The Szlenk index of $C(K,X)$

Ryan M Causey

Published 2016-04-26Version 1

Given any Banach space $X$ and any $w^*$-compact subset $K$ of $X^*$, we compute the Szlenk index of the $w^*$-closed, convex hull of $K$ as a function of the Szlenk index of $K$. As a consequence, for any compact, Hausdorff topological space $K$ and any Banach space $X$, we compute the the Szlenk index of $C(K,X)$ as a function of the Szlenk index of $X$ and the Cantor-Bendixson index of $K$. Also as an application, we compute the Szlenk index of any injective tensor product $X\hat{\otimes}_\epsilon Y$ in terms of $Sz(X)$ and $Sz(Y)$. As another application, we give a complete characterization of those ordinals which occur as the Szlenk index of a Banach space, as well as those ordinals which occur as the Bourgain $\ell_1$ or $c_0$ index of a Banach space.