Compactness in the Lebesgue–Bochner spaces Lp(μ;X)Lp(μ;X)

Let (Ω,μ)(Ω,μ) be a finite measure space, XX a Banach space, and let 1≤p<∞1≤p<∞. The aim of this paper is to give an elementary proof of the Diaz–Mayoral theorem that a subset V⊆Lp(μ;X)V⊆Lp(μ;X) is relatively compact in Lp(μ;X)Lp(μ;X) if and only if it is uniformly pp-integrable, uniformly tight, and scalarly relatively compact.