Mutual proximinality of two sets in Lp(µ, X)

Abstract

In this paper, we study mutual best approximation of two sets in the p–Bochner function spaces L^p(µ,X),  1 ≤ p < ∞. We give some sufficient conditions on M and K as subsets of the Banach space X to guarantee that L^p(and L^p( are mutually proximinal subsets in L^p(µ,X). The main issue in this work is that we state and prove a distance formula for mutual best approximation between the two sets L^p(and L^p( in L^p(µ,X), for  1 ≤ p < ∞. Then we use this formula to obtain some other results related to our study.