On summability in conjugate Banach spaces

Abstract

Generalizations of the Banach-Saks property were used by several authors to characterize reflexive Banach spaces (cf. [11], [12], and [16]). We give a characterization of separable conjugate Banach spaces by a similar summability condition. As a consequence, we obtain analogous characterizations of separable second conjugate Banach spaces and of quasi-reflexive spaces. Nonseparable conjugate Banach spaces possessing a smooth predual are also characterized in terms of a summability condition.