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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2510.08346 |
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| _version_ | 1866911200274546688 |
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| author | Ruf, Thomas |
| author_facet | Ruf, Thomas |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_08346 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Separability and Submetrizability in Locally Convex Spaces Ruf, Thomas Functional Analysis 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. |
| title | Separability and Submetrizability in Locally Convex Spaces |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2510.08346 |