作者：WalterRudin[美]沃尔特·鲁丁
页数：436
出版社：机械工业出版社
出版日期：2020
ISBN：9787111654742
电子书格式：pdf/epub/txt
内容简介
CONTENTSPrefaceAbout thd AuthorPart I General TheoryTopological Vector SpacesIntroductionSeparation propertiesLinear mappingsFinite-dimensional spacesMetrizationBoundedness and continuitySeminorms and local convexityQuotient spacesExamplesExercises2 CompletenessBaire categoryThe Banach-Steinhaus theoremThe open mapping theoremThe closed graph theoremBilinear mappingsExercises3 ConvexityThe Hahn-Banach theoremsWeak topologiesCompact convex setsVector-valued integrationHolomorphic functionsExercises4 Duality in Banach SpacesThe normed dual of a normed spaceAdjointsCompact operatorsExercises5 Some ApplicationsA continuity theoremClosed subspaces of fi-spacesThe range of a vector-valued measureA generalized Stone-Weierstrass theoremTwo interpolation theoremsKakutani””s fixed point theoremHaar measure on compact groupsUncomplemented subspacesSums of Poisson kernelsTwo more fixed point theoremsExercisesPart II Distributions and Fourier Transform6 Test Functions and DistributionsIntroductionTest function spacesCalculus with distributionsLocalizationSupports of distributionsDistributions as derivativesConvolutionsExercises7 Fourier TransformsBasic propertiesTempered d]str]but]onsPaley-Wiener theoremsSobolev””s lemmaExercises8 Applications to Differential EquationsFundamental solutionsElliptic equationsExercises9 Tauberian TheoryWiener””s theoremThe prime number theoremThe renewal equationExercisesPart III Banach Algebras and Spectral Theory10 Banach AlgebrasIntroductionComplex homomorphismsBasic properties of spectraSymbolic calculusThe group of invertible elementsLomonosov””s invariant subspace theoremExercises11 Commutative Banach AlgebrasIdeals and homomorphismsGelfand transformsInvolutionsApplications to noncommutative algebrasPositive functionalsExercises12 Bounded Operators on a Hilbert SpaceBasic factsBounded operatorsA commutativity theoremResolutions of the identityThe spectral theoremEigenvalues of normal operatorsPositive operators and square rootsThe group of invertible operatorsA characterization of B最-algebrasAn ergodic theoremExercises13 Unbounded OperatorsIntroductionGraphs and symmetric operatorsThe Cayley transformResolutions of the identityThe spectral theoremSemigroups of operatorsExercisesAppendix A Compactness and ContinuityAppendix B Notes and CommentsBibliographyList of Spe SymbolsIndex
作者简介
沃尔特·鲁丁（Walter Rudin），1953年于杜克大学获得数学博士学位。曾先后执教于麻省理工学院、罗切斯特大学、威斯康星大学麦迪逊分校、耶鲁大学等。他的主要研究兴趣集中在调和分析和复变函数上。除本书外，他还著有《Functional Analysis》（泛函分析）和《Principles of Mathematical Analysis》（数学分析原理）等其他名著。这些教材已被翻译成十几种语言，在世界各地广泛使用。
目录
About thd Author
Part I General Theory
Topological Vector Spaces
Introduction
Separation properties
Linear mappings
Finite-dimensional spaces
Metrization
Boundedness and continuity
Seminorms and local convexity
Quotient spaces
Examples
Exercises
2 Completeness
Baire category
The Banach-Steinhaus theorem
The open mapping theorem
The closed graph theorem
Bilinear mappings
Exercises
3 Convexity
The Hahn-Banach theorems
Weak topologies
Compact convex sets
Vector-valued integration
Holomorphic functions
Exercises
4 Duality in Banach Spaces
The normed dual of a normed space
Adjoints
Compact operators
Exercises
5 Some Applications
A continuity theorem
Closed subspaces of fi-spaces
The range of a vector-valued measure
A generalized Stone-Weierstrass theorem
Two interpolation theorems
Kakutani’s fixed point theorem
Haar measure on compact groups
Uncomplemented subspaces
Sums of Poisson kernels
Two more fixed point theorems
Exercises
Part II Distributions and Fourier Transform
6 Test Functions and Distributions
Introduction
Test function spaces
Calculus with distributions
Localization
Supports of distributions
Distributions as derivatives
Convolutions
Exercises
7 Fourier Transforms
Basic properties
Tempered d]str]but]ons
Paley-Wiener theorems
Sobolev’s lemma
Exercises
8 Applications to Differential Equations
Fundamental solutions
Elliptic equations
Exercises
9 Tauberian Theory
Wiener’s theorem
The prime number theorem
The renewal equation
Exercises
Part III Banach Algebras and Spectral Theory
10 Banach Algebras
Introduction
Complex homomorphisms
Basic properties of spectra
Symbolic calculus
The group of invertible elements
Lomonosov’s invariant subspace theorem
Exercises
11 Commutative Banach Algebras
Ideals and homomorphisms
Gelfand transforms
Involutions
Applications to noncommutative algebras
Positive functionals
Exercises
12 Bounded Operators on a Hilbert Space
Basic facts
Bounded operators
A commutativity theorem
Resolutions of the identity
The spectral theorem
Eigenvalues of normal operators
Positive operators and square roots
The group of invertible operators
A characterization of B最-algebras
An ergodic theorem
Exercises
13 Unbounded Operators
Introduction
Graphs and symmetric operators
The Cayley transform
Resolutions of the identity
The spectral theorem
Semigroups of operators
Exercises
Appendix A Compactness and Continuity
Appendix B Notes and Comments
Bibliography
List of Special Symbols
Index
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