Some remarks on James–Schreier spaces

The James–Schreier spaces Vp, where 1⩽p<∞, were recently introduced by Bird and Laustsen (in press) [5], as an amalgamation of James' quasi-reflexive Banach space on the one hand and Schreier's Banach space giving a counterexample to the Banach–Saks property on the other. The purpose of this note is to answer some questions left open in Bird and Laustsen (in press) [5]. Specifically, we prove that (i) the standard Schauder basis for the first James–Schreier space V1 is shrinking, and (ii) any two Schreier or James–Schreier spaces with distinct indices are non-isomorphic. The former of these results implies that V1 does not have Pełczyński's property (u) and hence does not embed in any Banach space with an unconditional Schauder basis.