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

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

1991-10-09

arxiv.org/abs/math/9201233v1

Mathematics

Functional Analysis

Scientific paper

A Banach space $X$ is reflexive if the Mackey topology $\tau(X^*,X)$ on $X^*$ agrees with the norm topology on $X^*$. Borwein [B] calls a Banach space $X$ {\it sequentially reflexive\/} provided that every $\tau(X^*,X)$ convergent {\it sequence\/} in $X^*$ 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 $X$.}

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