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Let (\Omega,\mu) be a finite measure space, X a Banach space, and let 1\lep<\infty. The aim of this paper is to give an elementary proof of theDiaz--Mayoral theorem that a subset V of L^p(\mu;X) is relatively compact ifand only if it is uniformly p-integrable, uniformly tight, and scalarlyrelatively compact.

Compactness in the Lebesgue-Bochner spaces L^p(\mu;X)