Nonstandard Analysis 1996/97
(Introduktionskurs i Ickestandardanalys för forskarstuderande
och avancerade grundnivåstuderande. Läsperiod II 1996/97.)
Background
In 1960 A Robinson gave a rigorous foundation for the use of
infinitesimals in analysis. But his invention - Nonstandard
Analysis - is much more. It is a powerful new tool for mathematical
research, which has led to many new insights into traditional
mathematics, and to solutions of unsolved problems in areas as
diverse as functional analysis, probability theory, complex function
theory, potential theory, mathematical physics, and mathematical
economics. The aim of the course is to make Robinson's discovery
available to students with a solid background in undergraduate
mathematics.
Contents
Ultrapower model of the hyperreal numbers with applications to
calculus and more advanced real analysis including differential equations.
Higher order models appropriate to discuss sets of sets, sets of functions
etc., including the notions of saturation, internal and external sets.
Applications to a topological context: nonstandard hulls, compactness, and
metric spaces, normed spaces, Hilbert spaces. Applications to nonstandard
measure theory.
Tentative text
A E Hurd, P A Loeb: An introduction to nonstandard real analysis
Instructor
Leif Arkeryd
<arkeryd@math.chalmers.se>
Lars Alexandersson <larsa@math.chalmers.se>
Last modified May 3, 1996