On the spaces dual to combinatorial Banach spaces

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 $\ddot{I}‰$ we define a quasi-norm $\Vert \cdot {\Vert }^{\mathcal{F}}$ 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.

22 pages