On the spaces dual to combinatorial Banach spaces

TL;DR

This paper introduces quasi-Banach spaces related to the duals of combinatorial Banach spaces, providing easier-to-handle quasi-norms that retain many properties of the dual spaces, including $ ext{l}_1$-saturation and lack of the Schur property.

Contribution

It defines new quasi-norms associated with combinatorial Banach space duals, especially for large families like Schreier families, and analyzes their properties.

Findings

Quasi-Banach spaces are $ ext{l}_1$-saturated.

They do not have the Schur property.

Applicable to Schreier families and similar large families.

Abstract

We present quasi-Banach spaces which are closely related to the duals of combinatorial Banach spaces. More precisely, for a compact family $\mathcal{F}$ of finite subsets of $\omega$ we define a quasi-norm $\lVert \cdot \rVert^\mathcal{F}$ whose Banach envelope is the dual norm for the combinatorial space generated by $\mathcal{F}$. Such quasi-norms seem to be much easier to handle than the dual norms and yet the quasi-Banach spaces induced by them share many properties with the dual spaces. We show that the quasi-Banach spaces induced by large families (in the sense of Lopez-Abad and Todorcevic) are $\ell_1$-saturated and do not have the Schur property. In particular, this holds for the Schreier families.