: hard ISBN 9780070542365
This classic text is written for graduate courses in functional analysis. This text is used in modern investigations in analysis and applied mathematics. This new edition includes up-to-date presentations of topics as well as more examples and exercises. New topics include Kakutani's fixed point theorem, Lamonosov's invariant subspace theorem, and an ergodic theorem.This text is part of the Walter Rudin Student Series in Advanced Mathematics.
Preface. PART ONE: GENERAL THEORY1. Topological Vector SpaceIntroductionSeparation 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 Lp-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 TWO: DISTRIBUTIONS AND FOURIER TRANSFORMS6. Test Functions and DistributionsIntroductionTest function spacesCalculus with distributionsLocalizationSupports of distributionsDistributions as derivativesConvolutionsExercises7. Fourier TransformsBasic propertiesTempered distributionsPaley-Wiener theoremsSobolev's lemmaExercises8. Applications to Differential EquationsFundamental solutionsElliptic equationsExercises9. Tauberian TheoryWiener's theoremThe prime number theoremThe renewal equationExercisesPART THREE: 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 Hillbert SpaceBasic factsBounded operatorsA commutativity theoremResolutions of the identityThe spectral theoremEigenvalues of normal operatorsPositive operators and square rootsThe group of invertible operatorsA characterization of B*-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 Special SymbolsIndex
: pbk ISBN 9780071009447
Written for undergraduate courses, this new edition includes coverage of current topics of research and contains more exercises and examples. New topics covered include: Kakutani's fixed point theorem; Lomonosov's invariant subspace theorem; and an ergodic theorem.
- Part1 General theory: topological vector spaces
- duality in Banach spaces
- some applications. Part 2 Distributions and Fourier transforms: test functions and distributions
- Fourier transforms
- applications to differential equations
- Tauberian theory. Part 3 Banach algebras and spectral theory: Banach algebras
- commutative Banach algebras
- bounded operators on a Hilbert space
- unbounded operators. Appendices: compactness and continuity
- notes and comments.
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