Simultaneous Approximations in Banach Space-Valued Bochner–Lebesgue Spaces with Variable Exponent. (2nd January 2019) - Wei, Haihua; Xu, Jingshi
Abstract: Let( A, 𝒜, μ ) be a σ -finite positive measure space and let X be a Banach space. DenoteL p ( · ) ( A, 𝒜, X ) the X- valued Bochner–Lebesgue spaces with variable exponentp ( · ), wherep ( · ) be a μ measurable function on A and take values in[ 1, ∞ ] . We establish some N -simultaneous proximinality results ofL p ( · ) ( A, 𝒜 0, Y ) inL p ( · ) ( A, 𝒜, X ) if1 < ess inf ⁡ y ∈ A p ( y ) andess sup ⁡ y ∈ A p ( y ) < ∞, where𝒜 0 be a sub-algebra of𝒜 and Y be a nonempty locally weakly compact convex subset of X .