Correspondences between model theory and Banach space theory

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 Šmulian in functional analysis. In this paper, we relate a {\em natural} Banach space $V$ to a formula $\varphi (x,y)$, and show that $\varphi$ 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-Šmulian theorem by a model theoretic approach using Ramsey theorems which is illustrative to show some correspondences between model theory and Banach space theory.

arXiv admin note: substantial text overlap with arXiv:1509.03193

References in corpus (1)

• Stability, NIP, and NSOP; Model Theoretic Properties of Formulas via Topological Properties of Function Spaces

Cited by in corpus (2)

• ${\aleph }_0$-categorical Banach spaces contain ${\ell }_p$ or $c_0$
• Model Theory for $C_p$-theorists