Infinitesimal Analysis

E. I. Gordon, A. G. Kusraev, S. S. Kutateladze (auth.)1402007388, 9781402007385

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

Table of contents :
Front Matter….Pages i-xiii
Excursus into the History of Calculus….Pages 1-9
Naive Foundations of Infinitesimal Analysis….Pages 10-34
Set-Theoretic Formalisms of Infinitesimal Analysis….Pages 35-115
Monads in General Topology….Pages 116-165
Infinitesimals and Subdifferentials….Pages 166-222
Technique of Hyperapproximation….Pages 223-280
Infinitesimals in Harmonic Analysis….Pages 281-366
Exercises and Unsolved Problems….Pages 367-379
Back Matter….Pages 380-422