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

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.

arXiv comment25 pages, no figures