University of Illinois Urbana-Champaign

Stable theories in functional analysis

Iovino, Jose Nicolas

This item is only available for download by members of the University of Illinois community. Students, faculty, and staff at the U of I may log in with your NetID and password to view the item. If you are trying to access an Illinois-restricted dissertation or thesis, you can request a copy through your library's Inter-Library Loan office or purchase a copy directly from ProQuest.

Permalink

https://hdl.handle.net/2142/19193

Description

Title
Stable theories in functional analysis
Author(s)
Iovino, Jose Nicolas
Issue Date
1994
Doctoral Committee Chair(s)
van den Dries, Lou
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
Language
eng
Abstract
  • Model theory is the logical analysis of mathematical structures. The class of structures considered in model theory includes, among others, all structures from algebra, number theory, and finite-dimensional analysis. A limitation, however, is that this class does not include the families of structures that are studied functional analysis, e.g., Banach spaces, or operator algebras. This limitation lies in the fact that the discourse in analysis is carried out almost entirely in higher order logic, and the traditional logic of model theory is first order logic. Higher order logics do not have a powerful model theory.
  • Recently, C. W. Henson has developed a model-theoretical apparatus that extends model theory to include important classes of structures from functional analysis preserving the desirable characteristics of first-order logic. One has, for instance, a compactness theorem and a Lowenheim-Skolem theorems for Banach spaces.
  • Some of the most remarkable results of Banach space theory in the last two decades have been proved by borrowing ideas from model theory (e.g., ultrapowers, stability). The apparatus described above provides a foundation for such contributions.
  • We have developed Henson's framework further, and introduced the fundamentals of classification theory (or stability theory) for structures based on Banach spaces.
Type of Resource
text
Permalink
http://hdl.handle.net/2142/19193
Copyright and License Information
Copyright 1994 Iovino, Jose Nicolas

Owning Collections

Graduate Dissertations and Theses at Illinois PRIMARY
Graduate Theses and Dissertations at Illinois
Dissertations and Theses - Mathematics