Super-reflexive spaces with bases

Abstract

Super-reflexivity is defined in such a way that all superreflexive Banach spaces are reflexive and a Banach space is super-reflexive if it is isomorphic to a Banach space that is either uniformly convex or uniformly non-square. It is shown that, if 0<2f< e< F and B is super-reflexive, then there are numbers r and s for which 1 < r < s < ¥ and, if (ei) is any normalized basic sequence inB with characteristic not less than e, then (formula presented), for all numbers (ai) such that Saiei is convergent. This also is true for unconditional basic subsets in nonseparable super-reflexive Banach spaces. Gurarii and Gurarii recently established the existence of f and r for uniformly smooth spaces, and the existence of F and s for uniformly convex spaces [Izv. Akad. Nauk SSSR Ser. Mat., 35 (1971), 210-215]. © 1972 Pacific Journal of Mathematics.