Book:Walter Rudin/Functional Analysis/Second Edition

Subject Matter

 * Functional Analysis

Contents
Preface

Part I: General Theory


 * 1 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 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 II: Distributions and Fourier Transforms


 * 6: Test Functions and Distributions


 * Introduction
 * Test functions 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 functions
 * 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 Homomorhpisms
 * 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*-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