Nonstandard analysis
type: Item
instance of: scholarly article
title: Nonstandard analysis
zbMATH Open document ID: 0797.26015
author: Dieter Landers
author: Lothar Rogge
published in: Springer-Lehrbuch
publication date: 1994-03-28T00:00:00Z
review text: The content of this book goes way beyond its title. The authors show the use of non-standard methods in analysis, topology, and stochastics, as well as develop enough of the underlying logic. The text should be successful in an advanced undergraduate or graduate course, because it presumes little background, proceeds slowly and carefully, is well- written and easy to understand, and contains a fair number of exercises with hints or solutions at the end. Still, a surprisingly large amount of material is covered. The non-standard reals are constructed as an ultrapower of the standard reals; this approach is then contrasted at the end of the text with Nelson's axiomatic development of a non-standard universe. The section on analysis begins with the most elementary properties of the hyperreals, and includes integration, differentiation, sequences of functions, and distributions. The topology part assumes no knowledge of the subject, and besides introducing point-set topology also develops metric and normed spaces. The material on stochastic processes begins with a lot of measure theory, both classical measure and Loeb measure, before culminating in Brownian motion.
Mathematics Subject Classification ID: 26E35
Mathematics Subject Classification ID: 03H05
Mathematics Subject Classification ID: 26-01
Mathematics Subject Classification ID: 28E05
Mathematics Subject Classification ID: 54J05
Mathematics Subject Classification ID: 60G99
Mathematics Subject Classification ID: 60J65
zbMATH DE Number: 542030
zbMATH Keywords: nonstandard analysis
zbMATH Keywords: point-set topology
zbMATH Keywords: stochastic processes
zbMATH Keywords: Loeb measure
zbMATH Keywords: Brownian motion
reviewed by: Robert S. Lubarsky
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 542030