On linearisation and existence of preduals
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866910464649199616 |
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| author | Kruse, Karsten |
| author_facet | Kruse, Karsten |
| contents | We study the problem of existence of preduals of locally convex Hausdorff spaces. We derive necessary and sufficient conditions for the existence of a predual with certain properties of a bornological locally convex Hausdorff space $X$. Then we turn to the case that $X=\mathcal{F}(Ω)$ is a space of scalar-valued functions on a non-empty set $Ω$ and characterise those among them which admit a special predual, namely a strong linearisation, i.e. 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 $f\in\mathcal{F}(Ω)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_12615 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On linearisation and existence of preduals Kruse, Karsten Functional Analysis Primary 46A08, 46A20 Secondary 46A70, 46B10, 46E10 We study the problem of existence of preduals of locally convex Hausdorff spaces. We derive necessary and sufficient conditions for the existence of a predual with certain properties of a bornological locally convex Hausdorff space $X$. Then we turn to the case that $X=\mathcal{F}(Ω)$ is a space of scalar-valued functions on a non-empty set $Ω$ and characterise those among them which admit a special predual, namely a strong linearisation, i.e. 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 $f\in\mathcal{F}(Ω)$. |
| title | On linearisation and existence of preduals |
| topic | Functional Analysis Primary 46A08, 46A20 Secondary 46A70, 46B10, 46E10 |
| url | https://arxiv.org/abs/2402.12615 |