AN INVITATION TO NONSTANDARD ANALYSIS

Summary

INTRODUCTION

Nonstandard Analysis – or the Theory of Infinitesimals as some prefer to call it – is now a little more than 25 years old (see Robinson (1961)). In its early days it was often presented as a surprising solution to the old and – it had seemed – impossible problem of providing infinitesimal methods in analysis with a logical foundation. It soon became clear, however, that the theory was much more than just a reformulation of the Calculus, when Bernstein and Robinson (1966) gave the first indication of its powers as a research tool by proving that all polynomially compact operators on Hilbert spaces have nontrivial invariant subspaces. Since then nonstandard techniques have been used to obtain new results in such diverse fields as Banach spaces, differential equations, probability theory, algebraic number theory, economics, and mathematical physics just to mention a few. Despite the wide variety of topics involved, these applications have enough themes in common that it is natural to regard them as examples of the same general method.

This paper is intended as an exposition of these recurrent themes and the theory uniting them. I have called it “An invititation to nonstandard analysis” because it is meant as an invitation – a friendly welcome requiring no other background than a smattering of general mathematical culture. My point of view is that of applied nonstandard analysis; I'm interested in the theory as a tool for studying and creating standard mathematical structures.