Copyright © 1991 Hindawi Publishing Corporation. 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 L ϕ ( μ , X ) is the set of all measurable functions f with values in X such that ∫ Ω ϕ ( ‖ c − 1 f ( t ) ‖ ) d μ ( t ) < ∞ for some c > 0 . One of the main results of this paper is: “For a closed subspace Y of X , L ϕ ( μ , Y ) is proximinal in L ϕ ( μ , X ) if and only if L 1 ( μ , Y ) is proximinal in L 1 ( μ , X ) ′ ​ ′ . As a result if Y is reflexive subspace of X , then L ϕ ( ϕ , Y ) is proximinal in L ϕ ( μ , X ) . Other results on proximinality of subspaces of L ϕ ( μ , X ) are proved.