Banach Spaces of Bounded Szlenk Index II

TL;DR

This paper constructs specific separable Banach spaces with prescribed Szlenk indices and demonstrates their universality for spaces with lower Szlenk bounds, using intrinsic embedding characterizations.

Contribution

It introduces a method to build Banach spaces with exact Szlenk indices and characterizes embedding conditions related to FDDs with upper estimates.

Findings

Existence of Banach spaces with Szlenk index ^{\u03b1} for <

Universal properties for spaces with Szlenk index up to ^{}

Intrinsic embedding criteria involving FDDs with upper estimates

Abstract

For every $\alpha<\omega_1$ we establish the existence of a separable Banach space whose Szlenk index is $\omega^{\alpha\omega+1}$ and which is universal for all separable Banach spaces whose Szlenk-index does not exceed $\omega^{\alpha\omega}$. In order to prove that result we provide an intrinsic characterization of which Banach spaces embed into a space admitting an FDD with upper estimates.