  Given any Banach space $X$ and any weak*-compact subset $K$ of $X^*$, we
compute the Szlenk index of the weak*-closed, convex hull of $K$ as a function
of the Szlenk index of $K$. Also as an application, we compute the Szlenk index
of any injective tensor product of two operators. In particular, we compute the
Szlenk index of an injective tensor product $\tens$ 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.
