Definable Types Over Banach Spaces

Abstract

We study connections between asymptotic structure in a Banach space and model theoretic properties of the space. We show that, in an asymptotic sense, a sequence $(x_n)$ in a Banach space X generates copies of one of the classical sequence spaces $\ell_p$ or $c_0$ inside X (almost isometrically) if and only if the quantifier-free types approximated by $(x_n)$ inside X are quantifier-free definable. More precisely, if $(x_n)$ is a bounded sequence X such that no normalized sequence of blocks of $(x_n)$ converges, then the following two conditions are equivalent. (1) There exists a sequence $(y_n)$ of blocks of $(x_n)$ such that for every finite dimensional subspace E of X, every quantifier-free type over $E +\overline{\rm span}\{y_n\mid n\in \mathbb{N}\}$ is quantifier-free definable. (2) One of the following two conditions holds: (a) there exists $1\le p< \infty$ such that for every $\epsilon>0$ and every finite dimensional subspace E of X there exists a sequence of blocks of $(x_n)$ which is $(1+\epsilon)$equivalent over E to the standard unit basis of $\ell_p$; (b) for every $\epsilon>0$ and every finite dimensional subspace E of X there exists a sequence of blocks of $(x_n)$ which is $(1+\epsilon)$-equivalent over E to the standard unit basis of $c_0$. Several byproducts of the proof are analyzed.