| Preface | p. vii |
| Convention and Symbols | p. xi |
| List of Theorems | p. xv |
| Nonstandard Analysis | p. 1 |
| Sets and Logic | p. 1 |
| Naive sets, first order formulas and ZFC | p. 1 |
| First order theory and consistency | p. 4 |
| Infinities, ordinals, cardinals and AC | p. 5 |
| Notes and exercises | p. 7 |
| The Nonstandard Universe | p. 9 |
| Elementary extensions and saturation | p. 9 |
| Superstructure, internal and external sets | p. 10 |
| Two principles | p. 13 |
| Internal extensions | p. 14 |
| Notes and exercises | p. 16 |
| The Ultraproduct Construction | p. 19 |
| Notes and exercises | p. 21 |
| Application: Elementary Calculus | p. 23 |
| Infinite, infinitesimals and the standard part | p. 23 |
| Overspill, underspill and limits | p. 24 |
| Infinitesimals and continuity | p. 26 |
| Notes and exercises | p. 30 |
| Application: Measure Theory | p. 33 |
| Classical measures | p. 33 |
| Internal measures and Loeb measures | p. 37 |
| Lebesgue measure, probability and liftings | p. 42 |
| Measure algebras and Kelley's Theorem | p. 48 |
| Notes and exercises | p. 51 |
| Application: Topology | p. 54 |
| Monads and topologies | p. 54 |
| Monads and separation axioms | p. 57 |
| Standard part and continuity | p. 58 |
| Robinson's characterization of compactness | p. 66 |
| The Baire Category Theorem | p. 70 |
| Stone-Cech compactification | p. 72 |
| Notes and exercises | p. 74 |
| Banach Spaces | p. 77 |
| Norms and Nonstandard Hulls | p. 77 |
| Seminormed linear spaces and quotients | p. 77 |
| Internal spaces and nonstandard hulls | p. 80 |
| Finite dimensional Banach spaces | p. 84 |
| Examples of Banach spaces | p. 86 |
| Notes and exercises | p. 91 |
| Linear Operators and Open Mappings | p. 94 |
| Bounded linear operators and dual spaces | p. 94 |
| Open mappings | p. 98 |
| Uniform boundedness | p. 104 |
| Notes and exercises | p. 105 |
| Helly's Theorem and the Hahn-Banach Theorem | p. 108 |
| Norming and Helly's Theorem | p. 108 |
| The Hahn-Banach Theorem | p. 116 |
| The Hahn-Banach Separation Theorem | p. 118 |
| Notes and exercises | p. 120 |
| General Nonstandard Hulls and Biduals | p. 121 |
| Nonstandard hulls by internal seminorms | p. 121 |
| Weak nonstandard hulls and biduals | p. 123 |
| Applications of weak nonstandard hulls | p. 127 |
| Weak compactness and separation | p. 130 |
| Weak* topology and Alaoglu's Theorem | p. 131 |
| Notes and exercises | p. 133 |
| Reflexive Spaces | p. 134 |
| Weak compactness and reflexivity | p. 134 |
| The Eberlein-Smulian Theorem | p. 136 |
| James' characterization of reflexivity | p. 141 |
| Finite representability and superreflexivity | p. 143 |
| Notes and exercises | p. 146 |
| Hilbert Spaces | p. 148 |
| Basic properties | p. 148 |
| Examples | p. 157 |
| Notes and exercises | p. 159 |
| Miscellaneous Topics | p. 161 |
| Compact operators | p. 161 |
| The Krein-Milman Theorem | p. 169 |
| Schauder bases | p. 171 |
| Schauder's Fixed Point Theorem | p. 174 |
| Notes and exercises | p. 179 |
| Banach Algebras | p. 181 |
| Normed Algebras and Nonstandard Hulls | p. 181 |
| Examples and basic properties | p. 181 |
| Spectra | p. 190 |
| Nonstandard hulls | p. 202 |
| Notes and exercises | p. 206 |
| C*-Algebras | p. 208 |
| Examples and basic properties | p. 208 |
| The Gelfand transform | p. 219 |
| The GNS construction | p. 233 |
| Notes and exercises | p. 240 |
| The Nonstandard Hull of a C*-Algebra | p. 242 |
| Basic properties | p. 242 |
| Notes and exercises | p. 248 |
| Von Neumann Algebras | p. 250 |
| Operator topologies and the bicommutant | p. 250 |
| Nonstandard hulls vs. von Neumann algebras | p. 254 |
| Weak nonstandard hulls and biduals | p. 258 |
| Notes and exercises | p. 264 |
| Some Applications of Projections | p. 267 |
| Infinite C*-algebras | p. 267 |
| P*-algebras | p. 271 |
| Notes and exercises | p. 274 |
| Selected Research Topics | p. 275 |
| Hilbert space-valued integrals | p. 275 |
| Reflexivity and fixed points | p. 285 |
| Arens product on a bidual | p. 291 |
| Noncommutative Loeb measures | p. 294 |
| Further questions and problems | p. 298 |
| Suggestions for Further Reading | p. 301 |
| Bibliography | p. 303 |
| Index | p. 309 |
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