On anti-coproximinal and strongly anti-coproximinal subspaces of function spaces
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915761443831808 |
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| author | Sohel, Shamim Ghosh, Souvik Sain, Debmalya Paul, Kallol |
| author_facet | Sohel, Shamim Ghosh, Souvik Sain, Debmalya Paul, Kallol |
| contents | The purpose of this article is to study the anti-coproximinal and strongly anti-coproximinal subspaces of the Banach space of all bounded (continuous) functions. We obtain a tractable necessary condition for a subspace to be stronsgly anti-coproximinal. We prove that for a subspace $\mathbb{Y}$ of a Banach space $\mathbb{X}$ to be strongly anti-coproximinal, $\mathbb Y$ must contain all w-ALUR points of $\mathbb{X}$ and intersect every maximal face of $B_{\mathbb{X}}.$ We also observe that the subspace $\mathbb{K}(\mathbb{X}, \mathbb{Y})$ of all compact operators between the Banach spaces $ \mathbb X $ and $ \mathbb Y$ is strongly anti-coproximinal in the space $\mathbb{L}(\mathbb{X}, \mathbb{Y})$ of all bounded linear operators between $ \mathbb X $ and $ \mathbb Y$, whenever $\mathbb{K}(\mathbb{X}, \mathbb{Y})$ is a proper subset of $\mathbb{L}(\mathbb{X}, \mathbb{Y}),$ and the unit ball $B_{\mathbb{X}}$ is the closed convex hull of its strongly exposed points. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_13464 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On anti-coproximinal and strongly anti-coproximinal subspaces of function spaces Sohel, Shamim Ghosh, Souvik Sain, Debmalya Paul, Kallol Functional Analysis Primary 41A65, Secondary 46B20, 46E15 The purpose of this article is to study the anti-coproximinal and strongly anti-coproximinal subspaces of the Banach space of all bounded (continuous) functions. We obtain a tractable necessary condition for a subspace to be stronsgly anti-coproximinal. We prove that for a subspace $\mathbb{Y}$ of a Banach space $\mathbb{X}$ to be strongly anti-coproximinal, $\mathbb Y$ must contain all w-ALUR points of $\mathbb{X}$ and intersect every maximal face of $B_{\mathbb{X}}.$ We also observe that the subspace $\mathbb{K}(\mathbb{X}, \mathbb{Y})$ of all compact operators between the Banach spaces $ \mathbb X $ and $ \mathbb Y$ is strongly anti-coproximinal in the space $\mathbb{L}(\mathbb{X}, \mathbb{Y})$ of all bounded linear operators between $ \mathbb X $ and $ \mathbb Y$, whenever $\mathbb{K}(\mathbb{X}, \mathbb{Y})$ is a proper subset of $\mathbb{L}(\mathbb{X}, \mathbb{Y}),$ and the unit ball $B_{\mathbb{X}}$ is the closed convex hull of its strongly exposed points. |
| title | On anti-coproximinal and strongly anti-coproximinal subspaces of function spaces |
| topic | Functional Analysis Primary 41A65, Secondary 46B20, 46E15 |
| url | https://arxiv.org/abs/2504.13464 |