# Book:Yuli Eidelman/Functional Analysis: An Introduction

## Yuli Eidelman, Vitali Milman and Antonis Tsolomitis: Functional Analysis: An Introduction

Published $\text {2004}$, American Mathematical Society

ISBN 978-0821836460.

### Subject Matter

Functional Analysis

### Contents

Preface

Introduction

Part I. Hilbert Spaces and Basic Operator Theory

1. Linear Spaces; noremd spaces; first examples
1.1 Linear Spaces
1.2 Normed spaces; first examples
1.2a Holder's inequality
1.2b Minkowski's inequality
1.3 Topological and geometrical notions
1.4 Quotient normed space
1.5 Completeness; completion
1.6 Exercises
2. Hilbert spaces
2.1 Basic notions; first examples
2.1a Cauchy-Schwartz inequality and the Hilbertian norm
2.1b Bessel's inequality
2.1c Complete systems
2.1d Gram-Schmidt orthogonalization procedure; orthogonal bases
2.1e Parseval's identity
2.2 Projection; orthogonal decompositions
2.2a Separable case
2.2b The distance from a point to a convex set
2.2c Orthogonal decomposition
2.3 Linear Functionals
2.3a Linear functionals in a general linear space
2.3b Bounded linear functionals
2.3c Bounded linear functionals in a Hilbert space
2.3d An example of non-separable Hilbert space
2.4 Exercises
3. The dual space
3.1 The Hahn-Banach theorem and its first consequences
3.1a Corollaries of The Hahn-Banach theorem
3.2 Examples of dual spaces
3.3 Exercises
4. Bounded linear operators
4.1 Completeness of the space of bounded linear operators
4.2 Examples of linear operators
4.3 Compact operators
4.3a Compact sets
4.3b The space of compact operators
4.4 Dual operators
4.5 Operators of finite rank
4.5a Compactness of the integral operator in $L_2$
4.6 Convergence in the space of bounded operators
4.7 Invertible operators
4.8 Exercises
5. Spectrum. Fredholm theory of compact operators
5.1 Clasification of spectrum
5.2 Fredholm theory of compact operators
5.3 Exercises
6.1 General properties
6.2a Spectral theory
6.2b Minimax principle
6.2c Applications to integral operators
6.3 Order in the space of self-adjoint operators
6.3a Properties of the ordering
6.4 Projection operators
6.4a Properties of projections in linear spaces
6.4b Orthoprojections
6.5 Exercises
7. Functions of operators; spectral decomposition
7.1 Spectral decomposition
7.1a The main inequality
7.1b Construction of the spectral integral
7.2 Hilbert theorem
7.3 Spectral family and spectrum of self-adjoint operators
7.4 Simple spectrum
7.5 Exercises

Part II. Basics of Functional Analysis

8. Spectral theory of unitary operators
8.1 Spectral properties of unitary operators
8.2 Exercises
9. The fundamental theorems and the basic methods
9.1 Auxiliary results
9.2 The Banach open mapping theorem
9.3 The closed graph theorem
9.4 The Banach-Steinhaus theorem
9.5 Bases in Banach spaces
9.6 Linear functionals; Hahn-Banach theorem
9.7 Separation of convex sets
9.8 The Eberlain-Schmulian theorem
9.9 Extremal points; the Krein-Milman theorem
9.10 Exercises
10. Banach algebras
10.1 Preliminaries
10.2 Gelfand's theorem on maximal ideals
10.3 Analytic functions
10.4 Gelfand map; the space of maximal ideals
10.4a The space of maximal ideals
10.6 Involutions; the Gelfand-Naimark theorem
10.7 Application to spectral theory
10.8 Application to a generalized limit and combinatorics
10.9 Exercises
11. Unbounded self-adjoint and symmetric operators in $H$
11.1 Basic notions and examples
11.2 More properties of symmetric operators
11.3 The spectrum $\map \sigma A$
11.4 Elements of the "graph" method
11.5 Cayley transform; spectral decomposition
11.6 Symmetric and self-adjoint extensions of a symmetric operator
11.7 Exercises
A. Solutions to exercises

Bibliography

Symbols index

Subject index