Banach spaces of bounded Szlenk index

TL;DR

This paper investigates classes of separable, reflexive Banach spaces with bounded Szlenk index, establishing the existence of universal spaces and embedding properties, and analyzing their descriptive set-theoretic complexity.

Contribution

It introduces universal spaces for classes with bounded Szlenk index and shows embedding results and descriptive set-theoretic properties of these classes.

Findings

Existence of separable, reflexive universal spaces for each class C_a.

Spaces in C_{omega^{a*omega}} embed into spaces with a basis within the same class.

Classes C_a are analytic in the Effros-Borel structure of subspaces of C[0,1].

Abstract

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].