On Hahn-Banach smoothness and related properties in Banach spaces
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929265656725504 |
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| author | Daptari, Soumitra |
| author_facet | Daptari, Soumitra |
| contents | In this paper, we study several variants of Hahn-Banach smoothness, viz., property-$(SU)$/$(HB)$/$(wU)$, where property-$(SU)$ and property-$(HB)$ are stronger notions and property-$(wU)$ is a weaker notion of Hahn-Banach smoothness. We characterize property-$(wU)$ and property-$(HB)$. It is observed that $L_1(μ)$ has property-$(wU)$ in $L_1(μ,(\mathbb{R}^2,\|.\|_2))$ but it does not have property-$(U)$ in $L_1(μ,(\mathbb{R}^2,\|.\|_2))$ for a non-atomic measure $μ$. We derive a sufficient condition when property-$(wU)$ is equivalent to property-$(U)$ of a subspace. It is observed that these properties are separably determined. Finally, finite-dimensional and finite co-dimensional subspaces of $c_0$, $\ell_p$ ($1\leq p<\infty$) having these properties are characterized. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_03453 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Hahn-Banach smoothness and related properties in Banach spaces Daptari, Soumitra Functional Analysis 46A22, 46B10, 46B25, 46B20, 46B45 In this paper, we study several variants of Hahn-Banach smoothness, viz., property-$(SU)$/$(HB)$/$(wU)$, where property-$(SU)$ and property-$(HB)$ are stronger notions and property-$(wU)$ is a weaker notion of Hahn-Banach smoothness. We characterize property-$(wU)$ and property-$(HB)$. It is observed that $L_1(μ)$ has property-$(wU)$ in $L_1(μ,(\mathbb{R}^2,\|.\|_2))$ but it does not have property-$(U)$ in $L_1(μ,(\mathbb{R}^2,\|.\|_2))$ for a non-atomic measure $μ$. We derive a sufficient condition when property-$(wU)$ is equivalent to property-$(U)$ of a subspace. It is observed that these properties are separably determined. Finally, finite-dimensional and finite co-dimensional subspaces of $c_0$, $\ell_p$ ($1\leq p<\infty$) having these properties are characterized. |
| title | On Hahn-Banach smoothness and related properties in Banach spaces |
| topic | Functional Analysis 46A22, 46B10, 46B25, 46B20, 46B45 |
| url | https://arxiv.org/abs/2403.03453 |