Nonstandard methods in geometric functional analysis
1992, American Mathematical Society Translations: Series 2
https://doi.org/10.1090/TRANS2/151/05Abstract
Nonstandard methods in the modern sense consist of the explicit or implicit appeal to two different models of set theory-"standard" and "nonstandard"-1s investigate concrete mathematical objects and problems. The main development of such methods dates to the last thirty years, and they have now crystallized in several directions (see 1291, l42l and the bibliography cited there). The main directions are now known as infinitesimal and Boolean analysis. In this paper we shall outline new applications of nonstandard methods to problems arising in the area of our personal interests, grouped together under the general heading of geometric functional analysis [48]; we shall also point out some promising directions of further research. $1. Infinitesimal analysis 1.1. Infinitesimal analysis, following its creator A. Robinson, is frequently referred to by the expressive but rather unfortunate phrase "nonstandard analysis"; nowadays one most frequently speaks of classical or Robinsonian nonstandard analysis. Infinitesimal analysis is characterized by the use of certain conceptions, long familiar in the practice of natural sciences but frowned upon in twentieth-century mathematics, involving the notions of actual infinitely large and infinitely small quantities.
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