On the Szlenk index and the weak∗-dentability index

ABSTRACT. We prove that if the Szlenk index Sz(X) and the weak*-dentability index δ∗(X) of a Banach space X are countable, then they are determined by the closed separable linear subspaces of X. From this we deduce the existence of an absolute function ψ from ω1 to ω1 (first uncountable ordinal) such that δ∗(X) is bounded above by ψ(Sz(X)), and that the condition Sz(X) < ω1 yields the existence of an equivalent norm on X whose dual norm is locally uniformly convex. As an other application, we compute Sz(C(K)), where K is a scattered compact space with K(ω1) = ∅. Finally we solve the three space problem for the condition Sz(X) < ω1. (*) This research has been supported by a grant “Lavoisier ” from the french “Ministère des Affaires Etrangères”. 1. INTRODUCTION. Let X be a Banach space. We will first define the two ordinal indices δ∗(X) and Sz(X). Weak*-dentability index, δ∗(X): Let F be a closed bounded subset of X∗. For ε> 0, F ′ε = {x ∗ ∈ F such that any weak*-slice of F containing x ∗ is of diameter> ε}