2023/04/25 by Mohammed Bachir, Bachir, Mohammed
Computer Science · Mathematics · #Advanced Banach Space Theory #Advanced Harmonic Analysis Research #FOS: Mathematics #Functional Analysis (math.FA) #Optimization and Variational Analysis
paper · pdf · doi:10.48550/arxiv.2304.12611
openalex publication_date 2023/04/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \citeBj 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 ℓ1-sum when the spaces have a same ``modulus". Applications to norm attaining bounded multilinear mappings and Lipschitz mappings will also be given.