$\alpha$-Large Families and Applications to Banach Space Theory

TL;DR

This paper introduces $oldsymbol{ ext{ extalpha}-}$large families of finite subsets of infinite sets, extending the concept of large families using transfinite Schreier families, and applies this to construct reflexive Banach spaces with specific spreading model properties.

Contribution

It defines $oldsymbol{ extalpha}- ext{large}$ families for all countable ordinals, proves their existence on the continuum, and constructs reflexive Banach spaces with prescribed spreading models.

Findings

Existence of $oldsymbol{ extalpha}- ext{large}$ families on $2^{eth_0}$

Construction of reflexive spaces with continuum density

Sequences generate $oldsymbol{ extell}_1^{ ext{ extalpha}}$ as spreading models

Abstract

The notion of $\alpha$-large families of finite subsets of an infinite set is defined for every countable ordinal number $\alpha$, extending the known notion of large families. The definition of the $\alpha$-large families is based on the transfinite hierarchy of the Schreier families $\mathcal{S}_{\alpha}, \alpha<\omega_1$. We prove the existence of such families on the cardinal number $2^{\aleph_0}$ and we study their properties. As an application, based on those families we construct a reflexive space $\mathfrak{X}_{2^{\aleph_0}}^{\alpha}$, $\alpha<\omega_1$ with density the continuum, such that every bounded non norm convergent sequence $\{x_k\}_k$ has a subsequence generating $\ell_1^{\alpha}$ as a spreading model.