Various notions of best approximation property in spaces of Bochner integrable functions

Abstract

We derive that for a separable proximinal subspace Y of X, Y is strongly proximinal (strongly ball proximinal) if and only if for 1 ≤ p < ∞, Lp(I, Y ) is strongly proximinal (strongly ball proximinal) in Lp(I, X). Case for p = ∞ follows from stronger assumption on Y in X (uniform proximinality). It is observed that for a separable proximinal subspace Y in X, Y is ball proximinal in X if and only if Lp(I, Y ) is ball proximinal in Lp(I, X) for 1 ≤ p ≤ ∞. Our observations also include the fact that for any (strongly) proximinal subspace Y of X, if every separable subspace of Y is ball (strongly) proximinal in X then Lp(I, Y ) is ball (strongly) proximinal in Lp(I, X) for 1 ≤ p < ∞. We introduce the notion of uniform proximinality of a closed convex set in a Banach space, which is wrongly defined in [16]. Several examples are given having this property, viz. any U -subspace of a Banach space, closed unit ball BX of a space with 3.2.I.P , closed unit ball of any Mideal of a space with 3.2.I.P. are uniformly proximinal. A new class of examples are given having this property.