A study on $\mr{F}$-simultaneous approximative $τ$-compactness property in Banach spaces
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arXiv
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| Format: | Preprint |
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2025
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| author | Das, Syamantak Paul, Tanmoy |
| author_facet | Das, Syamantak Paul, Tanmoy |
| contents | Veselý (1997) studied Banach spaces that admit $f$-centers for finite subsets of the space. In this work, we introduce the concept of $\mr{F}$-simultaneous approximative $τ$-compactness property ($τ$-$\mr{F}$-SACP in short) for triplets $(X, V,\mf{F})$, where $X$ is a Banach space, $V$ is a $τ$-closed subset of $X$, $\mf{F}$ is a subfamily of closed and bounded subsets of $X$, $\mr{F}$ is a collection of functions, and $τ$ is the norm or weak topology on $X$. We characterize reflexive spaces with the Kadec-Klee property using triplets with $τ$-$\mr{F}$-SACP. We investigate the relationship between $τ$-$\mr{F}$-SACP and the continuity properties of the restricted $f$-center map. The study further examines $τ$-$\mr{F}$-SACP in the context of $CLUR$ spaces and explores various characterizations of $τ$-$\mr{F}$-SACP, including connections to reflexivity, Fréchet smoothness, and the Kadec-Klee property. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_03347 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A study on $\mr{F}$-simultaneous approximative $τ$-compactness property in Banach spaces Das, Syamantak Paul, Tanmoy Functional Analysis 41A28, 46B20, 41A65, 41A50 Veselý (1997) studied Banach spaces that admit $f$-centers for finite subsets of the space. In this work, we introduce the concept of $\mr{F}$-simultaneous approximative $τ$-compactness property ($τ$-$\mr{F}$-SACP in short) for triplets $(X, V,\mf{F})$, where $X$ is a Banach space, $V$ is a $τ$-closed subset of $X$, $\mf{F}$ is a subfamily of closed and bounded subsets of $X$, $\mr{F}$ is a collection of functions, and $τ$ is the norm or weak topology on $X$. We characterize reflexive spaces with the Kadec-Klee property using triplets with $τ$-$\mr{F}$-SACP. We investigate the relationship between $τ$-$\mr{F}$-SACP and the continuity properties of the restricted $f$-center map. The study further examines $τ$-$\mr{F}$-SACP in the context of $CLUR$ spaces and explores various characterizations of $τ$-$\mr{F}$-SACP, including connections to reflexivity, Fréchet smoothness, and the Kadec-Klee property. |
| title | A study on $\mr{F}$-simultaneous approximative $τ$-compactness property in Banach spaces |
| topic | Functional Analysis 41A28, 46B20, 41A65, 41A50 |
| url | https://arxiv.org/abs/2501.03347 |