Some remarks on James–Schreier spaces

Abstract

The James–Schreier spaces V p , 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 V 1 is shrinking, and (ii) any two Schreier or James–Schreier spaces with distinct indices are non-isomorphic. The former of these results implies that V 1 does not have Pełczyński's property (u) and hence does not embed in any Banach space with an unconditional Schauder basis.