# Book:Yuli Eidelman/Functional Analysis: An Introduction

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## Yuli Eidelman, Vitali Milman and Antonis Tsolomitis:

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

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

- ISBN 978-0821836460.

### Subject Matter

### 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. Self-adjoint operators**

*6.1 General properties*

*6.2 Self-adjoint compact operators*

- 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

*10.5 Radicals*

*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**