On J. Borwein's concept of sequentially reflexive Banach spaces

Abstract

A Banach space $X$ is reflexive if the Mackey topology $\tau (X^{\ast },X)$ on $X^{\ast }$ agrees with the norm topology on $X^{\ast }$. Borwein [B] calls a Banach space $X$ {\it sequentially reflexive\/} provided that every $\tau (X^{\ast },X)$ convergent {\it sequence\/} in $X^{\ast }$ is norm convergent. The main result in [B] is that $X$ is sequentially reflexive if every separable subspace of $X$ has separable dual, and Borwein asks for a characterization of sequentially reflexive spaces. Here we answer that question by proving \proclaim Theorem. {\sl A Banach space $X$ is sequentially reflexive if and only if ${\ell }_1$ is not isomorphic to a subspace of $X$.}