Abstract :
LetXbe a Banach space and letYbe a closed subspace ofX. Let 1⩽p⩽∞ and let us denote byLp(μ, X) the Banach space of allX-valued Bochnerp-integrable (essentially bounded forp=∞) functions on a certain positive completeσ-finite measure space (Ω, Σ, μ), endowed with the usualp-norm. In this paper we give a negative answer to the following question: “IfYis proximinal inX, isLp(μ, Y) proximinal inLp(μ, X)?” We also show that the answer is affirmative for separable spacesY. Some consequences of this are obtained.
