L¹(μ, X) as a constrained subspace of its bidual

Abstract

In this note we consider the property of being constrained in the bidual, for the space of Bochner integrable functions. For a Banach space X having the Radon-Nikodym property and constrained in its bidual and for Y ⊂ X, under a natural assumption on Y, we show that L¹ (μ, X/Y) is constrained in its bidual and L¹ (μ, Y) is a proximinal subspace of L¹(μ, X). As an application of these results, we show that, if L¹(μ, X) admits generalized centers for finite sets and if Y ⊂ X is reflexive, then L¹(μ, X/Y) also admits generalized centers for finite sets.