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

Abstract

Let (A, $$\\mathscr{A}$$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 $${\\tilde L^{p(\\cdot),\\varphi }}(A,Y)$$L˜p(⋅),φ(A,Y) and $${\\tilde L^{p(\\cdot),\\varphi }}(A,X)$$L˜p(⋅),φ(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 $${\\tilde L^{p(\\cdot),\\varphi }}(A,Y)$$L˜p(⋅),φ(A,Y) when $$f \\in {\\tilde L^{p(\\cdot),\\varphi }}(A,X)$$f∈L˜p(⋅),φ(A,X). Then we prove that $${\\tilde L^{p(\\cdot),\\varphi }}(A,Y)$$L˜p(⋅),φ(A,Y) is proximinal in $${\\tilde L^{p(\\cdot),\\varphi }}(A,X)$$L˜p(⋅),φ(A,X) if Y is weakly $$\\mathcal{K}$$K-analytic and proximinal in X. Finally, we establish a connection between the proximinality of $${\\tilde L^{p(\\cdot),\\varphi }}(A,Y)$$L˜p(⋅),φ(A,Y) in $${\\tilde L^{p(\\cdot),\\varphi }}(A,X)$$L˜p(⋅),φ(A,X) and the proximinality of L1(A, Y) in L1(A, X).