Compactness in Vector-valued Banach Function Spaces

We give a new proof of a recent characterization by Diaz and Mayoral of compactness in the Lebesgue-Bochner spaces L X , where X is a Banach space and 1 ≤ p < ∞, and extend the result to vector-valued Banach function spaces EX , where E is a Banach function space with order continuous norm. Let X be a Banach space. The problem of describing the compact sets in the Lebesgue-Bochner spaces LpX , 1 ≤ p <∞, goes back to the work of Riesz, Fréchet, Vitali in the scalar-valued case, cf. [7], and has been considered by many authors, cf. [2, 4, 5, 11, 12]. In a recent paper, Diaz and Mayoral [5] proved that if the underlying measure space is finite, then a subset K of LpX is relatively compact if and only if K is uniformly p-integrable, scalarly relatively compact, and either uniformly tight or flatly concentrated. Their proof relies on the Diestel-RuessSchachermayer characterization [6] of weak compactness in L1X and the notion of Bocce oscillation, which was studied recently by Girardi [8] and Balder-GirardiJalby [3] in the context of compactness in L1X . The purpose of this note is to present an extension of the Diaz-Mayoral result to vector-valued Banach function spaces EX , with a proof based on Prohorov’s tightness theorem. We begin with some preliminaries on Banach lattices and Banach function spaces. Our terminology is standard and follows [9]. A Banach lattice E is said to have order continuous norm if every net in E which decreases to 0 converges to 0. Every separable Banach function space E has this property. Indeed, because such spaces are Dedekind complete [9, Lemma 2.6.1] and cannot contain an isomorphic copy of l, this follows from [9, Corollary 2.4.3]. A subset F of a Banach lattice E is called almost order bounded if for every ε > 0 there exists an element xε ∈ E+ such that F ⊆ [−xε, xε] + B(ε), where [−xε, xε] := {y ∈ E : −xε ≤ y ≤ xε} and B(ε) := {x ∈ X : ‖x‖ < ε}. It follows from [9, Theorem 2.4.2] that every almost order bounded set in a Banach lattice with order continuous norm is relatively weakly compact. Lemma 1. Let E be a Banach lattice and let I be a dense ideal in E. If the set A ⊆ E is almost order bounded, then for every ε > 0 there exists an element xε ∈ I + such that A ⊆ [0, xε] +B(ε). 2000 Mathematics Subject Classification. Primary: 46E40, Secondary: 46E30, 46B50, 47D06, 60B05.