2023/03/19 by Teena Thomas, Thomas, Teena
Computer Science · Mathematics · #41A65 (Primary) 46E15 #46A55 #46B20 (Secondary) #Advanced Banach Space Theory #FOS: Mathematics #Fixed Point Theorems Analysis #Functional Analysis (math.FA) #Optimization and Variational Analysis
paper · pdf · doi:10.48550/arxiv.2303.10676
openalex publication_date 2023/03/19 · openalex created_date 2023/03/22 · openalex updated_date 2026/07/28
For a compact Hausdorff space S, we prove that the closed unit ball of a closed linear subalgebra of the space of real-valued continuous functions on S, denoted by C(S), satisfies property-(P1) (the set-valued generalization of strong proximinality) for the non-empty closed bounded subsets of the bidual of C(S). Various stability results related to property-(P1) and semi-continuity properties of restricted Chebyshev-center maps are also established. As a consequence, we derive that if Y is a proximinal finite co-dimensional subspace of c0 then the closed unit ball of Y satisfies property-(P1) for the non-empty closed bounded subsets of ℓ∞ and the restricted Chebyshev-center map of the closed unit ball of Y is Hausdorff metric continuous on the class of non-empty closed bounded subsets of ℓ∞. We also investigate a variant of the transitivity property, similar to the one discussed in [C. R. Jayanarayanan and T. Paul, Strong proximinality and intersection properties of balls in Banach spaces, J. Math. Anal. Appl., 426(2):1217--1231, 2015], for property-(P1).