Details

Autor(en) / Beteiligte

• Loeb, Peter A[Herausgeber]
• Wolff, Manfred P.H[Herausgeber]

Titel

Nonstandard Analysis for the Working Mathematician [electronic resource]

Auflage

1st ed. 2000

Ort / Verlag

Dordrecht : Springer Netherlands

Erscheinungsjahr

2000

Link zum Volltext

• SpringerLink Books - AutoHoldings

Beschreibungen/Notizen

• Bibliographic Level Mode of Issuance: Monograph
• I An Introduction to Nonstandard Analysis by Peter A. Loeb -- 1 A Simple Introduction to Nonstandard Analysis -- 2 An Introduction to General Nonstandard Analysis -- 3 Topology and Measure Theory -- II Functional Analysis by Manfred P. H. Wolff -- 4 Functional Analysis -- III Measure and Probability Theory and Applications by Horst Osswald and Yeneng Sun -- 5 Measure Theory and Integration -- 6 Probability Theory -- 7 Conventional Operations on Nonstandard Constructions -- IV Economics and Nonstandard Analysis by Yeneng Sun -- 8 Nonstandard Analysis in Mathematical Economics.
• This book is addressed to mathematicians working in analysis and its applications. The aim is to provide an understandable introduction to the basic theory of nonstandard analysis in Part I, and then to illuminate some of its most striking applications. Much of the book, in particular Part I, can be used in a graduate course; problems are posed in all chapters. After Part I, each chapter takes up a different field for the application of nonstandard analysis, beginning with a gentle introduction that even non-experts can read with profit. The remainder of each chapter is then addressed to experts, showing how to use nonstandard analysis in the search for solutions of open problems and how to obtain rich new structures that produce deep insight into the field under consideration. The applications discussed here are in functional analysis including operator theory, probability theory including stochastic processes, and economics including game theory and financial mathematics. In all of these areas, the intuitive notion of an infinitely small or infinitely large quantity plays an essential and helpful role in the creative process. For example, Brownian motion is often thought of as a random walk with infinitesimal increments; the spectrum of a selfadjoint operator is viewed as the set of "almost eigenvalues"; an ideal economy consists of an infinite number of agents each having an infinitesimal influence on the economy. Already at the level of calculus, one often views the integral as an infinitely large sum of infinitesimal quantities.
• English
• Description based on publisher supplied metadata and other sources.