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).
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Wei, H., Xu, J. Proximinality in Banach Space-Valued Grand Bochner-Lebesgue Spaces with Variable Exponent. Math Notes 105, 618–624 (2019). https://doi.org/10.1134/S0001434619030349
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DOI: https://doi.org/10.1134/S0001434619030349