Separability and Submetrizability in Locally Convex Spaces
Thomas Ruf
TL;DR
This work links separability in locally convex spaces to dual-space metrizability by introducing countable separation. It proves that $X$ is separable exactly when $\sigma(X',X)$ admits a coarser metrizable locally convex topology, with a symmetric dual statement, and situates these results within the Mackey-Arens duality framework. The authors derive practical criteria and corollaries, such as normed-space separability being equivalent to weak$^*$ metrizability of the dual unit ball, and clarify how separability transfers between $X$ and $X'$ under duality constraints. Overall, the paper provides a precise duality between separability and metrizability and extends known conditions through the notion of countable separation and submetrizability of dual topologies.
Abstract
We introduce the property of countable separation for a locally convex Hausdorff space $X$ and relate it to the existence of a metrizable coarser topology. Building on this, we demonstrate how the separability of $X$ is equivalent to the existence of a locally convex topology on the dual $X'$ that is metrizable and coarser than the weak topology $σ(X', X)$. This result generalizes known conditions for separability and provides a precise duality between separability and metrizability. We also show how to derive new and known conditions for the separability of $X$ from this characterization.