On the spaces dual to combinatorial Banach spaces

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.