Proximinality in Banach Space-Valued Grand Bochner-Lebesgue Spaces with Variable Exponent

Abstract

Let (A, \(\mathscr{A}\), µ) be a σ-finite complete measure space, and let p(·) be a µ-measurable function on A which takes values in (1, ∞). Let Y be a subspace of a Banach space X. By \({\tilde L^{p(\cdot),\varphi }}(A,Y)\) and \({\tilde L^{p(\cdot),\varphi }}(A,X)\) we denote the grand Bochner-Lebesgue spaces with variable exponent p(·) whose functions take values in Y and X, respectively. First, we estimate the distance of f from \({\tilde L^{p(\cdot),\varphi }}(A,Y)\) when \(f \in {\tilde L^{p(\cdot),\varphi }}(A,X)\). Then we prove that \({\tilde L^{p(\cdot),\varphi }}(A,Y)\) is proximinal in \({\tilde L^{p(\cdot),\varphi }}(A,X)\) if Y is weakly \(\mathcal{K}\)-analytic and proximinal in X. Finally, we establish a connection between the proximinality of \({\tilde L^{p(\cdot),\varphi }}(A,Y)\) in \({\tilde L^{p(\cdot),\varphi }}(A,X)\) and the proximinality of L¹(A, Y) in L¹(A, X).