Weak vs. Norm Compactness in L1: The Bocce Criterion

. We present a new simple proof that if a relatively weakly compact subset of L1 satisfies the Bocce criterion (an oscillation condition), then it is relatively norm compact. The converse of this fact is easy to verify. A direct consequence is that, for a bounded linear operator T from L1 into a Banach space X, T is Dunford-Pettis if and only if the subset T (B(X )) of L1 satisfies the Bocce criterion. A relatively weakly compact subset of L 1 is relatively norm compact if and only if it satisfies the Bocce criterion (an oscillation condition) [G1]. We shall present a new simple proof that if a relatively weakly compact subset of L 1 satisfies the Bocce criterion, then it is relatively norm compact. The converse is easy to verify. Recall that a Banach space X has the complete continuity property (CCP) if each bounded linear operators from L 1 into X is Dunford-Pettis (i.e. maps weakly convergent sequences onto norm convergent sequences). The CCP is a weakening of the Radon-Nikod'..