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.
Auflage
Sprache
Verlagsort
Verlagsgruppe
McGraw-Hill Education - Europe
Maße
Höhe: 279 mm
Breite: 203 mm
Dicke: 30 mm
Gewicht
ISBN-13
978-0-07-054236-5 (9780070542365)
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Schweitzer Klassifikation
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
