Table of Contents
Preface vii
Convention and Symbols xi
List of Theorems xv
1 Nonstandard Analysis 1
1.1 Sets and Logic 1
1.1.1 Naive sets, first order formulas and ZFC 1
1.1.2 First order theory and consistency 4
1.1.3 Infinities, ordinals, cardinals and AC 5
1.1.4 Notes and exercises 7
1.2 The Nonstandard Universe 9
1.2.1 Elementary extensions and saturation 9
1.2.2 Superstructure, internal and external sets 10
1.2.3 Two principles 13
1.2.4 Internal extensions 14
1.2.5 Notes and exercises 16
1.3 The Ultraproduct Construction 19
1.3.1 Notes and exercises 21
1.4 Application: Elementary Calculus 23
1.4.1 Infinite, infinitesimals and the standard part 23
1.4.2 Overspill, underspill and limits 24
1.4.3 Infinitesimals and continuity 26
1.4.4 Notes and exercises 30
1.5 Application: Measure Theory 33
1.5.1 Classical measures 33
1.5.2 Internal measures and Loeb measures 37
1.5.3 Lebesgue measure, probability and liftings 42
1.5.4 Measure algebras and Kelley's Theorem 48
1.5.5 Notes and exercises 51
1.6 Application: Topology 54
1.6.1 Monads and topologies 54
1.6.2 Monads and separation axioms 57
1.6.3 Standard part and continuity 58
1.6.4 Robinson's characterization of compactness 66
1.6.5 The Baire Category Theorem 70
1.6.6 Stone-Cech compactification 72
1.6.7 Notes and exercises 74
2 Banach Spaces 77
2.1 Norms and Nonstandard Hulls 77
2.1.1 Seminormed linear spaces and quotients 77
2.1.2 Internal spaces and nonstandard hulls 80
2.1.3 Finite dimensional Banach spaces 84
2.1.4 Examples of Banach spaces 86
2.1.5 Notes and exercises 91
2.2 Linear Operators and Open Mappings 94
2.2.1 Bounded linear operators and dual spaces 94
2.2.2 Open mappings 98
2.2.3 Uniform boundedness 104
2.2.4 Notes and exercises 105
2.3 Helly's Theorem and the Hahn-Banach Theorem 108
2.3.1 Norming and Helly's Theorem 108
2.3.2 The Hahn-Banach Theorem 116
2.3.3 The Hahn-Banach Separation Theorem 118
2.3.4 Notes and exercises 120
2.4 General Nonstandard Hulls and Biduals 121
2.4.1 Nonstandard hulls by internal seminorms 121
2.4.2 Weak nonstandard hulls and biduals 123
2.4.3 Applications of weak nonstandard hulls 127
2.4.4 Weak compactness and separation 130
2.4.5 Weak* topology and Alaoglu's Theorem 131
2.4.6 Notes and exercises 133
2.5 Reflexive Spaces 134
2.5.1 Weak compactness and reflexivity 134
2.5.2 The Eberlein-Smulian Theorem 136
2.5.3 James' characterization of reflexivity 141
2.5.4 Finite representability and superreflexivity 143
2.5.5 Notes and exercises 146
2.6 Hilbert Spaces 148
2.6.1 Basic properties 148
2.6.2 Examples 157
2.6.3 Notes and exercises 159
2.7 Miscellaneous Topics 161
2.7.1 Compact operators 161
2.7.2 The Krein-Milman Theorem 169
2.7.3 Schauder bases 171
2.7.4 Schauder's Fixed Point Theorem 174
2.7.5 Notes and exercises 179
3 Banach Algebras 181
3.1 Normed Algebras and Nonstandard Hulls 181
3.1.1 Examples and basic properties 181
3.1.2 Spectra 190
3.1.3 Nonstandard hulls 202
3.1.4 Notes and exercises 206
3.2 C*-Algebras 208
3.2.1 Examples and basic properties 208
3.2.2 The Gelfand transform 219
3.2.3 The GNS construction 233
3.2.4 Notes and exercises 240
3.3 The Nonstandard Hull of a C*-Algebra 242
3.3.1 Basic properties 242
3.3.2 Notes and exercises 248
3.4 Von Neumann Algebras 250
3.4.1 Operator topologies and the bicommutant 250
3.4.2 Nonstandard hulls vs. von Neumann algebras 254
3.4.3 Weak nonstandard hulls and biduals 258
3.4.4 Notes and exercises 264
3.5 Some Applications of Projections 267
3.5.1 Infinite C*-algebras 267
3.5.2 P*-algebras 271
3.5.3 Notes and exercises 274
4 Selected Research Topics 275
4.1 Hilbert space-valued integrals 275
4.2 Reflexivity and fixed points 285
4.3 Arens product on a bidual 291
4.4 Noncommutative Loeb measures 294
4.5 Further questions and problems 298
Suggestions for Further Reading 301
Bibliography 303
Index 309
