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| Format: | Preprint |
| Published: |
2023
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| Online Access: | https://arxiv.org/abs/2304.12611 |
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| _version_ | 1866912490956259328 |
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| author | Bachir, Mohammed |
| author_facet | Bachir, Mohammed |
| contents | We give a class of bounded closed sets $C$ in a Banach space satisfying a generalized and stronger form of the Bishop-Phelps property studied by Bourgain in \cite{Bj} for dentable sets. A version of the {\it ``Bishop-Phelps-Bollobás"} theorem will be also given. The density and the residuality of bounded linear operators attaining their maximum on $C$ (known in the literature) will be replaced, for this class of sets, by being the complement of a $σ$-porous set. The result of the paper is applicable for both linear operators and non-linear mappings. When we apply our result to subsets (from this class) whose closed convex hull is the closed unit ball, we obtain a new class of Banach spaces involving property $(A)$ introduced by Lindenstrauss. We also establish that this class of Banach spaces is stable under $\ell_1$-sum when the spaces have a same ``modulus". Applications to norm attaining bounded multilinear mappings and Lipschitz mappings will also be given. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_12611 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A strong Bishop-Phelps property and a new class of Banach spaces with the property $(A)$ of Lindenstrauss Bachir, Mohammed Functional Analysis We give a class of bounded closed sets $C$ in a Banach space satisfying a generalized and stronger form of the Bishop-Phelps property studied by Bourgain in \cite{Bj} for dentable sets. A version of the {\it ``Bishop-Phelps-Bollobás"} theorem will be also given. The density and the residuality of bounded linear operators attaining their maximum on $C$ (known in the literature) will be replaced, for this class of sets, by being the complement of a $σ$-porous set. The result of the paper is applicable for both linear operators and non-linear mappings. When we apply our result to subsets (from this class) whose closed convex hull is the closed unit ball, we obtain a new class of Banach spaces involving property $(A)$ introduced by Lindenstrauss. We also establish that this class of Banach spaces is stable under $\ell_1$-sum when the spaces have a same ``modulus". Applications to norm attaining bounded multilinear mappings and Lipschitz mappings will also be given. |
| title | A strong Bishop-Phelps property and a new class of Banach spaces with the property $(A)$ of Lindenstrauss |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2304.12611 |