Banach spaces of bounded Szlenk index

For a countable ordinal a we denote by C_a the class of separable, reflexive Banach spaces whose Szlenk index and the Szlenk index of their dual are bounded by a. We show that each C_a admits a separable, reflexive universal space. We also show that spaces in the class C_{omega^{a*omega}} embed into spaces of the same class with a basis. As a consequence we deduce that each C_a is analytic in the Effros-Borel structure of subspaces of C[0,1].