Nonstandard Analysis

✍ Scribed by Martin Väth

• Publisher: Birkhäuser Verlag
• Year: 2007
• Tongue: English
• Leaves: 262
• Edition: 1
• Category: Library

✦ Synopsis

This book presents an introduction into Robinson's nonstandard analysis. Nonstandard analysis is the application of model theory in analysis. However, the reader is not expected to have any background in model theory; instead, some background in analysis, topology, or functional analysis would be useful - although the book is as much self-contained as possible and can be understood after a basic calculus course. Unlike some other texts, it does not attempt to teach elementary calculus on the basis of nonstandard analysis, but it points to some applications in more advanced analysis. Such applications can hardly be obtained by standard methods such as a deeper investigation of Hahn-Banach limits or of finitely additive measures.

✦ Table of Contents

Cover......Page 1
Title Page......Page 5
Contents......Page 7
Preface......Page 9
1.1 General Remarks......Page 11
1.2 Archimedean Fields and Infinitesimals......Page 14
2.1 Superstructures......Page 23
2.2 Formal Language......Page 27
2.3 Interpretations......Page 29
3.1 Elementary Embeddings and the Transfer Principle......Page 33
3.2 The Standard Definition Principle......Page 38
3.3 The Internal Definition Principle......Page 44
3.4 Existence of External Sets......Page 50
4.1 Ultrafilters......Page 54
4.2 Ultrapowers......Page 58
4.3 Embedding in a Superstructure......Page 61
5.1 Hyperreal and Hypernatural Numbers......Page 69
5.2 Interpretation of the Standard Part Homomorphism......Page 80
§6 The Permanence Principle and -finite Sets......Page 84
7.1 Sequences......Page 95
7.2 Sets......Page 100
7.3 Functions......Page 102
§8 Enlargements, Saturation, and Concurrency......Page 113
§9 Saturated Models......Page 123
9.1 Models for Enlargements......Page 124
9.2 Compact Enlargements......Page 125
9.3 Polysaturated Models......Page 132
10.1 Linear Functionals and Operators......Page 135
10.2 Hahn-Banach and Banach-Mazur Limits......Page 139
§11 Additive Measures......Page 150
12.1 Topological Spaces......Page 159
12.2 Filters in Nonstandard Analysis......Page 160
12.3 Topologies in Nonstandard Analysis......Page 164
12.4 Functions in Nonstandard Topology......Page 177
13.1 Uniform Spaces......Page 180
13.2 Nonstandard Hulls......Page 188
§14 Topological Vector Spaces......Page 195
§15 Loeb Measures......Page 207
§16 Distributions......Page 215
A Some Important -Values......Page 221
B Solutions to the Exercises......Page 227
Bibliography......Page 251
Index......Page 255