Applied Nonstandard Analysis

By: MARTIN DAVIS

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Paperback | 26 July 2005

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208 Pages

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20.9 x 13.6 x 1.27

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Geared toward upper-level undergraduates and graduate students, this text explores the applications of nonstandard analysis without assuming any knowledge of mathematical logic. It develops the key techniques of nonstandard analysis at the outset from a single, powerful construction; then, beginning with a nonstandard construction of the real number system, it leads students through a nonstandard treatment of the basic topics of elementary real analysis, topological spaces, and Hilbert space.Important topics include nonstandard treatments of equicontinuity, nonmeasurable sets, and the existence of Haar measure. The focus on compact operators on a Hilbert space includes the Bernstein-Robinson theorem on invariant subspaces, which was first proved with nonstandard methods. Ever mindful of the needs of readers with little background in these subjects, the text offers a straightforward treatment that provides a strong foundation for advanced studies of analysis

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You Can Find This Book In

Non-Fiction Mathematics Calculus & Mathematical Analysis Numerical Analysis

Glossary of Special Symbols	p. xi
Introduction	p. 1
Why Nonstandard Analysis?	p. 1
Infinitesimals as Ideal Elements	p. 1
The Role of Logic	p. 2
Three Techniques	p. 2
Mathematical Logic and Rigor	p. 3
Numbering of Theorems	p. 4
Universes and Languages	p. 5
Sets and Relations	p. 5
Filters	p. 8
Individuals and Superstructures	p. 11
Universes	p. 15
Languages	p. 20
Semantics	p. 22
Los' Theorem	p. 24
Concurrence, Infinite Integers, Internal Sets	p. 33
Recapitulation	p. 41
Exercises	p. 42
Real Numbers and Hyperreal Numbers	p. 43
Ordered Fields	p. 43
Nonstandard Theory of Archimedean Fields	p. 50
The Real Numbers	p. 53
The Hyperreal Numbers	p. 55
Real Sequences and Functions	p. 56
Prolongation Theorems	p. 61
Nonstandard Differential Calculus	p. 63
Additivity	p. 67
The Existence of Nonmeasurable Sets	p. 71
Exercises	p. 74
Topological and Metric Spaces	p. 75
Topological Spaces	p. 75
Mappings and Products	p. 79
Topological Groups	p. 82
The Existence of Haar Measure	p. 84
Metric Spaces	p. 87
Uniform Convergence	p. 94
Uniform Continuity and Equicontinuity	p. 96
Compact Mappings	p. 101
Normed Linear Spaces	p. 103
Linear Spaces	p. 103
Compact Operators	p. 108
Integration of Banach Space Valued Functions	p. 111
Differential Calculus	p. 116
Hilbert Space	p. 121
Unitary Spaces	p. 121
Orthogonal Projections	p. 130
The Bernstein-Robinson Theorem	p. 140
The Spectral Theorem for Compact Hermitian Operators	p. 150
Noncompact Hermitian Operators	p. 158
Index	p. 179
Table of Contents provided by Ingram. All Rights Reserved.

Applied Nonstandard Analysis

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