# Book:Walter Rudin/Functional Analysis/Second Edition

## Walter Rudin: Functional Analysis (2nd Edition)

Published $1991$, McGraw-Hill

ISBN 978-0070619883.

### 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
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