Correspondences between model theory and Banach space theory

TL;DR

This paper explores deep connections between model theory and Banach space theory, linking stability properties of formulas to geometric properties of Banach spaces, and offers a model-theoretic proof of a classical functional analysis theorem.

Contribution

It establishes a correspondence between model-theoretic stability notions and Banach space properties, and provides a novel model-theoretic proof of the Eberlein-ulean-Smulian theorem.

Findings

Stable formulas correspond to reflexive Banach spaces

NIP formulas relate to Rosenthal Banach spaces

NSOP formulas connect with weakly sequentially complete spaces

Abstract

In \cite{K3} we pointed out the correspondence between a result of Shelah in model theory, i.e. a theory is unstable if and only if it has IP or SOP, and the well known compactness theorem of Eberlein and \v{S}mulian in functional analysis. In this paper, we relate a {\em natural} Banach space $V$ to a formula $\phi(x,y)$, and show that $\phi$ is stable (resp NIP, NSOP) if and only if $V$ is reflexive (resp Rosenthal, weakly sequentially complete) Banach space. Also, we present a proof of the Eberlein-\v{S}mulian theorem by a model theoretic approach using Ramsey theorems which is illustrative to show some correspondences between model theory and Banach space theory.