International Journal of Mathematics and Mathematical Sciences
Volume 14 (1991), Issue 2, Pages 245-252
doi:10.1155/S0161171291000273
Best approximation in Orlicz spaces
Department of Mathematics, Kuwait University, P.O. BOX 5969, Safat 130, Kuwait
Received 17 April 1989; Revised 23 December 1989
Copyright © 1991 H. Al-Minawi and S. Ayesh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.