Equicontinuous and relatively weakly compact subsets of the space of Bochner integrable functionsL 1(?, X)

Abstract

continuous duals of both (C(S,X*),fll) and (C(S,X*),fl2) can be represented as L i (/~, X), the space of Bochner integrable functions from f2 into X. One consequence of this representation was a characterization of the subsets of L 1 (#, X) which are equicontinuous with respect to the fll and f12 topologies. A natural question to ask is whether or not these two classes of equicontinuous subsets of Li (/~, X) are in fact the same. A positive answer to this question would of course imply that the topologies fll and f12 coincide. In this paper, a positive answer is obtained by investigating the relationships between the equicontinuous subsets and the class of subsets of L~ (#, X) which are relatively compact in the weak topology a(L 1 (#, X), L1 (#, X)*) on Li (/A X). More specifically, it is shown that the ill- and flE-equicontinuous subsets both coincide with the so-called 66a-sets studied in [1]. A bounded set K c Li( #, X) is said to be a 6S~-set if it is