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| Glossary of Special Symbols xi Introduction 1 Why Nonstandard Analysis? 2 Infinitesimals as Ideal Elements 3 The Role of Logic 4 Three Techniques 5 Mathematical Logic and Rigor 6 Numbering of Theorems Chapter I Universes and Languages 1 Sets and Relations 2 Filters 3 Individuals and Superstructures 4 Universes 5 Languages 6 Semantics 7 J~os' Theorem 8 Concurrence Infinite Integers Internal Sets 9 Recapitulation Exercises Chapter 2 Real Numbers and Hyperreal Numbers 1 Ordered Fields 2 Nonstandard Theory of Archimedean Fields 3 The Real Numbers 4 The Hyperreal Numbers. 5 Real Sequences and Functions 6 Prolongation Theorems 61 7 Nonstandard Differential Calculus 8 Additivity 9 The Existence of Nonmeasurable Sets Exercises Chapter 3 Topological and Metric Spaces 1 Topological Spaces 2 Mappings and Products 3 Topological Groups 4 The Existence of Haar Measure 5 Metric Spaces 87 6 Uniform Convergence 7 Uniform Continuity and Equicontinuity 8 Compact Mappings Chapter 4 Normed Linear Spaces 1 Linear Spaces 2 Compact Operators 3 Integration of Banach Space Valued Functions 4 Differential Calculus Chapter 5 Hilbert Space 1 Unitary Spaces 2 Orthogonal Projections 3 The Bernstein-Robinson Theorem 4 The Spectral Theorem for Compact Hermitian Operators. 5 Noncompact Hermitian Operators Index Errata |
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