On linearisation and uniqueness of preduals
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866913733226266624 |
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| author | Kruse, Karsten |
| author_facet | Kruse, Karsten |
| contents | We study strong linearisations and the uniqueness of preduals of locally convex Hausdorff spaces of scalar-valued functions. Strong linearisations are special preduals. A locally convex Hausdorff space $\mathcal{F}(Ω)$ of scalar-valued functions on a non-empty set $Ω$ is said to admit a strong linearisation if there are a locally convex Hausdorff space $Y$, a map $δ\colonΩ\to Y$ and a topological isomorphism $T\colon\mathcal{F}(Ω)\to Y_{b}'$ such that $T(f)\circ δ= f$ for all $f\in\mathcal{F}(Ω)$. We give sufficient conditions that allow us to lift strong linearisations from the scalar-valued to the vector-valued case, covering many previous results on linearisations, and use them to characterise the bornological spaces $\mathcal{F}(Ω)$ with (strongly) unique predual in certain classes of locally convex Hausdorff spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_09167 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On linearisation and uniqueness of preduals Kruse, Karsten Functional Analysis Primary 46A20, 46E40 Secondary 46A08, 46B10, 46E10, We study strong linearisations and the uniqueness of preduals of locally convex Hausdorff spaces of scalar-valued functions. Strong linearisations are special preduals. A locally convex Hausdorff space $\mathcal{F}(Ω)$ of scalar-valued functions on a non-empty set $Ω$ is said to admit a strong linearisation if there are a locally convex Hausdorff space $Y$, a map $δ\colonΩ\to Y$ and a topological isomorphism $T\colon\mathcal{F}(Ω)\to Y_{b}'$ such that $T(f)\circ δ= f$ for all $f\in\mathcal{F}(Ω)$. We give sufficient conditions that allow us to lift strong linearisations from the scalar-valued to the vector-valued case, covering many previous results on linearisations, and use them to characterise the bornological spaces $\mathcal{F}(Ω)$ with (strongly) unique predual in certain classes of locally convex Hausdorff spaces. |
| title | On linearisation and uniqueness of preduals |
| topic | Functional Analysis Primary 46A20, 46E40 Secondary 46A08, 46B10, 46E10, |
| url | https://arxiv.org/abs/2307.09167 |