SPACES OF VECTOR-VALUED INTEGRABLE FUNCTIONS AND LOCALIZATION OF BOUNDED SUBSETS

We study the structure of bounded sets in the space L(1) {E} of absolu tely integrable Lusin-measurable functions with values in a locally co nvex space E. The main idea is to extend the notion of property (B) of Pietsch, defined within the context of vector-valued sequences, to sp aces of vector-valued functions. We prove that this extension, that at first sight looks more restrictive, coincides with the original prope rty (B) for quasicomplete spaces. Then we show that when dealing with a locally convex space, property (B) provides the link to prove the eq uivalence between Radon-Nikodym property (the existence of a density f unction for certain vector measures) and the integral representation o f continuous linear operators T:L(1) --> E, a fact well-known for Bana ch spaces. We also study the relationship between Radon-Nikodym proper ty and the characterization of the dual of L(1){E} as the space L(infi nity){E(b)'}.