Infinitesimal Analysis

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Springer Science & Business Media, Jun 30, 2002 - Mathematics - 422 pages
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Infinitesimal analysis, once a synonym for calculus, is now viewed as a technique for studying the properties of an arbitrary mathematical object by discriminating between its standard and nonstandard constituents. Resurrected by A. Robinson in the early 1960's with the epithet 'nonstandard', infinitesimal analysis not only has revived the methods of infinitely small and infinitely large quantities, which go back to the very beginning of calculus, but also has suggested many powerful tools for research in every branch of modern mathematics.

The book sets forth the basics of the theory, as well as the most recent applications in, for example, functional analysis, optimization, and harmonic analysis. The concentric style of exposition enables this work to serve as an elementary introduction to one of the most promising mathematical technologies, while revealing up-to-date methods of monadology and hyperapproximation.

This is a companion volume to the earlier works on nonstandard methods of analysis by A.G. Kusraev and S.S. Kutateladze (1999), ISBN 0-7923-5921-6 and Nonstandard Analysis and Vector Lattices edited by S.S. Kutateladze (2000), ISBN 0-7923-6619-0

  

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Contents

Excursus into the History of Calculus
1
11 G W Leibniz and I Newton
2
12 L Euler
5
14 J DAlembert and L Carnot
6
15 B Bolzano A Cauchy and K Weierstrass
7
17 A Robinson
9
Naive Foundations of Infinitesimal Analysis
10
22 Preliminaries on Standard and Nonstandard Reals
16
Infinitesimals and Subdifferentials
166
52 Classical Approximating and Regularizing Cones
170
53 Kuratowski and Rockafellar Limits
180
54 Approximation Given a Set of Infinitesimals
189
55 Approximation to Composites
199
56 Infinitesimal Subdifferentials
204
57 Infinitesimal Optimality
219
Technique of Hyperapproximation
223

23 Basics of Calculus on the Real Axis
23
SetTheoretic Formalisms of Infinitesimal Analysis
35
31 The Language of Set Theory
37
32 ZermeloFraenkel Set Theory
47
33 Nelson Internal Set Theory
64
34 External Set Theories
72
35 Credenda of Infinitesimal Analysis
80
36 Von NeumannGodelBernays Theory
85
37 Nonstandard Class Theory
94
38 Consistency of NCT
101
39 Relative Internal Set Theory
106
Monads in General Topology
116
42 Monads and Topological Spaces
123
Nearstandardness and Compactness
126
44 Infinite Proximity in Uniform Space
129
45 Prenearstandardness Compactness and Total Boundedness
133
46 Relative Monads
140
47 Compactness and Subcontinuity
148
48 Cyclic and Extensional Filters
151
49 Essential and Proideal Points of Cyclic Monads
156
410 Descending Compact and Precompact Spaces
159
411 Proultrafilters and Extensional Filters
160
61 Nonstandard Hulls
224
62 Discrete Approximation in Banach Space
233
63 Loeb Measure
242
64 Hyperapproximation of Measure Space
252
65 Hyperapproximation of Integral Operators
262
Infinitesimals in Harmonic Analysis
281
72 A Nonstandard Hull of a Hyperfinite Group
294
73 The Case of a Compact Nonstandard Hull
307
74 Hyperapproximation of Locally Compact Abelian Groups
316
75 Examples of Hyperapproximation
327
76 Discrete Approximation of Function Spaces on a Locally Compact Abelian Group
340
77 Hyperapproximation of Pseudodifferential Operators
355
Exercises and Unsolved Problems
367
82 Hyperapproximation and Spectral Theory
369
83 Combining Nonstandard Methods
371
84 Convex Analysis and Extremal Problems
374
85 Miscellany
376
Appendix
380
References
385
Notation Index
414
Subject Index
417
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