# Book:Martin Schechter/Principles of Functional Analysis/Second Edition

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## Martin Schechter:

## Martin Schechter: *Principles of Functional Analysis (2nd Edition)*

Published $\text {2002}$, **American Mathematical Society**

- ISBN 978-0821828953.

### Subject Matter

### Contents

**Preface to the Revised Edition**

**From the Preface to the First Edition**

**Chapter 1. Basic Notions**

- 1.1 A problem from differential equations

- 1.2 An examination of the results

- 1.3 Examples of Banach spaces

- 1.4 Fourier series

- 1.5 Problems

**Chapter 2. Duality**

- 2.1 The Riesz representation theorem

- 2.2 The Hahn-Banach theorem

- 2.3 Consequences of Hahn-Banach theorem

- 2.4 Examples of dual spaces

- 2.5 Problems

**Chapter 3. Linear Operators**

- 3.1 Basic properties

- 3.2 The adjoint operator

- 3.3 Annihilators

- 3.4 The inverse operator

- 3.5 Operators with closed ranges

- 3.6 The uniform boundedness principle

- 3.7 The open mapping theorem

- 3.8 Problems

**Chapter 4. The Riesz Theory for Compact Operators**

- 4.1 A type of integral equation

- 4.2 Operators of finite rank

- 4.3 Compact operators

- 4.4 The adjoint of a compact operator

- 4.5 Problems

**Chapter 5. Fredholm Operators**

- 5.1 Orientation

- 5.2 Further properties

- 5.3 Perturbation theory

- 5.4 The adjoint operator

- 5.5 A special case

- 5.6 Semi-Fredholm operators

- 5.7 Products of operators

- 5.8 Problems

**Chapter 6. Spectral Theory**

- 6.1 The spectrum and resolvent sets

- 6.2 The spectral mapping theorem

- 6.3 Operational calculus

- 6.4 Spectral projections

- 6.5 Complexification

- 6.6 The complex Hahn-Banach theorem

- 6.7 A geometric lemma

- 6.8 Problems

**Chapter 7. Unbounded Operators**

- 7.1 Unbounded Fredholm operators

- 7.2 Further properties

- 7.3 Operators with closed ranges

- 7.4 Total subsets

- 7.5 The essential spectrum

- 7.6 Unbounded semi-Fredholm operators

- 7.7 The adjoint of a product of operators

**Chapter 8. Reflexive Banach Spaces**

- 8.1 Properties of reflexive spaces

- 8.2 Saturated subspaces

- 8.3 Separable spaces

- 8.4 Weak convergence

- 8.5 Examples

- 8.6 Completing a normed vector space

- 8.7 Problems

**Chapter 9. Banach Algebras**

- 9.1 Introduction

- 9.2 An example

- 9.3 Commutative algebras

- 9.4 Properties of maximal ideals

- 9.5 Partially ordered sets

- 9.6 Riesz operators

- 9.7 Fredholm perturbations

- 9.8 Semi-Fredholm perturbations

- 9.9 Remarks

- 9.10 Problems

**Chapter 10. Semigroups**

- 10.1 A differential equation

- 10.2 Uniqueness

- 10.3 Unbounded operators

- 10.4 The infinitesimal operator

- 10.5 An approximation theorem

- 10.6 Problems

**Chapter 11. Hilbert Space**

- 11.1 When is a Banach space a Hilbert space

- 11.2 Normal operators

- 11.3 Approximation by operators of finite rank

- 11.4 Integral operators

- 11.5 Hyponormal operators

- 11.6 Problems

**Chapter 12. Bilinear Forms**

- 12.1 The numerical range

- 12.2 The associated operator

- 12.3 Symmetric forms

- 12.4 Closed forms

- 12.5 Closed extensions

- 12.6 Closable operators

- 12.7 Some proofs

- 12.8 Some representation theorems

- 12.9 Dissipative operators

- 12.10 The case of a line or a strip

- 12.11 Selfadjoint extensions

- 12.12 Problems

**Chapter 13. Selfadjoint Operators**

- 13.1 Orthogonal projections

- 13.2 Square roots of operators

- 13.3 A decomposition of operators

- 13.4 Spectral resolution

- 13.5 Some consequences

- 13.6 Unbounded selfadjoint operators

- 13.7 Problems

**Chapter 14. Measures of Operators**

- 14.1 A seminorm

- 14.2 Perturbation classes

- 14.3 Related measures

- 14.4 Measures of noncompactness

- 14.5 The quotient space

- 14.6 Strictly singular operators

- 14.7 Norm perturbations

- 14.8 Perturbation functions

- 14.9 Factored perturbation functions

- 14.10 Problems

**Chapter 15. Examples and Applications**

- 15.1 A few remarks

- 15.2 A differential operator

- 15.3 Does $A$ have a closed extension

- 15.4 The closure of $A$

- 15.5 Another approach

- 15.6 The Fourier transform

- 15.7 Multiplication by a function

- 15.8 More general operators

- 15.9 $B$-Compactness

- 15.10 The adjoint of $\bar A$

- 15.11 An integral operator

- 15.12 Problems

**Appendix A. Glossary**

**Appendix B. Major Theorems**

**Bibliography**

**Index**