Applications of Model Theory to Functional Analysis

Jose Iovino

• Publisher: Dover Publications
• Year: 2014
• Print ISBN: 9780486780849
• eBook ISBN: 9780486798615
• Topic: Mathematics
• Subtopic: Logic in Mathematics
• Index: Mathematics

"The text is well written and easy to read. A great tool for any person interested in learning relations between functional analysis and model theory." — MathSciNet
During the last two decades, methods that originated within mathematical logic have exhibited powerful applications to Banach space theory, particularly set theory and model theory. This volume constitutes the first self-contained introduction to techniques of model theory in Banach space theory. The area of research has grown rapidly since this monograph's first appearance, but much of this material is still not readily available elsewhere. For instance, this volume offers a unified presentation of Krivine's theorem and the Krivine-Maurey theorem on stable Banach spaces, with emphasis on the connection between these results and basic model-theoretic notions such as types, indiscernible sequences, and ordinal ranks.
Suitable for advanced undergraduates and graduate students of mathematics, this exposition does not presuppose expertise in either model theory or Banach space theory. Numerous exercises and historical notes supplement the text.

CHAPTER 1

Preliminaries: Banach Space Models

1. Banach Space Structures and Banach Space Ultrapowers

A Banach space is finite dimensional if and only if the unit ball is compact, i.e., if and only if for every bounded family (x_i)i

I and every ultrafilter

on the set I, the

-limit

exists. If X is an infinite dimensional Banach space and

is an ultrafilter on a set I, there is a canonical way of expanding X to a larger Banach space

by adding for every bounded family (xi)i

I in X an element

such that

This is accomplished through the construction of Banach space ultrapower, which we now define.

Let (X_i)i

I be a family of normed spaces. Define

is naturally a vector space. An ultrafilter

on I induces a seminorm on

by defining

The set

of families (x_i) in

such that

is obviously a closed subspace of

. We define

The space

is called the

-ultraproduct of (X_i)i

I. If X_i = X for every i

I, the space

is called the

-ultrapower of X and is denoted

If (x_i) is a family in

, let us denote by

the equivalence class of (x_i) in

is an ultrapower of a normed space X, the map

where x_i = x for every i

I, is an isometric embedding of X into

Hence, we may regard X as a subspace of

This embedding is generally not surjective; it is, however, when the ultrafilter

is principal or the space X is finite dimensional.

1.1. EXERCISE. An ultrafilter

is said to be countably incomp...

Table of contents

• Cover
• Title Page
• Copyright Page
• Contents
• Chapter 0. Introduction
• Chapter 1. Preliminaries: Banach Space Models
• Chapter 2. Semidefinability of Types
• Chapter 3. Maurey Strong Types and Convolutions
• Chapter 4. Fundamental Sequences
• Chapter 5. Quantifier-Free Types Over Banach Spaces
• Chapter 6. Digression: Ramsey’s Theorem for Analysis
• Chapter 7. Spreading Models
• Chapter 8. lp- and c0-Types
• Chapter 9. Extensions of Operators by Ultrapowers
• Chapter 10. Where Does the Number p Come From?
• Chapter 11. Block Representability of lp in Types
• Chapter 12. Krivine’s Theorem
• Chapter 13. Stable Banach Spaces
• Chapter 14. Block Representability of lp in Types Over Stable Spaces
• Chapter 15. lp-Subspaces of Stable Banach Spaces
• Historical Remarks
• Bibliography
• Index of Notation
• Index