On Banach spaces with angelic Mackey duals

TL;DR

This paper proves that sequentially reflexive Banach spaces have Mackey duals that are angelic, extending previous results on compactness properties and providing a new characterization of these spaces.

Contribution

It establishes that the Mackey dual of a sequentially reflexive Banach space is angelic and characterizes such spaces via topological properties of their duals.

Findings

Mackey duals of sequentially reflexive Banach spaces are angelic.

Relative sequential compactness is stronger than relative compactness in Mackey duals.

Characterization of sequentially reflexive spaces via topology on 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 $\tau (X^{*}, X)$.