Bird, Alistair and Laustsen, Niels Jakob and Zsak, Andras (2010) Some remarks on James-Schreier spaces. Journal of Mathematical Analysis and Applications, 371 (2). pp. 609-613. ISSN 0022-247X
Abstract
The James-Schreier spaces V_p, where p is a real number greater than or equal to 1, were recently introduced by Bird and Laustsen 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 by Bird and Laustsen. 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 Pelczynski's property (u) and hence does not embed in any Banach space with an unconditional Schauder basis.