Elementary Functional Analysis

Preface

Contents

Hilbert Space Preliminaries

1.1 Normed Linear Spaces

1.2 Orthogonality

1.3 Hilbert Space Geometry

1.4 Linear Functionals

1.5 Orthonormal Bases

1.6 Exercises

Operator Theory Basics

2.1 Bounded Linear Operators

2.2 Adjoints of Hilbert Space Operators

2.3 Adjoints of Banach Space Operators

2.4 Exercises

The Big Three

3.1 The HahnÒBanach Theorem

3.2 Principle of Uniform Boundedness

3.3 Open Mapping and Closed Graph Theorems

3.4 Quotient Spaces

3.5 Banach and the Scottish Caf ï e

3.6 Exercises

Compact Operators

4.1 Finite-Dimensional Spaces

4.2 Compact Operators

4.3 A Preliminary Spectral Theorem

4.4 The Invariant Subspace Problem

4.5 Introduction to the Spectrum

4.6 The Fredholm Alternative

4.7 Exercises

Banach and C*- Algebras

5.1 First Examples

5.2 Results on Spectra

5.3 Ideals and Homomorphisms

5.4 Commutative Banach Algebras

5.5 Weak Topologies

5.6 The Gelfand Transform

5.7 The Continuous Functional Calculus

5.8 Fredholm Operators

5.9 Exercises

The Spectral Theorem

6.1 Normal Operators Are Multiplication Operators

6.2 Spectral Measures

6.3 Exercises

Real Analysis Topics

A.1 Measures

A.2 Integration

A.3 Lp Spaces

A.4 The StoneÒWeierstrass Theorem

A.5 Positive Linear Functionals on C( X)

References

Index