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2953:{\displaystyle \dim \ker \left(T-\lambda \operatorname {Id} _{X}\right)=\dim {\big (}X/\operatorname {Im} \left(T-\lambda \operatorname {Id} _{X}\right){\big )}=\dim \ker \left(T^{*}-\lambda \operatorname {Id} _{X^{*}}\right)=\dim {\big (}X^{*}/\operatorname {Im} \left(T^{*}-\lambda \operatorname {Id} _{X^{*}}\right){\big )}}
3547:
into a
Fredholm integral equation. Existence of the solution and spectral properties then follow from the theory of compact operators; in particular, an elliptic boundary value problem on a bounded domain has infinitely many isolated eigenvalues. One consequence is that a solid body can vibrate only
4495:
Somewhat confusingly, compact operators are sometimes referred to as "completely continuous" in older literature, even though they are not necessarily completely continuous by the definition of that phrase in modern terminology.
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292:, it is true that any compact operator is a limit of finite-rank operators, so that the class of compact operators can be defined alternatively as the closure of the set of finite-rank operators in the
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is closed. This is essential in developing the spectral properties of compact operators. One can notice the similarity between this property and the fact that, if
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of all bounded operators on the space. Indeed, the compact operators on an infinite-dimensional separable
Hilbert space form a maximal ideal, so the
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a linear operator. Then the following statements are equivalent, and some of them are used as the principal definition by different authors
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335:. The method of approximation by finite-rank operators is basic in the numerical solution of such equations. The abstract idea of
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of the operator, and the series on the right hand side converges in the operator norm. The singular values can
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at isolated frequencies, given by the eigenvalues, and arbitrarily high vibration frequencies always exist.
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William McLean, Strongly
Elliptic Systems and Boundary Integral Equations, Cambridge University Press, 2000
4223:. While all trace-class operators are compact operators, the converse is not necessarily true. For example
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is a compact operator; indeed, the class of compact operators is a natural generalization of the class of
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5185:. Texts in Mathematical Sciences. Vol. 12 (2nd ed.). New York: Springer-Verlag. p. 292.
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5290:. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY:
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A bounded linear operator between Banach spaces is compact if and only if its adjoint is compact (
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For
Hilbert spaces, another equivalent definition of compact operators is given as follows.
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is the unknown function to be solved for) behaves much like as in finite dimensions. The
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on an infinite-dimensional Banach space has spectrum that is either a finite subset of
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4472:, §VI.3). Compact operators on a Banach space are always completely continuous. If
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3796:{\displaystyle T=\sum _{n=1}^{\infty }\lambda _{n}\langle f_{n},\cdot \rangle g_{n}}
7284:
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6180:
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5328:. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer.
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230:
5482:
319:, where integral operators supply concrete examples of such operators. A typical
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128:
35:
3416:, which asserts that the existence of solution of linear equations of the form
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6586:
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6463:
6323:
6205:
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4861: â bounded operator with kernel and cokernel both having finite dimension
4156:{\displaystyle T=\sum _{n=1}^{N}\lambda _{n}\langle f_{n},\cdot \rangle g_{n}}
3505:
2037:
5373:
5343:
5249:
5234:. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press.
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3317:{\displaystyle E_{r}=\left\{\lambda \in \sigma (T):|\lambda |>r\right\}}
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5261:. Texts in Applied Mathematics. Vol. 13 (2nd ed.). New York:
5137:
5118:
2204:
4935:
Functional analysis, Sobolev spaces and partial differential equations
3551:
The compact operators from a Banach space to itself form a two-sided
3504:. Moreover, in either case the non-zero elements of the spectrum are
3965:
only at zero. If the sequence becomes stationary at zero, that is
315:
The origin of the theory of compact operators is in the theory of
5119:"A counterexample to the approximation problem in Banach spaces"
4810:{\displaystyle (Tf)(x)=\int _{\Omega }k(x,y)f(y)\,\mathrm {d} y}
2333:{\displaystyle \operatorname {Im} ({\operatorname {Id} _{X}}-T)}
6853:
6138:
5486:
1807:{\displaystyle B(Y,Z)\circ K(X,Y)\circ B(W,X)\subseteq K(W,Z),}
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3957:
is a sequence of positive numbers with limit zero, called the
3656:{\displaystyle ({\mathcal {H}},\langle \cdot ,\cdot \rangle )}
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4887:
4681:{\displaystyle (Tf)(x)=\int _{0}^{x}f(t)g(t)\,\mathrm {d} t.}
233:, but the definition can be extended to more general spaces.
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296:. Whether this was true in general for Banach spaces (the
4346:{\textstyle \sum _{n=1}^{\infty }|\lambda _{n}|=\infty }
3699:{\displaystyle T\colon {\mathcal {H}}\to {\mathcal {H}}}
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If a linear operator is compact, then it is continuous.
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is the limit of finite rank operators. Notably, this "
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are orthonormal sets (not necessarily complete), and
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are
Hilbert spaces, then every compact operator from
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5356:
Topological Vector Spaces, Distributions and
Kernels
4863:
Pages displaying wikidata descriptions as a fallback
4849:
Pages displaying wikidata descriptions as a fallback
4074:, a finite-dimensional range, and can be written as
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is Banach, these statements are also equivalent to:
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7015:
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6304:
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5833:
5777:
5719:
5676:
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5524:
5411:. Unsourced material may be challenged and removed.
4999:, (2005) London Mathematical Society Student Texts
4216:{\displaystyle \operatorname {Tr} (|T|)<\infty }
300:) was an unsolved question for many years; in 1973
7209:Spectral theory of ordinary differential equations
6094:Spectral theory of ordinary differential equations
5164:Introductory functional analysis with applications
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4456:
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4345:
4281:
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4167:An important subclass of compact operators is the
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5992:SchröderâBernstein theorems for operator algebras
5259:An introduction to partial differential equations
4543:converging to zero, the multiplication operator (
3950:{\displaystyle \lambda _{1},\lambda _{2},\ldots }
3571:. More generally, the compact operators form an
3388:{\displaystyle T-\lambda \operatorname {Id} _{X}}
2655:{\displaystyle T-\lambda \operatorname {Id} _{X}}
2619:{\displaystyle T-\lambda \operatorname {Id} _{X}}
3412:A crucial property of compact operators is the
203:, and so continuous. Some authors require that
5230:Narici, Lawrence; Beckenstein, Edward (2011).
3527:An important example of a compact operator is
1818:of sets is taken element-wise. In particular,
6865:
6150:
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4916:
4369:be Banach spaces. A bounded linear operator
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2800:
2750:
8:
4480:, then every completely continuous operator
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4121:
3898:
3866:
3846:
3814:
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3761:
3647:
3635:
940:{\displaystyle (Tx_{n})_{n\in \mathbb {N} }}
304:gave a counter-example, building on work by
2282:{\displaystyle {\operatorname {Id} _{X}}-T}
1589:{\displaystyle (T_{n})_{n\in \mathbf {N} }}
1489:{\displaystyle (T_{n})_{n\in \mathbf {N} }}
867:{\displaystyle (x_{n})_{n\in \mathbb {N} }}
6900:
6872:
6858:
6850:
6157:
6143:
6135:
5505:
5491:
5483:
2196:{\displaystyle T^{*}\colon X^{*}\to X^{*}}
5471:Learn how and when to remove this message
5136:
5087:. Vol. 96 (2nd ed.). New York:
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4707: : Ω à Ω â
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3484:(1918). It shows that a compact operator
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2691:then the following are finite and equal:
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2509:is any bounded linear operator then both
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266:in an infinite-dimensional setting. When
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7162:Group algebra of a locally compact group
5038:
4256:{\textstyle \lambda _{n}={\frac {1}{n}}}
3024:, which would necessarily be the origin.
4883:
3904:{\displaystyle \{g_{1},g_{2},\ldots \}}
3852:{\displaystyle \{f_{1},f_{2},\ldots \}}
3492:which includes 0, or the spectrum is a
1448:given a sequence of compact operators
1241:{\displaystyle \operatorname {Id} _{X}}
1195:denotes the space of compact operators
331:; the compactness property is shown by
6296:Uniform boundedness (BanachâSteinhaus)
5017:
4893:
4699:More generally, if Ω is any domain in
4505:Every finite rank operator is compact.
4469:
3352:{\displaystyle \lambda \in \sigma (T)}
3139:{\displaystyle \lambda \in \sigma (T)}
2036:is a Banach space and there exists an
1445:(in the norm topology). Equivalently,
5825:Spectral theory of normal C*-algebras
5623:Spectral theory of normal C*-algebras
5358:. Mineola, N.Y.: Dover Publications.
4997:A Short Course on Banach Space Theory
4983:
4905:
4070:, then the operator has finite rank,
1703:" is false for general Banach spaces
199:). Such an operator is necessarily a
7:
5820:Spectral theory of compact operators
5409:adding citations to reliable sources
5292:McGraw-Hill Science/Engineering/Math
4927:
4925:
4873:Spectral theory of compact operators
3512:with finite multiplicities (so that
3478:spectral theory of compact operators
5257:Renardy, M.; Rogers, R. C. (2004).
5183:Fundamentals of Functional Analysis
4692:is indeed compact follows from the
3408:Origins in integral equation theory
977:the image of any bounded subset of
644:the image of any bounded subset of
5972:CohenâHewitt factorization theorem
4938:. H.. Brézis. New York: Springer.
4800:
4763:
4668:
4340:
4312:
4276:
4210:
3746:
3579:Compact operator on Hilbert spaces
3324:is finite, and for every non-zero
2150:is a compact linear operator, and
2094:is necessarily finite-dimensional.
1130:is the space of bounded operators
947:contains a converging subsequence.
460:is a relatively compact subset of
27:Type of continuous linear operator
25:
5977:Extensions of symmetric operators
4840:Compact operator on Hilbert space
4025:{\displaystyle N\in \mathbb {N} }
3714:if it can be written in the form
3585:Compact operator on Hilbert space
339:is derived from this connection.
323:gives rise to a compact operator
7318:
7317:
7244:Topological quantum field theory
6834:
6833:
5795:Positive operator-valued measure
5385:
5207:. New York: Wiley-Interscience.
5061:. Springer-Verlag. Section 2.4.
3997:{\displaystyle \lambda _{N+k}=0}
3455:{\displaystyle (\lambda K+I)u=f}
1580:
1480:
6821:With the approximation property
6079:RayleighâFaberâKrahn inequality
5396:needs additional citations for
5081:A Course in Functional Analysis
5059:A course in functional analysis
4357:Completely continuous operators
3545:elliptic boundary value problem
3480:then follows, and it is due to
3075:{\displaystyle 0\in \sigma (T)}
816:{\displaystyle T(U)\subseteq V}
387:if there exists a neighborhood
6284:Open mapping (BanachâSchauder)
4795:
4789:
4783:
4771:
4752:
4746:
4743:
4734:
4663:
4657:
4651:
4645:
4621:
4615:
4612:
4603:
4576:), define the linear operator
4451:
4435:
4411:
4398:
4333:
4318:
4273:
4204:
4200:
4192:
4188:
3686:
3650:
3622:
3440:
3425:
3346:
3340:
3299:
3291:
3284:
3278:
3133:
3127:
3104:{\displaystyle \lambda \neq 0}
3069:
3063:
2981:
2975:
2684:{\displaystyle \lambda \neq 0}
2583:{\displaystyle \lambda \neq 0}
2502:{\displaystyle S\colon X\to X}
2493:
2327:
2306:
2241:
2235:
2180:
2143:{\displaystyle T\colon X\to X}
2134:
2056:
1929:is bounded and compact, then:
1913:
1867:
1861:
1834:
1828:
1798:
1786:
1777:
1765:
1756:
1744:
1735:
1723:
1683:
1569:
1555:
1506:
1469:
1455:
1432:
1420:
1397:
1385:
1359:
1347:
1338:
1332:
1309:
1297:
1288:
1282:
1205:
1182:
1170:
1140:
1117:
1105:
920:
903:
847:
833:
804:
798:
571:the image of the unit ball of
527:
447:
441:
363:
61:
1:
7040:Uniform boundedness principle
5987:Limiting absorption principle
5085:Graduate Texts in Mathematics
5003:, Cambridge University Press.
3047:is infinite-dimensional then
5613:Singular value decomposition
5320:; Wolff, Manfred P. (1999).
4847: â mathematical theorem
4282:{\displaystyle n\to \infty }
4063:{\displaystyle k=1,2,\dots }
3543:, can be used to convert an
2626:is closed and the kernel of
2448:is finite-dimensional, then
2293:of index 0. In particular,
1932:the closure of the range of
781:{\displaystyle V\subseteq Y}
6505:Radially convex/Star-shaped
6490:Pre-compact/Totally bounded
6044:Hearing the shape of a drum
5727:Decomposition of a spectrum
3612:on an infinite-dimensional
1548:are Banach) and given that
1365:{\displaystyle K(X)=K(X,X)}
1315:{\displaystyle B(X)=B(X,X)}
7370:
7183:Invariant subspace problem
6191:Continuous linear operator
5632:Special Elements/Operators
5181:Kutateladze, S.S. (1996).
4868:Strictly singular operator
4854:Fredholm integral equation
3582:
2987:{\displaystyle \sigma (T)}
321:Fredholm integral equation
7344:Compactness (mathematics)
7313:
6903:
6829:
6536:Algebraic interior (core)
6278:Vector-valued HahnâBanach
6166:Topological vector spaces
6104:Superstrong approximation
5967:Banach algebra cohomology
5800:Projection-valued measure
5785:Borel functional calculus
5557:Projection-valued measure
5322:Topological Vector Spaces
5232:Topological Vector Spaces
5166:. John Wiley & Sons.
4917:Schaefer & Wolff 1999
4528:{\displaystyle \ell ^{p}}
3520:has a finite-dimensional
3472:is a given function, and
3166:is an eigenvalue of both
2247:{\displaystyle T\in K(X)}
2040:bounded compact operator
826:for any bounded sequence
381:topological vector spaces
107:, with the property that
7152:Spectrum of a C*-algebra
6366:Topological homomorphism
6226:Topological vector space
5696:Spectrum of a C*-algebra
5567:Spectrum of a C*-algebra
5162:Kreyszig, Erwin (1978).
5041:, Theorems 2.5-3, 2.5-5.
4703:and the integral kernel
4457:{\displaystyle (Tx_{n})}
3535:, which, along with the
3524:for all complex λ â 0).
3500:which has 0 as its only
3159:{\displaystyle \lambda }
2554:{\displaystyle T\circ S}
2528:{\displaystyle S\circ T}
2065:{\displaystyle T:X\to X}
1922:{\displaystyle T:X\to Y}
1883:Any compact operator is
1410:is a closed subspace of
536:{\displaystyle T:X\to Y}
372:{\displaystyle T:X\to Y}
70:{\displaystyle T:X\to Y}
7249:Noncommutative geometry
6124:WienerâKhinchin theorem
6059:Kuznetsov trace formula
6034:Almost Mathieu operator
5852:Banach function algebra
5841:Amenable Banach algebra
5598:GelfandâNaimark theorem
5552:Noncommutative topology
4417:{\displaystyle (x_{n})}
3468:is a compact operator,
1088:{\displaystyle X,Y,Z,W}
343:Equivalent formulations
7305:TomitaâTakesaki theory
7280:Approximation property
7224:Calculus of variations
6424:Absolutely convex/disk
6099:SturmâLiouville theory
5997:ShermanâTakeda theorem
5877:TomitaâTakesaki theory
5652:Hermitian/Self-adjoint
5603:Gelfand representation
4817:is a compact operator.
4811:
4713:HilbertâSchmidt kernel
4682:
4529:
4478:reflexive Banach space
4464:is norm-convergent in
4458:
4418:
4347:
4316:
4283:
4257:
4217:
4157:
4110:
4064:
4026:
3998:
3951:
3905:
3853:
3797:
3750:
3700:
3657:
3606:
3456:
3389:
3353:
3318:
3238:
3237:{\displaystyle r>0}
3209:
3181:
3160:
3140:
3105:
3076:
3041:
3020:, and has at most one
3009:
2988:
2954:
2685:
2662:is finite-dimensional.
2656:
2620:
2584:
2561:are compact operators.
2555:
2529:
2503:
2468:
2441:
2420:
2399:
2377:
2355:
2334:
2283:
2248:
2197:
2144:
2118:is a Banach space and
2112:
2087:
2066:
2030:
2009:is finite-dimensional.
2002:
1976:
1947:
1923:
1874:
1841:
1808:
1701:approximation property
1693:
1692:{\displaystyle X\to Y}
1667:
1635:
1610:
1590:
1542:
1516:
1515:{\displaystyle X\to Y}
1490:
1439:
1438:{\displaystyle B(X,Y)}
1404:
1403:{\displaystyle K(X,Y)}
1366:
1316:
1266:
1242:
1215:
1214:{\displaystyle X\to Y}
1189:
1188:{\displaystyle K(X,Y)}
1150:
1149:{\displaystyle X\to Y}
1124:
1123:{\displaystyle B(X,Y)}
1089:
1039:
1014:
992:
966:
941:
889:
868:
817:
782:
755:
734:
707:
681:
659:
634:
608:
586:
568:is a compact operator;
561:
537:
505:
475:
454:
424:
402:
373:
298:approximation property
281:
251:
223:
193:
175:(subsets with compact
169:
145:
121:
97:
71:
7300:BanachâMazur distance
7263:Generalized functions
6459:Complemented subspace
6273:hyperplane separation
5593:GelfandâMazur theorem
4812:
4683:
4530:
4459:
4419:
4383:completely continuous
4348:
4296:
4284:
4258:
4218:
4158:
4090:
4065:
4027:
3999:
3952:
3906:
3854:
3798:
3730:
3701:
3658:
3607:
3457:
3390:
3354:
3319:
3239:
3210:
3208:{\displaystyle T^{*}}
3182:
3161:
3141:
3106:
3077:
3042:
3010:
2989:
2955:
2686:
2657:
2621:
2585:
2556:
2530:
2504:
2469:
2442:
2421:
2400:
2378:
2356:
2335:
2284:
2249:
2198:
2145:
2113:
2088:
2067:
2031:
2003:
1977:
1948:
1924:
1887:, but not vice versa.
1875:
1842:
1809:
1694:
1668:
1636:
1611:
1591:
1543:
1517:
1491:
1440:
1405:
1367:
1317:
1267:
1243:
1216:
1190:
1151:
1125:
1090:
1040:
1015:
993:
967:
942:
890:
869:
818:
783:
762:and a compact subset
756:
735:
708:
682:
660:
635:
609:
587:
562:
538:
511:be normed spaces and
506:
476:
455:
425:
403:
374:
282:
264:finite-rank operators
252:
236:Any bounded operator
224:
194:
170:
146:
122:
98:
72:
18:Approximation problem
7045:Kakutani fixed-point
7030:Riesz representation
6709:Locally convex space
6259:Closed graph theorem
6211:Locally convex space
6069:Proto-value function
6048:Dirichlet eigenvalue
5962:Abstract index group
5847:Approximate identity
5810:Rigged Hilbert space
5686:KreinâRutman theorem
5532:Involution/*-algebra
5405:improve this article
4845:Fredholm alternative
4731:
4715:, then the operator
4600:
4512:
4432:
4395:
4293:
4267:
4227:
4179:
4081:
4036:
4008:
3969:
3915:
3863:
3811:
3721:
3670:
3619:
3596:
3422:
3414:Fredholm alternative
3363:
3328:
3248:
3222:
3192:
3171:
3150:
3115:
3089:
3051:
3031:
2999:
2969:
2695:
2669:
2630:
2594:
2568:
2539:
2513:
2481:
2452:
2431:
2410:
2389:
2367:
2345:
2297:
2258:
2223:
2154:
2122:
2102:
2077:
2044:
2020:
1992:
1987:, then the range of
1966:
1937:
1901:
1873:{\displaystyle B(X)}
1855:
1840:{\displaystyle K(X)}
1822:
1717:
1677:
1651:
1625:
1616:with respect to the
1600:
1552:
1526:
1500:
1452:
1414:
1379:
1326:
1276:
1256:
1225:
1199:
1164:
1134:
1099:
1061:
1029:
1004:
982:
956:
900:
879:
830:
792:
766:
745:
724:
697:
671:
649:
624:
598:
576:
551:
515:
489:
465:
453:{\displaystyle T(U)}
435:
414:
392:
351:
271:
241:
207:
183:
159:
135:
111:
105:normed vector spaces
81:
49:
7229:Functional calculus
7188:Mahler's conjecture
7167:Von Neumann algebra
6881:Functional analysis
6689:Interpolation space
6221:Operator topologies
5872:Von Neumann algebra
5608:Polar decomposition
5318:Schaefer, Helmut H.
5287:Functional Analysis
5205:Functional Analysis
5020:, pp. 173â177.
4986:, pp. 103â115.
4932:Brézis, H. (2011).
4641:
3541:LaxâMilgram theorem
2467:{\displaystyle M+N}
1666:{\displaystyle X,Y}
1541:{\displaystyle X,Y}
1095:are Banach spaces,
504:{\displaystyle X,Y}
222:{\displaystyle X,Y}
96:{\displaystyle X,Y}
32:functional analysis
7254:Riemann hypothesis
6953:Topological vector
6719:(Pseudo)Metrizable
6551:Minkowski addition
6403:Sublinear function
6002:Unbounded operator
5931:Essential spectrum
5910:SchurâHorn theorem
5900:BauerâFike theorem
5895:AlonâBoppana bound
5888:Finite-Dimensional
5862:Nuclear C*-algebra
5706:Spectral asymmetry
5420:"Compact operator"
5138:10.1007/BF02392270
4807:
4688:That the operator
4678:
4627:
4525:
4454:
4414:
4343:
4279:
4263:tends to zero for
4253:
4213:
4175:, i.e., such that
4153:
4060:
4022:
3994:
3947:
3901:
3849:
3793:
3696:
3653:
3602:
3537:GĂ„rding inequality
3494:countably infinite
3452:
3385:
3349:
3314:
3234:
3205:
3177:
3156:
3136:
3101:
3072:
3037:
3005:
2984:
2950:
2681:
2652:
2616:
2590:then the range of
2580:
2551:
2525:
2499:
2464:
2437:
2416:
2395:
2373:
2351:
2330:
2279:
2244:
2193:
2140:
2108:
2083:
2062:
2026:
1998:
1972:
1943:
1919:
1892:Schauder's theorem
1870:
1847:forms a two-sided
1837:
1804:
1689:
1663:
1631:
1606:
1586:
1538:
1512:
1486:
1435:
1400:
1362:
1312:
1262:
1238:
1211:
1185:
1146:
1120:
1085:
1057:In the following,
1035:
1010:
988:
962:
937:
885:
864:
813:
778:
751:
730:
703:
690:relatively compact
677:
655:
630:
617:relatively compact
604:
582:
557:
533:
501:
471:
450:
420:
398:
369:
317:integral equations
277:
247:
219:
189:
165:
153:relatively compact
141:
117:
93:
67:
7331:
7330:
7234:Integral operator
7011:
7010:
6847:
6846:
6566:Relative interior
6312:Bilinear operator
6196:Linear functional
6132:
6131:
6109:Transfer operator
6084:Spectral geometry
5769:Spectral abscissa
5749:Approximate point
5691:Normal eigenvalue
5481:
5480:
5473:
5455:
5365:978-0-486-45352-1
5335:978-1-4612-7155-0
5301:978-0-07-054236-5
5272:978-0-387-00444-0
5214:978-0-471-55604-6
5192:978-0-7923-3898-7
5173:978-0-471-50731-4
5098:978-0-387-97245-9
5068:978-3-540-96042-3
4945:978-0-387-70914-7
4859:Fredholm operator
4835:Compact embedding
4387:weakly convergent
4251:
4173:nuclear operators
3605:{\displaystyle T}
3529:compact embedding
3180:{\displaystyle T}
3040:{\displaystyle X}
3008:{\displaystyle T}
2440:{\displaystyle N}
2419:{\displaystyle M}
2398:{\displaystyle X}
2384:are subspaces of
2376:{\displaystyle N}
2354:{\displaystyle M}
2291:Fredholm operator
2111:{\displaystyle X}
2098:Now suppose that
2086:{\displaystyle X}
2029:{\displaystyle X}
2001:{\displaystyle T}
1975:{\displaystyle T}
1946:{\displaystyle T}
1885:strictly singular
1634:{\displaystyle T}
1609:{\displaystyle T}
1265:{\displaystyle X}
1250:identity operator
1038:{\displaystyle Y}
1013:{\displaystyle T}
991:{\displaystyle X}
965:{\displaystyle Y}
888:{\displaystyle X}
754:{\displaystyle X}
740:of the origin in
733:{\displaystyle U}
706:{\displaystyle Y}
680:{\displaystyle T}
658:{\displaystyle X}
633:{\displaystyle Y}
607:{\displaystyle T}
585:{\displaystyle X}
560:{\displaystyle T}
474:{\displaystyle Y}
423:{\displaystyle X}
409:of the origin in
401:{\displaystyle U}
337:Fredholm operator
280:{\displaystyle Y}
250:{\displaystyle T}
192:{\displaystyle Y}
168:{\displaystyle Y}
144:{\displaystyle X}
120:{\displaystyle T}
16:(Redirected from
7361:
7349:Linear operators
7321:
7320:
7239:Jones polynomial
7157:Operator algebra
6901:
6874:
6867:
6860:
6851:
6837:
6836:
6811:Uniformly smooth
6480:
6472:
6439:Balanced/Circled
6429:Absorbing/Radial
6159:
6152:
6145:
6136:
6114:Transform theory
5834:Special algebras
5815:Spectral theorem
5778:Spectral Theorem
5618:Spectral theorem
5507:
5500:
5493:
5484:
5476:
5469:
5465:
5462:
5456:
5454:
5413:
5389:
5381:
5377:
5352:TrÚves, François
5347:
5313:
5276:
5253:
5226:
5196:
5177:
5158:
5140:
5124:Acta Mathematica
5110:
5072:
5042:
5036:
5030:
5027:
5021:
5015:
5004:
4995:N.L. Carothers,
4993:
4987:
4981:
4958:
4957:
4929:
4920:
4914:
4908:
4903:
4897:
4891:
4864:
4850:
4816:
4814:
4813:
4808:
4803:
4767:
4766:
4687:
4685:
4684:
4679:
4671:
4640:
4635:
4534:
4532:
4531:
4526:
4524:
4523:
4463:
4461:
4460:
4455:
4450:
4449:
4423:
4421:
4420:
4415:
4410:
4409:
4352:
4350:
4349:
4344:
4336:
4331:
4330:
4321:
4315:
4310:
4288:
4286:
4285:
4280:
4262:
4260:
4259:
4254:
4252:
4244:
4239:
4238:
4222:
4220:
4219:
4214:
4203:
4195:
4162:
4160:
4159:
4154:
4152:
4151:
4133:
4132:
4120:
4119:
4109:
4104:
4069:
4067:
4066:
4061:
4031:
4029:
4028:
4023:
4021:
4003:
4001:
4000:
3995:
3987:
3986:
3956:
3954:
3953:
3948:
3940:
3939:
3927:
3926:
3910:
3908:
3907:
3902:
3891:
3890:
3878:
3877:
3858:
3856:
3855:
3850:
3839:
3838:
3826:
3825:
3802:
3800:
3799:
3794:
3792:
3791:
3773:
3772:
3760:
3759:
3749:
3744:
3705:
3703:
3702:
3697:
3695:
3694:
3685:
3684:
3662:
3660:
3659:
3654:
3631:
3630:
3611:
3609:
3608:
3603:
3561:quotient algebra
3461:
3459:
3458:
3453:
3394:
3392:
3391:
3386:
3384:
3383:
3358:
3356:
3355:
3350:
3323:
3321:
3320:
3315:
3313:
3309:
3302:
3294:
3260:
3259:
3243:
3241:
3240:
3235:
3214:
3212:
3211:
3206:
3204:
3203:
3186:
3184:
3183:
3178:
3165:
3163:
3162:
3157:
3145:
3143:
3142:
3137:
3110:
3108:
3107:
3102:
3081:
3079:
3078:
3073:
3046:
3044:
3043:
3038:
3014:
3012:
3011:
3006:
2993:
2991:
2990:
2985:
2959:
2957:
2956:
2951:
2949:
2948:
2942:
2938:
2937:
2936:
2935:
2934:
2914:
2913:
2893:
2888:
2887:
2878:
2877:
2862:
2858:
2857:
2856:
2855:
2854:
2834:
2833:
2804:
2803:
2797:
2793:
2792:
2791:
2762:
2754:
2753:
2738:
2734:
2733:
2732:
2690:
2688:
2687:
2682:
2661:
2659:
2658:
2653:
2651:
2650:
2625:
2623:
2622:
2617:
2615:
2614:
2589:
2587:
2586:
2581:
2560:
2558:
2557:
2552:
2534:
2532:
2531:
2526:
2508:
2506:
2505:
2500:
2473:
2471:
2470:
2465:
2446:
2444:
2443:
2438:
2425:
2423:
2422:
2417:
2404:
2402:
2401:
2396:
2382:
2380:
2379:
2374:
2360:
2358:
2357:
2352:
2339:
2337:
2336:
2331:
2320:
2319:
2318:
2288:
2286:
2285:
2280:
2272:
2271:
2270:
2253:
2251:
2250:
2245:
2202:
2200:
2199:
2194:
2192:
2191:
2179:
2178:
2166:
2165:
2149:
2147:
2146:
2141:
2117:
2115:
2114:
2109:
2092:
2090:
2089:
2084:
2071:
2069:
2068:
2063:
2035:
2033:
2032:
2027:
2007:
2005:
2004:
1999:
1981:
1979:
1978:
1973:
1961:if the range of
1952:
1950:
1949:
1944:
1928:
1926:
1925:
1920:
1879:
1877:
1876:
1871:
1846:
1844:
1843:
1838:
1813:
1811:
1810:
1805:
1698:
1696:
1695:
1690:
1672:
1670:
1669:
1664:
1642:is then compact.
1640:
1638:
1637:
1632:
1615:
1613:
1612:
1607:
1595:
1593:
1592:
1587:
1585:
1584:
1583:
1567:
1566:
1547:
1545:
1544:
1539:
1521:
1519:
1518:
1513:
1495:
1493:
1492:
1487:
1485:
1484:
1483:
1467:
1466:
1444:
1442:
1441:
1436:
1409:
1407:
1406:
1401:
1371:
1369:
1368:
1363:
1321:
1319:
1318:
1313:
1271:
1269:
1268:
1263:
1247:
1245:
1244:
1239:
1237:
1236:
1220:
1218:
1217:
1212:
1194:
1192:
1191:
1186:
1155:
1153:
1152:
1147:
1129:
1127:
1126:
1121:
1094:
1092:
1091:
1086:
1044:
1042:
1041:
1036:
1019:
1017:
1016:
1011:
997:
995:
994:
989:
971:
969:
968:
963:
946:
944:
943:
938:
936:
935:
934:
918:
917:
894:
892:
891:
886:
873:
871:
870:
865:
863:
862:
861:
845:
844:
822:
820:
819:
814:
787:
785:
784:
779:
760:
758:
757:
752:
739:
737:
736:
731:
712:
710:
709:
704:
686:
684:
683:
678:
664:
662:
661:
656:
639:
637:
636:
631:
613:
611:
610:
605:
591:
589:
588:
583:
566:
564:
563:
558:
542:
540:
539:
534:
510:
508:
507:
502:
480:
478:
477:
472:
459:
457:
456:
451:
429:
427:
426:
421:
407:
405:
404:
399:
378:
376:
375:
370:
286:
284:
283:
278:
258:that has finite
256:
254:
253:
248:
228:
226:
225:
220:
201:bounded operator
198:
196:
195:
190:
174:
172:
171:
166:
150:
148:
147:
142:
126:
124:
123:
118:
102:
100:
99:
94:
76:
74:
73:
68:
40:compact operator
21:
7369:
7368:
7364:
7363:
7362:
7360:
7359:
7358:
7354:Operator theory
7334:
7333:
7332:
7327:
7309:
7273:Advanced topics
7268:
7192:
7171:
7130:
7096:HilbertâSchmidt
7069:
7060:GelfandâNaimark
7007:
6957:
6892:
6878:
6848:
6843:
6825:
6587:B-complete/Ptak
6570:
6514:
6478:
6470:
6449:Bounding points
6412:
6354:Densely defined
6300:
6289:Bounded inverse
6235:
6169:
6163:
6133:
6128:
6089:Spectral method
6074:Ramanujan graph
6022:
6006:
5982:Fredholm theory
5950:
5945:Shilov boundary
5941:Structure space
5919:Generalizations
5914:
5905:Numerical range
5883:
5867:Uniform algebra
5829:
5805:Riesz projector
5790:Min-max theorem
5773:
5759:Direct integral
5715:
5701:Spectral radius
5672:
5627:
5581:
5572:Spectral radius
5520:
5514:Spectral theory
5511:
5477:
5466:
5460:
5457:
5414:
5412:
5402:
5390:
5366:
5350:
5336:
5316:
5302:
5280:
5273:
5265:. p. 356.
5263:Springer-Verlag
5256:
5242:
5229:
5215:
5199:
5193:
5180:
5174:
5161:
5113:
5099:
5089:Springer-Verlag
5077:Conway, John B.
5075:
5069:
5055:Conway, John B.
5053:
5050:
5045:
5037:
5033:
5028:
5024:
5016:
5007:
4994:
4990:
4982:
4961:
4946:
4931:
4930:
4923:
4915:
4911:
4904:
4900:
4892:
4885:
4881:
4862:
4848:
4831:
4758:
4729:
4728:
4598:
4597:
4564:For some fixed
4559:
4555:
4551:
4540:
4535:and a sequence
4515:
4510:
4509:
4502:
4441:
4430:
4429:
4428:, the sequence
4401:
4393:
4392:
4359:
4322:
4291:
4290:
4265:
4264:
4230:
4225:
4224:
4177:
4176:
4143:
4124:
4111:
4079:
4078:
4034:
4033:
4006:
4005:
3972:
3967:
3966:
3959:singular values
3931:
3918:
3913:
3912:
3882:
3869:
3861:
3860:
3830:
3817:
3809:
3808:
3783:
3764:
3751:
3719:
3718:
3668:
3667:
3617:
3616:
3594:
3593:
3587:
3581:
3563:, known as the
3420:
3419:
3410:
3375:
3361:
3360:
3326:
3325:
3268:
3264:
3251:
3246:
3245:
3220:
3219:
3195:
3190:
3189:
3169:
3168:
3148:
3147:
3113:
3112:
3087:
3086:
3049:
3048:
3029:
3028:
2997:
2996:
2967:
2966:
2926:
2921:
2905:
2904:
2900:
2879:
2846:
2841:
2825:
2824:
2820:
2783:
2773:
2769:
2724:
2714:
2710:
2693:
2692:
2667:
2666:
2642:
2628:
2627:
2606:
2592:
2591:
2566:
2565:
2537:
2536:
2511:
2510:
2479:
2478:
2474:is also closed.
2450:
2449:
2429:
2428:
2408:
2407:
2387:
2386:
2365:
2364:
2343:
2342:
2310:
2295:
2294:
2262:
2256:
2255:
2221:
2220:
2183:
2170:
2157:
2152:
2151:
2120:
2119:
2100:
2099:
2075:
2074:
2042:
2041:
2018:
2017:
1990:
1989:
1964:
1963:
1935:
1934:
1899:
1898:
1853:
1852:
1820:
1819:
1715:
1714:
1675:
1674:
1649:
1648:
1647:Conversely, if
1623:
1622:
1598:
1597:
1568:
1558:
1550:
1549:
1524:
1523:
1498:
1497:
1468:
1458:
1450:
1449:
1412:
1411:
1377:
1376:
1324:
1323:
1274:
1273:
1254:
1253:
1228:
1223:
1222:
1197:
1196:
1162:
1161:
1132:
1131:
1097:
1096:
1059:
1058:
1055:
1027:
1026:
1023:totally bounded
1002:
1001:
980:
979:
954:
953:
951:If in addition
919:
909:
898:
897:
896:, the sequence
877:
876:
846:
836:
828:
827:
790:
789:
764:
763:
743:
742:
722:
721:
717:there exists a
695:
694:
669:
668:
647:
646:
622:
621:
596:
595:
574:
573:
549:
548:
513:
512:
487:
486:
463:
462:
433:
432:
412:
411:
390:
389:
349:
348:
345:
329:function spaces
269:
268:
239:
238:
205:
204:
181:
180:
157:
156:
133:
132:
129:bounded subsets
109:
108:
79:
78:
47:
46:
44:linear operator
28:
23:
22:
15:
12:
11:
5:
7367:
7365:
7357:
7356:
7351:
7346:
7336:
7335:
7329:
7328:
7326:
7325:
7314:
7311:
7310:
7308:
7307:
7302:
7297:
7292:
7290:Choquet theory
7287:
7282:
7276:
7274:
7270:
7269:
7267:
7266:
7256:
7251:
7246:
7241:
7236:
7231:
7226:
7221:
7216:
7211:
7206:
7200:
7198:
7194:
7193:
7191:
7190:
7185:
7179:
7177:
7173:
7172:
7170:
7169:
7164:
7159:
7154:
7149:
7144:
7142:Banach algebra
7138:
7136:
7132:
7131:
7129:
7128:
7123:
7118:
7113:
7108:
7103:
7098:
7093:
7088:
7083:
7077:
7075:
7071:
7070:
7068:
7067:
7065:BanachâAlaoglu
7062:
7057:
7052:
7047:
7042:
7037:
7032:
7027:
7021:
7019:
7013:
7012:
7009:
7008:
7006:
7005:
7000:
6995:
6993:Locally convex
6990:
6976:
6971:
6965:
6963:
6959:
6958:
6956:
6955:
6950:
6945:
6940:
6935:
6930:
6925:
6920:
6915:
6910:
6904:
6898:
6894:
6893:
6879:
6877:
6876:
6869:
6862:
6854:
6845:
6844:
6842:
6841:
6830:
6827:
6826:
6824:
6823:
6818:
6813:
6808:
6806:Ultrabarrelled
6798:
6792:
6787:
6781:
6776:
6771:
6766:
6761:
6756:
6747:
6741:
6736:
6734:Quasi-complete
6731:
6729:Quasibarrelled
6726:
6721:
6716:
6711:
6706:
6701:
6696:
6691:
6686:
6681:
6676:
6671:
6670:
6669:
6659:
6654:
6649:
6644:
6639:
6634:
6629:
6624:
6619:
6609:
6604:
6594:
6589:
6584:
6578:
6576:
6572:
6571:
6569:
6568:
6558:
6553:
6548:
6543:
6538:
6528:
6522:
6520:
6519:Set operations
6516:
6515:
6513:
6512:
6507:
6502:
6497:
6492:
6487:
6482:
6474:
6466:
6461:
6456:
6451:
6446:
6441:
6436:
6431:
6426:
6420:
6418:
6414:
6413:
6411:
6410:
6405:
6400:
6395:
6390:
6389:
6388:
6383:
6378:
6368:
6363:
6362:
6361:
6356:
6351:
6346:
6341:
6336:
6331:
6321:
6320:
6319:
6308:
6306:
6302:
6301:
6299:
6298:
6293:
6292:
6291:
6281:
6275:
6266:
6261:
6256:
6254:BanachâAlaoglu
6251:
6249:AndersonâKadec
6245:
6243:
6237:
6236:
6234:
6233:
6228:
6223:
6218:
6213:
6208:
6203:
6198:
6193:
6188:
6183:
6177:
6175:
6174:Basic concepts
6171:
6170:
6164:
6162:
6161:
6154:
6147:
6139:
6130:
6129:
6127:
6126:
6121:
6116:
6111:
6106:
6101:
6096:
6091:
6086:
6081:
6076:
6071:
6066:
6061:
6056:
6051:
6041:
6039:Corona theorem
6036:
6030:
6028:
6024:
6023:
6021:
6020:
6018:Wiener algebra
6014:
6012:
6008:
6007:
6005:
6004:
5999:
5994:
5989:
5984:
5979:
5974:
5969:
5964:
5958:
5956:
5952:
5951:
5949:
5948:
5938:
5936:Pseudospectrum
5933:
5928:
5926:Dirac spectrum
5922:
5920:
5916:
5915:
5913:
5912:
5907:
5902:
5897:
5891:
5889:
5885:
5884:
5882:
5881:
5880:
5879:
5869:
5864:
5859:
5854:
5849:
5843:
5837:
5835:
5831:
5830:
5828:
5827:
5822:
5817:
5812:
5807:
5802:
5797:
5792:
5787:
5781:
5779:
5775:
5774:
5772:
5771:
5766:
5761:
5756:
5751:
5746:
5745:
5744:
5739:
5734:
5723:
5721:
5717:
5716:
5714:
5713:
5708:
5703:
5698:
5693:
5688:
5682:
5680:
5674:
5673:
5671:
5670:
5665:
5657:
5649:
5641:
5635:
5633:
5629:
5628:
5626:
5625:
5620:
5615:
5610:
5605:
5600:
5595:
5589:
5587:
5583:
5582:
5580:
5579:
5577:Operator space
5574:
5569:
5564:
5559:
5554:
5549:
5544:
5539:
5537:Banach algebra
5534:
5528:
5526:
5525:Basic concepts
5522:
5521:
5512:
5510:
5509:
5502:
5495:
5487:
5479:
5478:
5393:
5391:
5384:
5379:
5378:
5364:
5348:
5334:
5314:
5300:
5278:
5271:
5254:
5241:978-1584888666
5240:
5227:
5213:
5197:
5191:
5178:
5172:
5159:
5131:(1): 309â317.
5111:
5097:
5073:
5067:
5049:
5046:
5044:
5043:
5031:
5022:
5005:
4988:
4959:
4944:
4921:
4909:
4898:
4882:
4880:
4877:
4876:
4875:
4870:
4865:
4856:
4851:
4842:
4837:
4830:
4827:
4826:
4825:
4818:
4806:
4802:
4797:
4794:
4791:
4788:
4785:
4782:
4779:
4776:
4773:
4770:
4765:
4761:
4757:
4754:
4751:
4748:
4745:
4742:
4739:
4736:
4697:
4694:Ascoli theorem
4677:
4674:
4670:
4665:
4662:
4659:
4656:
4653:
4650:
4647:
4644:
4639:
4634:
4630:
4626:
4623:
4620:
4617:
4614:
4611:
4608:
4605:
4562:
4557:
4553:
4549:
4538:
4522:
4518:
4506:
4501:
4498:
4453:
4448:
4444:
4440:
4437:
4413:
4408:
4404:
4400:
4385:if, for every
4358:
4355:
4342:
4339:
4335:
4329:
4325:
4320:
4314:
4309:
4306:
4303:
4299:
4278:
4275:
4272:
4250:
4247:
4242:
4237:
4233:
4212:
4209:
4206:
4202:
4198:
4194:
4190:
4187:
4184:
4165:
4164:
4150:
4146:
4142:
4139:
4136:
4131:
4127:
4123:
4118:
4114:
4108:
4103:
4100:
4097:
4093:
4089:
4086:
4059:
4056:
4053:
4050:
4047:
4044:
4041:
4020:
4016:
4013:
3993:
3990:
3985:
3982:
3979:
3975:
3946:
3943:
3938:
3934:
3930:
3925:
3921:
3900:
3897:
3894:
3889:
3885:
3881:
3876:
3872:
3868:
3848:
3845:
3842:
3837:
3833:
3829:
3824:
3820:
3816:
3805:
3804:
3790:
3786:
3782:
3779:
3776:
3771:
3767:
3763:
3758:
3754:
3748:
3743:
3740:
3737:
3733:
3729:
3726:
3710:is said to be
3708:
3707:
3693:
3688:
3683:
3678:
3675:
3652:
3649:
3646:
3643:
3640:
3637:
3634:
3629:
3624:
3601:
3583:Main article:
3580:
3577:
3573:operator ideal
3565:Calkin algebra
3533:Sobolev spaces
3451:
3448:
3445:
3442:
3439:
3436:
3433:
3430:
3427:
3409:
3406:
3405:
3404:
3382:
3378:
3374:
3371:
3368:
3348:
3345:
3342:
3339:
3336:
3333:
3312:
3308:
3305:
3301:
3297:
3293:
3289:
3286:
3283:
3280:
3277:
3274:
3271:
3267:
3263:
3258:
3254:
3233:
3230:
3227:
3216:
3202:
3198:
3176:
3155:
3135:
3132:
3129:
3126:
3123:
3120:
3100:
3097:
3094:
3083:
3071:
3068:
3065:
3062:
3059:
3056:
3036:
3025:
3004:
2983:
2980:
2977:
2974:
2960:
2947:
2941:
2933:
2929:
2924:
2920:
2917:
2912:
2908:
2903:
2899:
2896:
2892:
2886:
2882:
2876:
2871:
2868:
2865:
2861:
2853:
2849:
2844:
2840:
2837:
2832:
2828:
2823:
2819:
2816:
2813:
2810:
2807:
2802:
2796:
2790:
2786:
2782:
2779:
2776:
2772:
2768:
2765:
2761:
2757:
2752:
2747:
2744:
2741:
2737:
2731:
2727:
2723:
2720:
2717:
2713:
2709:
2706:
2703:
2700:
2680:
2677:
2674:
2663:
2649:
2645:
2641:
2638:
2635:
2613:
2609:
2605:
2602:
2599:
2579:
2576:
2573:
2562:
2550:
2547:
2544:
2524:
2521:
2518:
2498:
2495:
2492:
2489:
2486:
2475:
2463:
2460:
2457:
2436:
2426:is closed and
2415:
2394:
2372:
2350:
2329:
2326:
2323:
2317:
2313:
2308:
2305:
2302:
2278:
2275:
2269:
2265:
2243:
2240:
2237:
2234:
2231:
2228:
2190:
2186:
2182:
2177:
2173:
2169:
2164:
2160:
2139:
2136:
2133:
2130:
2127:
2107:
2096:
2095:
2082:
2061:
2058:
2055:
2052:
2049:
2025:
2014:
2013:
2012:
2011:
2010:
1997:
1971:
1959:
1942:
1918:
1915:
1912:
1909:
1906:
1888:
1881:
1869:
1866:
1863:
1860:
1836:
1833:
1830:
1827:
1803:
1800:
1797:
1794:
1791:
1788:
1785:
1782:
1779:
1776:
1773:
1770:
1767:
1764:
1761:
1758:
1755:
1752:
1749:
1746:
1743:
1740:
1737:
1734:
1731:
1728:
1725:
1722:
1712:
1688:
1685:
1682:
1662:
1659:
1656:
1645:
1644:
1643:
1630:
1605:
1582:
1578:
1575:
1571:
1565:
1561:
1557:
1537:
1534:
1531:
1511:
1508:
1505:
1482:
1478:
1475:
1471:
1465:
1461:
1457:
1434:
1431:
1428:
1425:
1422:
1419:
1399:
1396:
1393:
1390:
1387:
1384:
1361:
1358:
1355:
1352:
1349:
1346:
1343:
1340:
1337:
1334:
1331:
1311:
1308:
1305:
1302:
1299:
1296:
1293:
1290:
1287:
1284:
1281:
1261:
1235:
1231:
1210:
1207:
1204:
1184:
1181:
1178:
1175:
1172:
1169:
1145:
1142:
1139:
1119:
1116:
1113:
1110:
1107:
1104:
1084:
1081:
1078:
1075:
1072:
1069:
1066:
1054:
1051:
1047:
1046:
1034:
1009:
987:
961:
949:
948:
933:
929:
926:
922:
916:
912:
908:
905:
884:
860:
856:
853:
849:
843:
839:
835:
824:
812:
809:
806:
803:
800:
797:
777:
774:
771:
750:
729:
715:
702:
676:
654:
642:
629:
603:
581:
569:
556:
532:
529:
526:
523:
520:
500:
497:
494:
470:
449:
446:
443:
440:
419:
397:
383:is said to be
368:
365:
362:
359:
356:
344:
341:
333:equicontinuity
276:
246:
218:
215:
212:
188:
164:
140:
116:
92:
89:
86:
66:
63:
60:
57:
54:
34:, a branch of
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
7366:
7355:
7352:
7350:
7347:
7345:
7342:
7341:
7339:
7324:
7316:
7315:
7312:
7306:
7303:
7301:
7298:
7296:
7295:Weak topology
7293:
7291:
7288:
7286:
7283:
7281:
7278:
7277:
7275:
7271:
7264:
7260:
7257:
7255:
7252:
7250:
7247:
7245:
7242:
7240:
7237:
7235:
7232:
7230:
7227:
7225:
7222:
7220:
7219:Index theorem
7217:
7215:
7212:
7210:
7207:
7205:
7202:
7201:
7199:
7195:
7189:
7186:
7184:
7181:
7180:
7178:
7176:Open problems
7174:
7168:
7165:
7163:
7160:
7158:
7155:
7153:
7150:
7148:
7145:
7143:
7140:
7139:
7137:
7133:
7127:
7124:
7122:
7119:
7117:
7114:
7112:
7109:
7107:
7104:
7102:
7099:
7097:
7094:
7092:
7089:
7087:
7084:
7082:
7079:
7078:
7076:
7072:
7066:
7063:
7061:
7058:
7056:
7053:
7051:
7048:
7046:
7043:
7041:
7038:
7036:
7033:
7031:
7028:
7026:
7023:
7022:
7020:
7018:
7014:
7004:
7001:
6999:
6996:
6994:
6991:
6988:
6984:
6980:
6977:
6975:
6972:
6970:
6967:
6966:
6964:
6960:
6954:
6951:
6949:
6946:
6944:
6941:
6939:
6936:
6934:
6931:
6929:
6926:
6924:
6921:
6919:
6916:
6914:
6911:
6909:
6906:
6905:
6902:
6899:
6895:
6890:
6886:
6882:
6875:
6870:
6868:
6863:
6861:
6856:
6855:
6852:
6840:
6832:
6831:
6828:
6822:
6819:
6817:
6814:
6812:
6809:
6807:
6803:
6799:
6797:) convex
6796:
6793:
6791:
6788:
6786:
6782:
6780:
6777:
6775:
6772:
6770:
6769:Semi-complete
6767:
6765:
6762:
6760:
6757:
6755:
6751:
6748:
6746:
6742:
6740:
6737:
6735:
6732:
6730:
6727:
6725:
6722:
6720:
6717:
6715:
6712:
6710:
6707:
6705:
6702:
6700:
6697:
6695:
6692:
6690:
6687:
6685:
6684:Infrabarreled
6682:
6680:
6677:
6675:
6672:
6668:
6665:
6664:
6663:
6660:
6658:
6655:
6653:
6650:
6648:
6645:
6643:
6642:Distinguished
6640:
6638:
6635:
6633:
6630:
6628:
6625:
6623:
6620:
6618:
6614:
6610:
6608:
6605:
6603:
6599:
6595:
6593:
6590:
6588:
6585:
6583:
6580:
6579:
6577:
6575:Types of TVSs
6573:
6567:
6563:
6559:
6557:
6554:
6552:
6549:
6547:
6544:
6542:
6539:
6537:
6533:
6529:
6527:
6524:
6523:
6521:
6517:
6511:
6508:
6506:
6503:
6501:
6498:
6496:
6495:Prevalent/Shy
6493:
6491:
6488:
6486:
6485:Extreme point
6483:
6481:
6475:
6473:
6467:
6465:
6462:
6460:
6457:
6455:
6452:
6450:
6447:
6445:
6442:
6440:
6437:
6435:
6432:
6430:
6427:
6425:
6422:
6421:
6419:
6417:Types of sets
6415:
6409:
6406:
6404:
6401:
6399:
6396:
6394:
6391:
6387:
6384:
6382:
6379:
6377:
6374:
6373:
6372:
6369:
6367:
6364:
6360:
6359:Discontinuous
6357:
6355:
6352:
6350:
6347:
6345:
6342:
6340:
6337:
6335:
6332:
6330:
6327:
6326:
6325:
6322:
6318:
6315:
6314:
6313:
6310:
6309:
6307:
6303:
6297:
6294:
6290:
6287:
6286:
6285:
6282:
6279:
6276:
6274:
6270:
6267:
6265:
6262:
6260:
6257:
6255:
6252:
6250:
6247:
6246:
6244:
6242:
6238:
6232:
6229:
6227:
6224:
6222:
6219:
6217:
6216:Metrizability
6214:
6212:
6209:
6207:
6204:
6202:
6201:Fréchet space
6199:
6197:
6194:
6192:
6189:
6187:
6184:
6182:
6179:
6178:
6176:
6172:
6167:
6160:
6155:
6153:
6148:
6146:
6141:
6140:
6137:
6125:
6122:
6120:
6117:
6115:
6112:
6110:
6107:
6105:
6102:
6100:
6097:
6095:
6092:
6090:
6087:
6085:
6082:
6080:
6077:
6075:
6072:
6070:
6067:
6065:
6062:
6060:
6057:
6055:
6052:
6049:
6045:
6042:
6040:
6037:
6035:
6032:
6031:
6029:
6025:
6019:
6016:
6015:
6013:
6009:
6003:
6000:
5998:
5995:
5993:
5990:
5988:
5985:
5983:
5980:
5978:
5975:
5973:
5970:
5968:
5965:
5963:
5960:
5959:
5957:
5955:Miscellaneous
5953:
5946:
5942:
5939:
5937:
5934:
5932:
5929:
5927:
5924:
5923:
5921:
5917:
5911:
5908:
5906:
5903:
5901:
5898:
5896:
5893:
5892:
5890:
5886:
5878:
5875:
5874:
5873:
5870:
5868:
5865:
5863:
5860:
5858:
5855:
5853:
5850:
5848:
5844:
5842:
5839:
5838:
5836:
5832:
5826:
5823:
5821:
5818:
5816:
5813:
5811:
5808:
5806:
5803:
5801:
5798:
5796:
5793:
5791:
5788:
5786:
5783:
5782:
5780:
5776:
5770:
5767:
5765:
5762:
5760:
5757:
5755:
5752:
5750:
5747:
5743:
5740:
5738:
5735:
5733:
5730:
5729:
5728:
5725:
5724:
5722:
5720:Decomposition
5718:
5712:
5709:
5707:
5704:
5702:
5699:
5697:
5694:
5692:
5689:
5687:
5684:
5683:
5681:
5679:
5675:
5669:
5666:
5664:
5661:
5658:
5656:
5653:
5650:
5648:
5645:
5642:
5640:
5637:
5636:
5634:
5630:
5624:
5621:
5619:
5616:
5614:
5611:
5609:
5606:
5604:
5601:
5599:
5596:
5594:
5591:
5590:
5588:
5584:
5578:
5575:
5573:
5570:
5568:
5565:
5563:
5560:
5558:
5555:
5553:
5550:
5548:
5545:
5543:
5540:
5538:
5535:
5533:
5530:
5529:
5527:
5523:
5519:
5515:
5508:
5503:
5501:
5496:
5494:
5489:
5488:
5485:
5475:
5472:
5464:
5453:
5450:
5446:
5443:
5439:
5436:
5432:
5429:
5425:
5422: â
5421:
5417:
5416:Find sources:
5410:
5406:
5400:
5399:
5394:This article
5392:
5388:
5383:
5382:
5375:
5371:
5367:
5361:
5357:
5353:
5349:
5345:
5341:
5337:
5331:
5327:
5323:
5319:
5315:
5311:
5307:
5303:
5297:
5293:
5289:
5288:
5283:
5282:Rudin, Walter
5279:
5277:(Section 7.5)
5274:
5268:
5264:
5260:
5255:
5251:
5247:
5243:
5237:
5233:
5228:
5224:
5220:
5216:
5210:
5206:
5202:
5198:
5194:
5188:
5184:
5179:
5175:
5169:
5165:
5160:
5156:
5152:
5148:
5144:
5139:
5134:
5130:
5126:
5125:
5120:
5116:
5112:
5108:
5104:
5100:
5094:
5090:
5086:
5082:
5078:
5074:
5070:
5064:
5060:
5056:
5052:
5051:
5047:
5040:
5039:Kreyszig 1978
5035:
5032:
5026:
5023:
5019:
5014:
5012:
5010:
5006:
5002:
4998:
4992:
4989:
4985:
4980:
4978:
4976:
4974:
4972:
4970:
4968:
4966:
4964:
4960:
4955:
4951:
4947:
4941:
4937:
4936:
4928:
4926:
4922:
4919:, p. 98.
4918:
4913:
4910:
4907:
4902:
4899:
4896:, Section 2.4
4895:
4890:
4888:
4884:
4878:
4874:
4871:
4869:
4866:
4860:
4857:
4855:
4852:
4846:
4843:
4841:
4838:
4836:
4833:
4832:
4828:
4823:
4822:Riesz's lemma
4819:
4804:
4792:
4786:
4780:
4777:
4774:
4768:
4759:
4755:
4749:
4740:
4737:
4727:) defined by
4726:
4722:
4718:
4714:
4710:
4706:
4702:
4698:
4695:
4691:
4675:
4672:
4660:
4654:
4648:
4642:
4637:
4632:
4628:
4624:
4618:
4609:
4606:
4595:
4591:
4587:
4583:
4579:
4575:
4571:
4568: â
4567:
4563:
4560:
4546:
4542:
4520:
4516:
4507:
4504:
4503:
4499:
4497:
4493:
4491:
4487:
4483:
4479:
4475:
4471:
4467:
4446:
4442:
4438:
4427:
4406:
4402:
4391:
4388:
4384:
4380:
4376:
4372:
4368:
4364:
4356:
4354:
4337:
4327:
4323:
4307:
4304:
4301:
4297:
4270:
4248:
4245:
4240:
4235:
4231:
4207:
4196:
4185:
4182:
4174:
4170:
4148:
4144:
4137:
4134:
4129:
4125:
4116:
4112:
4106:
4101:
4098:
4095:
4091:
4087:
4084:
4077:
4076:
4075:
4073:
4057:
4054:
4051:
4048:
4045:
4042:
4039:
4014:
4011:
3991:
3988:
3983:
3980:
3977:
3973:
3964:
3960:
3944:
3941:
3936:
3932:
3928:
3923:
3919:
3895:
3892:
3887:
3883:
3879:
3874:
3870:
3843:
3840:
3835:
3831:
3827:
3822:
3818:
3788:
3784:
3777:
3774:
3769:
3765:
3756:
3752:
3741:
3738:
3735:
3731:
3727:
3724:
3717:
3716:
3715:
3713:
3676:
3673:
3666:
3665:
3664:
3644:
3641:
3638:
3632:
3615:
3614:Hilbert space
3599:
3590:
3586:
3578:
3576:
3574:
3570:
3566:
3562:
3558:
3554:
3549:
3546:
3542:
3538:
3534:
3530:
3525:
3523:
3519:
3515:
3511:
3507:
3503:
3499:
3495:
3491:
3487:
3483:
3482:Frigyes Riesz
3479:
3475:
3471:
3467:
3462:
3449:
3446:
3443:
3437:
3434:
3431:
3428:
3417:
3415:
3407:
3402:
3398:
3397:proper subset
3380:
3376:
3372:
3369:
3366:
3359:the range of
3343:
3337:
3334:
3331:
3310:
3306:
3303:
3295:
3287:
3281:
3275:
3272:
3269:
3265:
3261:
3256:
3252:
3231:
3228:
3225:
3217:
3200:
3196:
3187:
3174:
3153:
3130:
3124:
3121:
3118:
3098:
3095:
3092:
3084:
3066:
3060:
3057:
3054:
3034:
3026:
3023:
3019:
3015:
3002:
2978:
2972:
2965:
2961:
2939:
2931:
2927:
2922:
2918:
2915:
2910:
2906:
2901:
2897:
2894:
2890:
2884:
2880:
2869:
2866:
2863:
2859:
2851:
2847:
2842:
2838:
2835:
2830:
2826:
2821:
2817:
2814:
2811:
2808:
2805:
2794:
2788:
2784:
2780:
2777:
2774:
2770:
2766:
2763:
2759:
2755:
2745:
2742:
2739:
2735:
2729:
2725:
2721:
2718:
2715:
2711:
2707:
2704:
2701:
2698:
2678:
2675:
2672:
2664:
2647:
2643:
2639:
2636:
2633:
2611:
2607:
2603:
2600:
2597:
2577:
2574:
2571:
2563:
2548:
2545:
2542:
2522:
2519:
2516:
2496:
2490:
2487:
2484:
2476:
2461:
2458:
2455:
2447:
2434:
2413:
2405:
2392:
2383:
2370:
2361:
2348:
2324:
2321:
2315:
2311:
2303:
2300:
2292:
2289: is a
2276:
2273:
2267:
2263:
2238:
2232:
2229:
2226:
2218:
2217:
2216:
2214:
2210:
2206:
2188:
2184:
2175:
2171:
2167:
2162:
2158:
2137:
2131:
2128:
2125:
2105:
2093:
2080:
2059:
2053:
2050:
2047:
2039:
2023:
2015:
2008:
1995:
1986:
1983:is closed in
1982:
1969:
1960:
1957:
1953:
1940:
1931:
1930:
1916:
1910:
1907:
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1896:
1895:
1893:
1889:
1886:
1882:
1864:
1858:
1850:
1831:
1825:
1817:
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1783:
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1768:
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1738:
1732:
1729:
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1706:
1702:
1686:
1680:
1660:
1657:
1654:
1646:
1641:
1628:
1619:
1618:operator norm
1603:
1596:converges to
1576:
1573:
1563:
1559:
1535:
1532:
1529:
1509:
1503:
1476:
1473:
1463:
1459:
1447:
1446:
1429:
1426:
1423:
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1391:
1388:
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1375:
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1373:
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1353:
1350:
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1294:
1291:
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1279:
1259:
1251:
1233:
1229:
1208:
1202:
1179:
1176:
1173:
1167:
1159:
1158:operator norm
1143:
1137:
1114:
1111:
1108:
1102:
1082:
1079:
1076:
1073:
1070:
1067:
1064:
1052:
1050:
1032:
1024:
1020:
1007:
998:
985:
976:
975:
974:
972:
959:
927:
924:
914:
910:
906:
895:
882:
854:
851:
841:
837:
825:
810:
807:
801:
795:
775:
772:
769:
761:
748:
727:
720:
719:neighbourhood
716:
713:
700:
691:
687:
674:
665:
652:
643:
640:
627:
618:
614:
601:
592:
579:
570:
567:
554:
546:
545:
544:
530:
524:
521:
518:
498:
495:
492:
483:
481:
468:
444:
438:
430:
417:
408:
395:
386:
382:
366:
360:
357:
354:
347:A linear map
342:
340:
338:
334:
330:
326:
322:
318:
313:
311:
307:
303:
299:
295:
294:norm topology
291:
290:Hilbert space
287:
274:
265:
261:
257:
244:
234:
232:
216:
213:
210:
202:
186:
178:
162:
154:
138:
130:
114:
106:
90:
87:
84:
64:
58:
55:
52:
45:
41:
37:
33:
19:
7285:Balanced set
7259:Distribution
7197:Applications
7090:
7050:KreinâMilman
7035:Closed graph
6745:Polynomially
6674:Grothendieck
6667:tame Fréchet
6617:Bornological
6477:Linear cone
6469:Convex cone
6444:Banach disks
6386:Sesquilinear
6348:
6241:Main results
6231:Vector space
6186:Completeness
6181:Banach space
6027:Applications
5857:Disk algebra
5711:Spectral gap
5586:Main results
5467:
5458:
5448:
5441:
5434:
5427:
5415:
5403:Please help
5398:verification
5395:
5355:
5321:
5286:
5258:
5231:
5204:
5182:
5163:
5128:
5122:
5080:
5058:
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5025:
5000:
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4991:
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4589:
4585:
4581:
4577:
4573:
4569:
4565:
4548:
4544:
4536:
4494:
4492:is compact.
4489:
4485:
4481:
4473:
4465:
4425:
4382:
4378:
4374:
4370:
4366:
4362:
4360:
4166:
4071:
3806:
3711:
3709:
3592:An operator
3591:
3588:
3550:
3526:
3517:
3513:
3509:
3497:
3489:
3485:
3473:
3469:
3465:
3463:
3418:
3411:
3400:
3167:
3016:is compact,
2995:
2427:
2385:
2363:
2341:
2212:
2097:
2073:
1988:
1984:
1962:
1933:
1891:
1708:
1704:
1621:
1248:denotes the
1056:
1048:
1000:
978:
952:
950:
875:
741:
693:
667:
645:
620:
594:
572:
547:
484:
461:
410:
388:
384:
379:between two
346:
324:
314:
306:Grothendieck
267:
237:
235:
39:
29:
7214:Heat kernel
7204:Hardy space
7111:Trace class
7025:HahnâBanach
6987:Topological
6739:Quasinormed
6652:FK-AK space
6546:Linear span
6541:Convex hull
6526:Affine hull
6329:Almost open
6269:HahnâBanach
6054:Heat kernel
5754:Compression
5639:Isospectral
5018:Conway 1990
4894:Conway 1985
4561:is compact.
4470:Conway 1985
4169:trace-class
3506:eigenvalues
3502:limit point
3022:limit point
1816:composition
155:subsets of
36:mathematics
7338:Categories
7147:C*-algebra
6962:Properties
6779:Stereotype
6637:(DF)-space
6632:Convenient
6371:Functional
6339:Continuous
6324:Linear map
6264:F. Riesz's
6206:Linear map
5732:Continuous
5547:C*-algebra
5542:B*-algebra
5431:newspapers
5201:Lax, Peter
5048:References
4984:Rudin 1991
4906:Enflo 1973
4381:is called
4032:and every
3963:accumulate
3496:subset of
3218:For every
2038:invertible
1814:where the
1156:under the
1053:Properties
788:such that
431:such that
7121:Unbounded
7116:Transpose
7074:Operators
7003:Separable
6998:Reflexive
6983:Algebraic
6969:Barrelled
6795:Uniformly
6754:Reflexive
6602:Barrelled
6598:Countably
6510:Symmetric
6408:Transpose
5518:-algebras
5374:853623322
5354:(2006) .
5344:840278135
5250:144216834
5147:0001-5962
5115:Enflo, P.
4954:695395895
4764:Ω
4760:∫
4723:(Ω;
4629:∫
4517:ℓ
4341:∞
4324:λ
4313:∞
4298:∑
4277:∞
4274:→
4232:λ
4211:∞
4186:
4141:⟩
4138:⋅
4122:⟨
4113:λ
4092:∑
4058:…
4015:∈
4004:for some
3974:λ
3945:…
3933:λ
3920:λ
3896:…
3844:…
3781:⟩
3778:⋅
3762:⟨
3753:λ
3747:∞
3732:∑
3687:→
3677::
3648:⟩
3645:⋅
3639:⋅
3636:⟨
3429:λ
3373:λ
3370:−
3338:σ
3335:∈
3332:λ
3296:λ
3276:σ
3273:∈
3270:λ
3201:∗
3154:λ
3125:σ
3122:∈
3119:λ
3096:≠
3093:λ
3061:σ
3058:∈
3018:countable
2973:σ
2932:∗
2919:λ
2916:−
2911:∗
2898:
2885:∗
2870:
2852:∗
2839:λ
2836:−
2831:∗
2818:
2812:
2781:λ
2778:−
2767:
2746:
2722:λ
2719:−
2708:
2702:
2676:≠
2673:λ
2640:λ
2637:−
2604:λ
2601:−
2575:≠
2572:λ
2546:∘
2520:∘
2494:→
2488::
2322:−
2304:
2274:−
2230:∈
2209:transpose
2189:∗
2181:→
2176:∗
2168::
2163:∗
2135:→
2129::
2057:→
1956:separable
1914:→
1781:⊆
1760:∘
1739:∘
1684:→
1577:∈
1507:→
1477:∈
1206:→
1141:→
928:∈
855:∈
808:⊆
773:⊆
528:→
364:→
302:Per Enflo
62:→
7323:Category
7135:Algebras
7017:Theorems
6974:Complete
6943:Schwartz
6889:glossary
6839:Category
6790:Strictly
6764:Schwartz
6704:LF-space
6699:LB-space
6657:FK-space
6627:Complete
6607:BK-space
6532:Relative
6479:(subset)
6471:(subset)
6398:Seminorm
6381:Bilinear
6119:Weyl law
6064:Lax pair
6011:Examples
5845:With an
5764:Discrete
5742:Residual
5678:Spectrum
5663:operator
5655:operator
5647:operator
5562:Spectrum
5461:May 2008
5310:21163277
5284:(1991).
5223:47767143
5203:(2002).
5117:(1973).
5107:21195908
5079:(1990).
5057:(1985).
4829:See also
4592:(;
4584:(;
4572:(;
4500:Examples
4484: :
4390:sequence
4373: :
3539:and the
3244:the set
2964:spectrum
2219:For any
1496:mapping
77:, where
7126:Unitary
7106:Nuclear
7091:Compact
7086:Bounded
7081:Adjoint
7055:Minâmax
6948:Sobolev
6933:Nuclear
6923:Hilbert
6918:Fréchet
6883: (
6804:)
6752:)
6694:K-space
6679:Hilbert
6662:Fréchet
6647:F-space
6622:Brauner
6615:)
6600:)
6582:Asplund
6564:)
6534:)
6454:Bounded
6349:Compact
6334:Bounded
6271: (
5660:Unitary
5445:scholar
5155:0402468
3712:compact
3557:algebra
3555:in the
3464:(where
2205:adjoint
2203:is the
1522:(where
385:compact
177:closure
7101:Normal
6938:Orlicz
6928:Hölder
6908:Banach
6897:Spaces
6885:topics
6816:Webbed
6802:Quasi-
6724:Montel
6714:Mackey
6613:Ultra-
6592:Banach
6500:Radial
6464:Convex
6434:Affine
6376:Linear
6344:Closed
6168:(TVSs)
5644:Normal
5447:
5440:
5433:
5426:
5418:
5372:
5362:
5342:
5332:
5308:
5298:
5269:
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5170:
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5105:
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5065:
4952:
4942:
4289:while
3807:where
3569:simple
3522:kernel
2406:where
1322:, and
1160:, and
999:under
666:under
593:under
310:Banach
231:Banach
6913:Besov
6774:Smith
6759:Riesz
6750:Semi-
6562:Quasi
6556:Polar
5737:Point
5452:JSTOR
5438:books
4879:Notes
4711:is a
4596:) by
4588:) to
4580:from
4476:is a
4424:from
3567:, is
3553:ideal
3395:is a
3146:then
2072:then
1849:ideal
288:is a
127:maps
42:is a
7261:(or
6979:Dual
6393:Norm
6317:form
6305:Maps
5668:Unit
5516:and
5424:news
5370:OCLC
5360:ISBN
5340:OCLC
5330:ISBN
5306:OCLC
5296:ISBN
5267:ISBN
5246:OCLC
5236:ISBN
5219:OCLC
5209:ISBN
5187:ISBN
5168:ISBN
5143:ISSN
5103:OCLC
5093:ISBN
5063:ISBN
4950:OCLC
4940:ISBN
4508:For
4365:and
4361:Let
4208:<
4072:i.e.
3859:and
3304:>
3229:>
3188:and
3111:and
2962:The
2535:and
2362:and
1707:and
485:Let
308:and
260:rank
229:are
103:are
38:, a
5407:by
5326:GTM
5133:doi
5129:130
4820:By
4719:on
4552:= t
4171:or
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3508:of
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3085:If
3027:If
2994:of
2867:dim
2815:ker
2809:dim
2743:dim
2705:ker
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2211:of
2207:or
2016:If
1954:is
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1894:).
1851:in
1252:on
1025:in
1021:is
874:in
692:in
688:is
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615:is
327:on
179:in
151:to
131:of
30:In
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4886:^
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3992:0
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1996:T
1985:Y
1970:T
1958:.
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1911:X
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641:;
628:Y
602:T
580:X
555:T
531:Y
525:X
522::
519:T
499:Y
496:,
493:X
469:Y
448:)
445:U
442:(
439:T
418:X
396:U
367:Y
361:X
358::
355:T
325:K
275:Y
245:T
217:Y
214:,
211:X
187:Y
163:Y
139:X
115:T
91:Y
88:,
85:X
65:Y
59:X
56::
53:T
20:)
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