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Introduction

Henson, C. Ward and Iovino, José and Kechris, Alexander S. and Odell, Edward (2003) Introduction. In: Analysis and Logic. London Mathematical Society Lecture Note Series. No.262. Cambridge University Press , Cambridge, pp. 3-9. ISBN 9781107360006. https://resolver.caltech.edu/CaltechAUTHORS:20180816-160145668

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Abstract

The ideas and methods of model theory are being applied today in nearly all parts of mathematics. Here we concentrate on a framework for applications in functional analysis. Model theory has already provided several tools for the research analyst, of which the most important are: (a) the Banach space ultraproduct and nonstandard hull constructions in functional analysis; (b) spaces of (model-theoretic) types as used in the geometry of Banach spaces; and (c) Loeb measure spaces in stochastic analysis and its applications. In this paper we explain a systematic model theoretic framework within which these tools (especially (a) and (b)) are naturally situated. Our main intended audience consists of analysts who are familiar with the ultraproduct construction, and this perspective has strongly influenced our presentation of the material. We expect that many model theorists will also find something of interest in this subject. In particular, we indicate the initial steps of a program for introducing the key ideas and methods of model theory into functional analysis in a systematic and comprehensive way. Applications of model theory in functional analysis have been pursued since the mid 1960s, beginning with the introduction of Banach space ultraproducts by Bretagnolle, Dacunha-Castelle, and Krivine [BDCK66] [Kri67] [DCK72] and nonstandard hulls by Luxemburg [Lux69b]. These two constructions Avere first used at about the same time and they are essentially the same; however, these initial steps led to largely independent lines of research that have still not been fully integrated.


Item Type:Book Section
Related URLs:
URLURL TypeDescription
https://doi.org/10.1017/cbo9781107360006.003DOIArticle
Additional Information:© 2003 Cambridge University Press.
Series Name:London Mathematical Society Lecture Note Series
Issue or Number:262
Record Number:CaltechAUTHORS:20180816-160145668
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20180816-160145668
Official Citation:Henson, C., Iovino, J., Kechris, A., & Odell, E. (2003). Introduction. In C. Finet & C. Michaux (Eds.), Analysis and Logic (London Mathematical Society Lecture Note Series, pp. 3-9). Cambridge: Cambridge University Press. doi:10.1017/CBO9781107360006.003
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:88885
Collection:CaltechAUTHORS
Deposited By: George Porter
Deposited On:17 Aug 2018 14:37
Last Modified:03 Oct 2019 20:11

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