Separability and submetrizability in locally convex spaces
Abstract
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 $$X'$$ X ′ that is metrizable and coarser than the weak topology $$\sigma (X', X)$$ 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.
Fields of Study & Subjects
Advanced Banach Space Theory
Advanced Topology and Set Theory
Economic theories and models
Publication details
- Journal / source
- Annals of Functional Analysis
- Volume
- 17
- Issue
- 3
- Publication date
- 29 June 2026
- Language
- EN