Book:Yuli Eidelman/Functional Analysis: An Introduction

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. 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