Book:Barbara D. MacCluer/Elementary Functional Analysis

Barbara D. MacCluer: Elementary Functional Analysis

Published $\text {2009}$, Springer

ISBN 978-0387855288

Subject Matter

• Functional Analysis

Contents

Preface

1. 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

2. 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

3. The Big Three

3.1 The Hahn-Banach 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 Café

3.6 Exercises

4. 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

5. 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

6. The Spectral Theorem

6.1 Normal Operators Are Multiplication Operators

6.2 Spectral Measures

6.3 Exercises

Appendix A: Real Analysis Topics

A.1 Measures

A.2 Integration

A.3 $L^p$ Spaces

A.4 The Stone-Weierstrass Theorem

A.5 Positive Linear Functionals on $\map C X$

References

Index