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 F$\mathcal {F}$ of finite subsets of ω$\omega$ we define a quasi‐norm ∥·∥F$\Vert \cdot \Vert ^\mathcal {F}$ whose Banach envelope is the dual norm for the combinatorial space generated by F$\mathcal {F}$. 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 ℓ1$\ell _1$‐saturated and do not have the Schur property. In particular, this holds for the Schreier families.