Analysis now

Gert K. Pedersen9780387967882, 9783540967880, 0387967885, 3540967885

Graduate students in mathematics, who want to travel light, will find this book invaluable; impatient young researchers in other fields will enjoy it as an instant reference to the highlights of modern analysis. Starting with general topology, it moves on to normed and seminormed linear spaces. From there it gives an introduction to the general theory of operators on Hilbert space, followed by a detailed exposition of the various forms the spectral theorem may take; from Gelfand theory, via spectral measures, to maximal commutative von Neumann algebras. The book concludes with two supplementary chapters: a concise account of unbounded operators and their spectral theory, and a complete course in measure and integration theory from an advanced point of view.

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Table of contents :
Title ……Page 1
Title page ……Page 2
Date-line ……Page 3
Dedication ……Page 4
Preface ……Page 6
Contents ……Page 10
1.1. Ordered Sets ……Page 14
1.2. Topology ……Page 21
1.3. Convergence ……Page 26
1.4. Continuity ……Page 30
1.5. Separation ……Page 36
1.6. Compactness ……Page 43
1.7. Local Compactness ……Page 49
2.1. Normed Spaces ……Page 56
2.2. Category ……Page 65
2.3. Dual Spaces ……Page 69
2.4. Weak Topologies ……Page 75
2.5. $w^ast$-Compactness ……Page 82
3.1. Inner Products ……Page 92
3.2. Operators on Hilbert Space ……Page 101
3.3. Compact Operators ……Page 118
3.4. The Trace ……Page 128
Chapter 4 Spectral Theory ……Page 140
4.1. Banach Algebras ……Page 141
4.2. The Gelfand Transform ……Page 150
4.3. Function Algebras ……Page 157
4.4. The Spectral Theorem, I ……Page 169
4.5. The Spectral Theorem, II ……Page 175
4.6. Operator Algebra ……Page 184
4.7. Maximal Commutative Algebras ……Page 193
Chapter 5 Unbounded Operators ……Page 204
5.1. Domains, Extensions, and Graphs ……Page 205
5.2. The Cayley Transform ……Page 216
5.3. Unlimited Spectral Theory ……Page 222
6.1. Radon Integrals ……Page 234
6.2. Measurability ……Page 241
6.3. Measures ……Page 248
6.4. $L^p$-spaces ……Page 252
6.5. Duality Theory ……Page 260
6.6. Product Integrals ……Page 268
Bibliography ……Page 280
List of Symbols ……Page 284
Index ……Page 286