**Graduate Studies in Mathematics**

Volume: 66;
2004;
322 pp;
Hardcover

MSC: Primary 46; 47;

Print ISBN: 978-0-8218-3646-0

Product Code: GSM/66

List Price: $65.00

Individual Member Price: $52.00

**Electronic ISBN: 978-1-4704-1150-3
Product Code: GSM/66.E**

List Price: $65.00

Individual Member Price: $52.00

#### Supplemental Materials

# Functional Analysis: An Introduction

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*Yuli Eidelman; Vitali Milman; Antonis Tsolomitis*

This textbook provides an introduction to the methods and language of functional analysis, including Hilbert spaces, Fredholm theory for compact operators, and spectral theory of self-adjoint operators. It also presents the basic theorems and methods of abstract functional analysis and a few applications of these methods to Banach algebras and the theory of unbounded self-adjoint operators.

The text corresponds to material for two semester courses (Part I and Part II, respectively) and is essentially self-contained. Prerequisites for the first part are minimal amounts of linear algebra and calculus. For the second part, some knowledge of topology and measure theory is recommended. Each of the 11 chapters is followed by numerous exercises, with solutions given at the end of the book.

The text is ideal for a one-year course. It will also provide a sound basis for further study. It is suitable for graduate students and researchers interested in operator theory and functional analysis.

#### Table of Contents

# Table of Contents

## Functional Analysis: An Introduction

- Cover Cover11 free
- Title iii4 free
- Copyright iv5 free
- Contents v6 free
- Preface xi12 free
- Introduction xiii14 free
- Part I. Hilbert Spaces and Basic Operator Theory 118 free
- 1 Linear spaces; normed spaces; first examples 320
- 2 Hilbert spaces 2542
- 3 The dual space 4764
- 4 Bounded linear operators 5572
- 5 Spectrum. Fredholm theory of compact operators 7592
- 6 Self-ad joint operators 87104
- 7 Functions of operators; spectral decomposition 105122

- Part II. Basics of Functional Analysis 119136
- 8 Spectral theory of unitary operators 121138
- 9 The fundamental theorems and the basic methods 127144
- 9.1 Auxiliary results 128145
- 9.2 The Banach open mapping theorem 131148
- 9.3 The closed graph theorem 132149
- 9.4 The Banach-Steinhaus theorem 135152
- 9.5 Bases in Banach spaces 139156
- 9.6 Linear functionals; the Hahn-Banach theorem 142159
- 9.7 Separation of convex sets 146163
- 9.8 The Eberlain-Schmulian theorem 152169
- 9.9 Extremal points; the Krein-Milman theorem 153170
- 9.10 Exercises 159176

- 10 Banach algebras 167184
- 10.1 Preliminaries 167184
- 10.2 Gelfand's theorem on maximal ideals 171188
- 10.3 Analytic functions 172189
- 10.4 Gelfand map; the space of maximal ideals 176193
- 10.5 Radicals 181198
- 10.6 Involutions; the Gelfand-Naimark theorem 184201
- 10.7 Application to spectral theory 189206
- 10.8 Application to a generalized limit and combinatorics 193210
- 10.9 Exercises 196213

- 11 Unbounded self-adjoint and symmetric operators in H 203220
- 11.1 Basic notions and examples 203220
- 11.2 More properties of symmetric operators 210227
- 11.3 The spectrum σ(A) 211228
- 11.4 Elements of the "graph method" 215232
- 11.5 Cayley transform; spectral decomposition 216233
- 11.6 Symmetric and self-adjoint extensions of a symmetric operator 221238
- 11.7 Exercises 225242

- A: Solutions to exercises 227244
- A.1 Solutions to the exercises of Chapter 1 227244
- A.2 Solutions to the exercises of Chapter 2 235252
- A.3 Solutions to the exercises of Chapter 3 250267
- A.4 Solutions to the exercises of Chapter 4 254271
- A.5 Solutions to the exercises of Chapter 5 263280
- A.6 Solutions to the exercises of Chapter 6 270287
- A.7 Solutions to the exercises of Chapter 7 277294
- A.8 Solutions to the exercises of Chapter 8 279296
- A.9 Solutions to the exercises of Chapter 9 282299
- A.10 Solutions to the exercises of Chapter 10 296313
- A.11 Solutions to the exercises of Chapter 11 309326

- Bibliography 311328
- Symbols index 313330
- Subject index 317334
- Back Cover Back Cover1344

#### Readership

Graduate students and research mathematicians interested in operator theory and functional analysis.

#### Reviews

This book contains a wealth of material. Each chapter concludes with a comprehensive set of exercises that serve to illustrate the theory. Solutions to the exercises are given in the final section.

-- Mathematical Reviews

This is a gentle introduction to functional analysis that is clearly written and comes with detailed, elegant and effective proofs and well-chosen examples. ... This book is written with great care and with much sympathy to the reader. It is pleasant to read... It is simply a good book to learn the foundations of functional analysis.

-- Zentralblatt MATH

Each chapter includes exercises, in total 195 of the them, all provided with solutions at the end of the book. The text is as self-contained as possible... The authors have taken special care to be brief and not to overload the students with the enormous amount of information available on the subject.

-- EMS Newsletter