Estimation of the Szlenk index of Banach Spaces via Schreier spaces

TL;DR

This paper constructs Banach spaces with specific Szlenk indices that are universal for classes of spaces with smaller indices, using embeddings into spaces with FDDs and Schreier space estimates.

Contribution

It introduces a method to estimate Szlenk indices of Banach spaces via Schreier spaces and constructs universal spaces for classes defined by these indices.

Findings

Existence of Banach spaces with prescribed Szlenk indices and bases.

Universal spaces for classes of Banach spaces with bounded Szlenk index.

Characterization of embeddings into spaces with FDDs and Schreier estimates.

Abstract

For each ordinal $\alpha<\omega_1$, we prove the existence of a space with a basis and Szlenk index $\omega^{\alpha+1}$ which is universal for the class of spaces with Szlenk index not exceeding $\omega^\alpha$. Our proof involves developing a characterization of which Banach spaces embed into spaces with an FDD with upper Schreier space estimates.