泛函分析（影印版）

图书标准信息

• 作者 [美]拉克斯 著
• 出版社 高等教育出版社
• 出版时间 2007-02
• 版次 1
• ISBN 9787040216493
• 定价 46.40元
• 装帧 平装
• 开本 16开
• 纸张 其他
• 页数 580页
• 正文语种 英语
• 原版书名 Functional Analysis
• 丛书 天元基金影印数学丛书

【内容简介】

《泛函分析（影印版）》是美国科学院院士PeterD．Lax在CotJrant数学所长期讲授泛函分析课程的教学经验基础上编写的。《泛函分析（影印版）》包括泛函分析的基本内容：Barlach空间、Hilbert空间和线性拓扑空间的基本概念和性质，线性拓扑空间中的凸集及其端点集的性质，有界线性算子的性质等。可作为本科生泛函分析课的教学内容；还包括泛函分析较深的内容：自伴算子的谱分解理论。紧算子的理论，交换Barlach代数的Gelfand理论，不变子空间的理论等。可作为研究生泛函分析课的教学内容。《泛函分析（影印版）》特别强调泛函分析与其他数学分支的联系及泛函分析理论的应用，可以使读者深刻地理解到：抽象的泛函分析理论有着丰富的数学背景。

【目录】

Foreword
1.LinearSpaces
Axiomsforlinearspaces-Infinite-dimensionalexamples-Subspace,linearspan-Quotientspace-Isomorphism-Convexsets-Extremesubsets

2.LinearMaps
2.1Algebraoflinearmaps,
Axiomsforlinearmaps-Sumsandcomposites-Invertiblelinearmaps-Nullspaceandrange-Invariantsubspaces
2.2.Indexofalinearmap,
Degeneratemaps-Pseudoinverse-IndexmProductformulafortheindex-Stabilityoftheindex

3.TheHahn,BanachTheorem
3.1Theextensiontheorem,
Positivehomogeneous,subadditivefunctionals-Extensionoflinearfunctionals-Gaugefunctionsofconvexsets
3.2GeometricHahn-Banachtheorem,
Thehyperplaneseparationtheorem
3.3ExtensionsoftheHahn-Banachtheorem,
TheAgnew-Morsetheorem-The
Bohnenblust-Sobczyk-Soukhomlinovtheorem

4.ApplicationsoftheHahn-Banachtheorem
4.1Extensionofpositivelinearfunctionals,
4.2Banachlimits.
4.3Finitelyadditiveinvariantsetfunctions,
Historicalnote,

5.NormedLinearSpaces
5.1Norms,
Normsforquotientspaces-Completenormedlinearspaces-ThespacesC,B-LpspacesandH61dersinequality-Sobolevspaces,embeddingtheorems-Separablespaces
5.2Noncompactnessoftheunitbail,
Uniformconvexity-TheMazur-Ulamtheoremonisometrics
5.3Isometrics,

6.HilbertSpace
6.1Scalarproduct,
SchwarzinequalityParallelogramidentity——Completeness,closure-e2,L
6.2Closestpointinaclosedconvexsubset,54Orthogonalcomplementofasubspace-Orthogonaldecomposition
6.3Linearfunctionals,
TheRiesz-Frechetrepresentationtheorem-Lax-Milgramlemma
6.4Linearspan,
Orthogonalprojection-Orthonormalbases,Gram-Schmidtprocess-IsometriesofaHilbertspace

7.ApplicationsofHilbertSpaceResults
7.1Radon-Nikodymtheorem,
7.2Dirichletsproblem,
UseoftheRiesz-Frechettheorem-UseoftheLax-MilgramtheoremUseoforthogonaldecomposition

8.DualsofNormedLinearSpaces
8.1Boundedlinearfunctionals,
Dualspace
8.2Extensionofboundedlinearfunctionals,
Dualcharacterizationofnorm-Dualcharacterizationofdistancefromasubspace-Dualcharacterizationoftheclosedlinearspanofaset
8.3Reflexivespaces,
ReflexivityofLp,18.4Supportfunctionofaset,
Dualcharacterizationofconvexhull-Dualcharacterizationofdistancefromaclosed,convexset

9.ApplicationsofDuality
9.1Completenessofweightedpowers,
9.2TheMuntzapproximationtheorem,
9.3Rungestheorem,
9.4Dualvariationalproblemsinfunctiontheory,
9.5ExistenceofGreensfunction,

10.WeakConvergence
10.1Uniformboundednessofweaklyconvergentsequences,101Principleofuniformboundedness-Weaklysequentiallyclosedconvexsets
10.2Weaksequentialcompactness,104Compactnessofunitballinreflexivespace
10.3Weak*convergence,105Hellystheorem

11.ApplicationsofWeakConvergence
11.1Approximationofthefunctionbycontinuousfunctions,108Toeplitzstheoremonsummability
11.2DivergenceofFourierseries,
11.3Approximatequadrature,
11.4Weakandstronganalyticityofvector-valuedfunctions,
11.5Existenceofsolutionsofpartialdifferentialequations,112Galerkinsmethod
11.6Therepresentationofanalyticfunctionswithpositiverealpart,115Hergiotz-Riesztheorem

12.TheWeakandWeak*Topologies
Comparisonwithweaksequentialtopology-Closedconvexsetsintheweaktopology——Weakcompactness-Alaoglustheorem

13.LocallyConvexTopologiesandtheKrein-MilmanTheorem
13.1Separationofpointsbylinearfunctionals,
13.2TheKrein-Milmantheorem,
13.3TheStone-Weierstrasstheorem,
13.4Choquetstheorem,

14.ExamplesofConvexSetsandTheirExtremePoints
14.1Positivefunctionals,
14.2Convexfunctions,
14.3Completelymonotonefunctions,
14.4TheoremsofCaratheodoryandBochner,
14.5AtheoremofKrein,
14.6Positiveharmonicfunctions,
14.7TheHamburgermomentproblem,
14.8G.Birkhoffsconjecture,
14.9DeFinettistheorem,
14.10Measure-preservingmappings,
Historicalnote,

15.BoundedLinearMaps
15.1Boundednessandcontinuity,
Normofaboundedlinearmap-Transpose
15.2Strongandweaktopologies,
Strongandweaksequentialconvergence
15.3Principleofuniformboundedness,
15.4Compositionofboundedmaps,
15.5Theopenmappingprinciple,
ClosedgraphtheoremHistoricalnote,

16.ExamplesofBoundedLinearMaps
16.1Boundednessofintegraloperators,
IntegraloperatorsofHilbert-Schmidttype-IntegraloperatorsofHolmgrentype
16.2TheconvexitytheoremofMarcelRiesz,
16.3Examplesofboundedintegraloperators,
TheFouriertransform,ParsevalstheoremandHausdorff-Younginequality-TheHilberttransformTheLaplacetransform-TheHilbert-Hankeltransform
……
A.Riesz-Kakutanirepresentationtheorem
B.Theoryofdistributions
C.ZornsLemma
AuthorIndex
SubjectIndex