On Banach spaces with angelic Mackey duals

Abstract

We show that if $X$ is a sequentially reflexive Banach space, then its Mackey dual $(X^{*},\tau (X^{*}, X))$ is an angelic space. This builds on a result of J. Howard which says that in the Mackey dual $(X^{*}, \tau (X^{*}, X))$ of a Banach space $X$, relative sequential compactness is, in general, strictly stronger than relative compactness and that the two notions of compactness are equivalent if $X$ is reflexive or separable. Our main result gives a characterization of the sequentially reflexive spaces as the Banach spaces $X$ for which the the finest locally convex topology on $X^{*}$ with the same precompact sets as the Mackey topology $\tau (X^{*}, X)$ is the bound extension of $\tau (X^{*}, X)$.