Various Notions of Best Approximation Property in Spaces of Bochner Integrable Functions

We show that a separable proximinal subspace of X, say Y is strongly proximinal (strongly ball proximinal) if and only if Lp(I, Y ) is strongly proximinal (strongly ball proximinal) in Lp(I,X), for 1 ≤ p <∞. The p =∞ case requires a stronger assumption, that of ’uniform proximinality’. Further, we show that a separable subspace Y is ball proximinal in X if and only if Lp(I, Y ) is ball proximinal in Lp(I,X) for 1 ≤ p ≤ ∞. We develop the notion of ’uniform proximinality’ of a closed convex set in a Banach space, rectifying one that was defined in a recent paper by P.-K Lin et al. [J. Approx. Theory 183 (2014), 72–81]. We also provide several examples having this property; viz. any U -subspace of a Banach space has this property. Recall the notion of 3.2.I.P. by Joram Lindenstrauss, a Banach space X is said to have 3.2.I.P. if any three closed balls which are pairwise intersecting actually intersect in X. It is proved the closed unit ball BX of a space with 3.2.I.P and closed unit ball of any M-ideal of a space with 3.2.I.P. are uniformly proximinal. A new class of examples are given having this property.