Correspondences between model theory and Banach space theory

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.