Contents -- Preface -- Convention and Symbols -- List of Theorems -- 1. Nonstandard Analysis -- 1.1 Sets and Logic -- 1.1.1 Naive sets, first order formulas and ZFC -- 1.1.2 First order theory and consistency -- 1.1.3 Infinities, ordinals, cardinals and AC -- 1.1.4 Notes and exercises -- Exercises -- 1.2 The Nonstandard Universe -- 1.2.1 Elementary extensions and saturation -- 1.2.2 Superstructure, internal and external sets -- 1.2.3 Two principles -- 1.2.4 Internal extensions -- 1.2.5 Notes and exercises -- Exercises -- 1.3 The Ultraproduct Construction -- 1.3.1 Notes and exercises -- Exercises -- 1.4 Application: Elementary Calculus -- 1.4.1 Infinite, infinitesimals and the standard part -- 1.4.2 Overspill, underspill and limits -- 1.4.3 Infinitesimals and continuity -- 1.4.4 Notes and exercises -- Exercises -- 1.5 Application: Measure Theory -- 1.5.1 Classical measures -- 1.5.2 Internal measures and Loeb measures -- 1.5.3 Lebesgue measure, probability and liftings -- 1.5.4 Measure algebras and Kelley's Theorem -- 1.5.5 Notes and exercises -- Exercises -- 1.6 Application: Topology -- 1.6.1 Monads and topologies -- 1.6.2 Monads and separation axioms -- 1.6.3 Standard part and continuity -- 1.6.4 Robinson's characterization of compactness -- 1.6.5 The Baire Category Theorem -- 1.6.6 Stone- Cech compactification -- 1.6.7 Notes and exercises -- Exercises -- 2. Banach Spaces -- 2.1 Norms and Nonstandard Hulls -- 2.1.1 Seminormed linear spaces and quotients -- 2.1.2 Internal spaces and nonstandard hulls -- 2.1.3 Finite dimensional Banach spaces -- 2.1.4 Examples of Banach spaces -- 2.1.5 Notes and exercises -- Exercises -- 2.2 Linear Operators and Open Mappings -- 2.2.1 Bounded linear operators and dual spaces -- 2.2.2 Open mappings -- 2.2.3 Uniform boundedness -- 2.2.4 Notes and exercises -- Exercises.

2.3 Helly's Theorem and the Hahn-Banach Theorem -- 2.3.1 Norming and Helly's Theorem -- 2.3.2 The Hahn-Banach Theorem -- 2.3.3 The Hahn-Banach Separation Theorem -- 2.3.4 Notes and exercises -- Exercises -- 2.4 General Nonstandard Hulls and Biduals -- 2.4.1 Nonstandard hulls by internal seminorms -- 2.4.2 Weak nonstandard hulls and biduals -- 2.4.3 Applications of weak nonstandard hulls -- 2.4.4 Weak compactness and separation -- 2.4.5 Weak* topology and Alaoglu's Theorem -- 2.4.6 Notes and exercises -- Exercises -- 2.5 Reexive Spaces -- 2.5.1 Weak compactness and reflexivity -- 2.5.2 The Eberlein-Smulian Theorem -- 2.5.3 James' characterization of reflexivity -- 2.5.4 Finite representability and superreflexivity -- 2.5.5 Notes and exercises -- Exercises -- 2.6 Hilbert Spaces -- 2.6.1 Basic properties -- 2.6.2 Examples -- 2.6.3 Notes and exercises -- Exercises -- 2.7 Miscellaneous Topics -- 2.7.1 Compact operators -- 2.7.2 The Krein-Milman Theorem -- 2.7.3 Schauder bases -- 2.7.4 Schauder's Fixed Point Theorem -- 2.7.5 Notes and exercises -- Exercises -- 3. Banach Algebras -- 3.1 Normed Algebras and Nonstandard Hulls -- 3.1.1 Examples and basic properties -- 3.1.2 Spectra -- 3.1.3 Nonstandard hulls -- 3.1.4 Notes and exercises -- Exercises -- 3.2 C*-Algebras -- 3.2.1 Examples and basic properties -- 3.2.2 The Gelfand transform -- 3.2.3 The GNS construction -- 3.2.4 Notes and exercises -- Exercises -- 3.3 The Nonstandard Hull of a C*-Algebra -- 3.3.1 Basic properties -- 3.3.2 Notes and exercises -- Exercises -- 3.4 Von Neumann Algebras -- 3.4.1 Operator topologies and the bicommutant -- 3.4.2 Nonstandard hulls vs. von Neumann algebras -- 3.4.3 Weak nonstandard hulls and biduals -- 3.4.4 Notes and exercises -- Exercises -- 3.5 Some Applications of Projections -- 3.5.1 Infinite C*-algebras -- 3.5.2 P*-algebras -- 3.5.3 Notes and exercises -- Exercises.

4. Selected Research Topics -- 4.1 Hilbert space-valued integrals -- 4.2 Reflexivity and fixed points -- 4.3 Arens product on a bidual -- 4.4 Noncommutative Loeb measures -- 4.5 Further questions and problems -- Suggestions for Further Reading -- Bibliography -- Index.