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


 * 5. Spectrum. Fredholm theory of compact operators


 * 6. Self-adjoint operators


 * 7. Functions of operators; spectral decomposition

'''Part II. Basics of Functional Analysis'''


 * 8. Spectral theory of unitary operators


 * 9. The fundamental theorems and the basic methods


 * 10. Banach algebras


 * 11. Unbounded self-adjoint and symmetric operators in $H$


 * A. Solutions to exercises

Bibliography

Symbols index

Subject index