Best approximation in Orlicz spaces

Abstract

Let X be a real Banach space and (Ω,μ) be a finite measure space and ϕ be a strictly icreasing convex continuous function on [0,∞) with ϕ(0)=0. The space Lϕ(μ,X) is the set of all measurable functions f with values in X such that ∫Ωϕ(‖c−1f(t)‖)dμ(t)<∞ for some c>0. One of the main results of this paper is: “For a closed subspace Y of X, Lϕ(μ,Y) is proximinal in Lϕ(μ,X) if and only if L1(μ,Y) is proximinal in L1(μ,X)′​′. As a result if Y is reflexive subspace of X, then Lϕ(ϕ,Y) is proximinal in Lϕ(μ,X). Other results on proximinality of subspaces of Lϕ(μ,X) are proved.