Functional Analysis

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 L^p-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