Elementary Functional Analysis

by Charles W Swartz

Overview

This text is an introduction to functional analysis which requires readers to have a minimal background in linear algebra and real analysis at the first-year graduate level. Prerequisite knowledge of general topology or Lebesgue integration is not required. The book explains the principles and applications of functional analysis and explores the development of the basic properties of normed linear, inner product spaces and continuous linear operators defined in these spaces. Though Lebesgue integral is not discussed, the book offers an in-depth knowledge on the numerous applications of the abstract results of functional analysis in differential and integral equations, Banach limits, harmonic analysis, summability and numerical integration. Also covered in the book are versions of the spectral theorem for compact, symmetric operators and continuous, self adjoint operators.

Product Details

ISBN-13:	9789814273343
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	07/14/2009
Edition description:	New Edition
Pages:	192
Product dimensions:	6.10(w) x 9.10(h) x 0.70(d)

Table of Contents

Preface v

1 Normed Linear and Banach Spaces 1

2 Linear Operators 9

3 Quotient Spaces 21

4 Finite Dimensional Normed Spaces 25

5 Inner Product and Hilbert Spaces 29

6 The Hahn-Banach Theorem 41

7 Applications of the Hahn-Banach Theorem to Normed Spaces 47

8 The Uniform Boundedness Principle 53

9 Weak Convergence 61

10 The Open Mapping and Closed Graph Theorems 69

11 Projections 73

12 Schauder Basis 77

13 Transpose and Adjoints of Continuous Linear Operators 83

14 Compact Operators 91

15 The Fredholm Alternative 97

16 The Spectrum of an Operator 103

17 Subdivisions of the Spectrum 107

18 The Spectrum of a Compact Operator 111

19 Symmetric Linear Operators 115

20 The Spectral Theorem for Compact Symmetric Operators 119

21 Symmetric Operators with Compact Inverse 129

22 Bounded Self Adjoint Operators 133

23 Orthogonal Projections 141

24 Sesquilinear Functionals 143

25 The Spectral Theorem for Bounded Self Adjoint Operators 145

26 An Operational Calculus 151

27 The Spectral Theorem for Normal Operators 159

Appendix A Functions of Bounded Variation 163

Appendix B The Riemann-Stieltjes Integral 167

Appendix C The Dual of C[a,b] 169

Appendix D The Baire Category Theorem 173

References 175

Index 179