031:X
4028:(
4025:B
4005:Y
3982:)
3979:Y
3976:,
3973:X
3970:(
3967:B
3944:)
3941:Y
3938:,
3935:X
3932:(
3929:B
3909:Y
3889:X
3861:L
3841:,
3832:n
3806:n
3800:2
3797:=
3791:)
3786:n
3782:v
3778:(
3775:L
3752:,
3746:2
3743:=
3735:n
3731:v
3707:,
3701:,
3698:2
3695:,
3692:1
3689:=
3686:n
3664:x
3661:n
3658:i
3654:e
3650:=
3645:n
3641:v
3616:X
3610:X
3607::
3604:L
3582:.
3579:x
3576:d
3571:|
3567:)
3564:x
3561:(
3558:P
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3531:=
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3497:]
3491:,
3482:[
3458:X
3409:)
3402:,
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3393:x
3389:,
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3376:,
3371:0
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3360:0
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3352:=
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3269:+
3264:2
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3181:x
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3150:(
3124:2
3076:)
3071:n
3066:R
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3030:(
3025:2
3021:H
3017::
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2962:)
2959:y
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2922:y
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2916:)
2913:f
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2887:L
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2864:d
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2858:c
2855:[
2852:C
2828:]
2825:b
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2816:[
2796:]
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2784:[
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2724:c
2721:[
2715:]
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2706:a
2703:[
2700::
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2641:1
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2592:.
2578:.
2562:F
2534:F
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2350:F
2323:F
2296:Y
2276:X
2251:.
2248:X
2221:1
2218:=
2215:i
2210:)
2204:i
2200:x
2194:i
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2171:r
2137:1
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2113:(
2108:=
2099:r
2077:X
2045:1
2042:=
2039:i
2034:)
2029:i
2025:x
2021:(
2016:=
2007:x
1984:.
1981:Y
1961:F
1927:F
1917:;
1905:)
1902:X
1899:(
1896:F
1890:F
1861:F
1839:F
1816:F
1793:Y
1787:X
1784::
1781:F
1750:Y
1744:X
1741::
1738:F
1718:,
1715:Y
1695:X
1656:F
1636:,
1633:Y
1613:)
1610:U
1607:(
1604:F
1584:X
1564:U
1544:Y
1538:X
1535::
1532:F
1453:.
1450:Y
1430:)
1427:B
1424:(
1421:F
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1377:X
1371:B
1335:Y
1329:X
1326::
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1271:.
1265:x
1254:1
1249:=
1246:1
1231:x
1213:)
1203:x
1196:x
1187:(
1183:L
1166:x
1157:=
1148:)
1138:x
1131:x
1122:(
1118:L
1106:x
1092:=
1086:x
1083:L
1060:,
1057:X
1051:x
1031:.
1019:h
996:X
990:h
970:1
961:)
958:0
955:(
952:L
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943:h
940:(
937:L
931:=
925:)
922:h
919:(
916:L
893:0
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836:,
833:x
813:M
793:.
790:x
770:L
750:h
730:.
724:h
718:M
709:)
706:h
703:(
700:L
694:=
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685:x
682:(
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673:)
670:h
667:+
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661:(
658:L
635:h
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484:f
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448::
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342:.
337:X
329:x
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282:,
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253:0
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227:L
203:Y
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163:.
160:Y
140:X
116:Y
96:X
72:Y
66:X
63::
60:L
20:)
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