Book:Yuli Eidelman/Functional Analysis: An Introduction
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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