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

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

Published $2002$, American Mathematical Society

ISBN 978-0821828953.

### Subject Matter

Functional Analysis

### 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
3.3 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
5.5 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
12.12 Problems
13.1 Orthogonal projections
13.2 Square roots of operators
13.3 A decomposition of operators
13.4 Spectral resolution
13.5 Some consequences
13.7 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