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.
Table of contents:
Preface. PART ONE: GENERAL THEORY 1. Topological Vector Space 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 Lp-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 TWO: DISTRIBUTIONS AND FOURIER TRANSFORMS 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 distributions 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 THREE: 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 Hillbert 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*-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
This classic text is written for graduate courses in functional analysis. This text is used in modern investigations in analysis and applied mathematics. This edition includes up-to-date presentations of topics as well as more examples and exercises. New topics include Kakutani's fixed point theorem, Lomonosov's invariant subspace theorem, and an ergodic theorem.
Table of contents:
Preface. PART ONE: GENERAL THEORY 1. Topological Vector Space 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 Lp-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 TWO: DISTRIBUTIONS AND FOURIER TRANSFORMS 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 distributions 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 THREE: 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 Hillbert 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*-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
This classic text is written for graduate courses in functional analysis. This text is used in modern investigations in analysis and applied mathematics. This edition includes up-to-date presentations of topics as well as more examples and exercises. New topics include Kakutani's fixed point theorem, Lomonosov's invariant subspace theorem, and an ergodic theorem.