Let (E,[...]) be a Banach function space over a probability measure space, and let (X,[...]) be a Banach space. Let E(X)[...] stand for the order continuous dual of a Koethe-Bochner space E(X ) (i.e., E(X )[...] consists of all linear functionals F on E(X ) such that for a net (f[sigma]) in E(X),[...] 0 in E implies F(f[sigma]) --> 0). We present a characterization of conditional sigma(E(X), E(X)[...])-compactness and relative sigma(E(X), E(X)[...] )-compactness in E(X ). We generalize J. Diestel, W. Ruess and W. Schachermayer's criterion for relative weak compactness in E(X ) as well as M. Talagrand's results on conditional weak compactness and weak sequential completeness in E(X ) by removing order continuity of (E, [...]). Applications to Orlicz-Bochner spaces are given.
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