Banach spaces of bounded Szlenk index
Volume 183 / 2007
Studia Mathematica 183 (2007), 63-97
MSC: 46B20, 54H05.
DOI: 10.4064/sm183-1-4
Abstract
For a countable ordinal $\alpha$ we denote by ${\cal C}_\alpha$ the class of separable, reflexive Banach spaces whose Szlenk index and the Szlenk index of their dual are bounded by $\alpha$. We show that each ${\cal C}_\alpha$ admits a separable, reflexive universal space. We also show that spaces in the class ${\cal C}_{\omega^{\alpha\cdot\omega}}$ embed into spaces of the same class with a basis. As a consequence we deduce that each ${\cal C}_\alpha$ is analytic in the Effros–Borel structure of subspaces of $C[0,1]$.
