Nonstandard Analysis

By: Martin Andreas VÃ¤th

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Hardcover | 23 October 2006

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264 Pages

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23.5 x 17.78 x 1.91

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This book presents an introduction into Robinson's nonstandard analysis, which is an application of model theory in analysis. Unlike some texts, it does not attempt to teach elementary calculus on the basis of nonstandard analysis, but points to some applications in more advanced analysis. These can hardly be obtained by standard methods such as a deeper investigation of Hahn-Banach limits or of finitely additive measures. The contents move from a discussion of the preliminaries to Nonstandard Models; Nonstandard Real Analysis; Enlargements and Saturated Models; Functionals, Generalized Limits, and Additive Measures; and finally Nonstandard Topology and Functional Analysis. The reader is not expected to have any background in model theory; some background in analysis, topology, or functional analysis is useful - although the book is as much self-contained as possible and can be understood after a basic calculus course.

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You Can Find This Book In

Non-Fiction Mathematics Calculus & Mathematical Analysis

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Non-Fiction Mathematics Calculus & Mathematical Analysis

Preface	p. vii
Preliminaries	p. 1
Introduction	p. 1
General Remarks	p. 1
Archimedean Fields and Infinitesimals	p. 4
Superstructures, Sentences, and Interpretations	p. 13
Superstructures	p. 13
Formal Language	p. 17
Interpretations	p. 19
Nonstandard Models	p. 23
The Three Fundamental Principles	p. 23
Elementary Embeddings and the Transfer Principle	p. 23
The Standard Definition Principle	p. 28
The Internal Definition Principle	p. 34
Existence of External Sets	p. 40
Nonstandard Ultrapower Models	p. 44
Ultrafilters	p. 44
Ultrapowers	p. 48
Embedding in a Superstructure	p. 51
Nonstandard Real Analysis	p. 59
Hyperreal Numbers	p. 59
Hyperreal and Hypernatural Numbers	p. 59
Interpretation of the Standard Part Homomorphism	p. 70
The Permanence Principle and *-finite Sets	p. 74
Calculus	p. 85
Sequences	p. 85
Sets	p. 90
Functions	p. 92
Enlargements and Saturated Models	p. 103
Enlargements, Saturation, and Concurrency	p. 103
Saturated Models	p. 113
Models for Enlargements	p. 114
Compact Enlargements	p. 115
Polysaturated Models	p. 122
Functionals, Generalized Limits, and Additive Measures	p. 125
Normed Spaces	p. 125
Linear Functionals and Operators	p. 125
Hahn-Banach and Banach-Mazur Limits	p. 129
Additive Measures	p. 140
Nonstandard Topology and Functional Analysis	p. 149
Topologies and Filters	p. 149
Topological Spaces	p. 149
Filters in Nonstandard Analysis	p. 150
Topologies in Nonstandard Analysis	p. 154
Functions in Nonstandard Topology	p. 167
Uniform Structures	p. 170
Uniform Spaces	p. 170
Nonstandard Hulls	p. 178
Topological Vector Spaces	p. 185
Miscellaneous	p. 197
Loeb Measures	p. 197
Distributions	p. 205
Some Important *-Values	p. 211
Solutions to the Exercises	p. 217
Bibliography	p. 241
Index	p. 245
Table of Contents provided by Ingram. All Rights Reserved.

Nonstandard Analysis

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