Spaces of Vector-Valued Integrable Functions and Localization of Bounded Subsets

Abstract

We study the structure of bounded sets in the space L1{E} of absolutely integrable Lusin-measurable functions with values in a locally convex space E. The main idea is to extend the notion of property (B) of Pietsch, defined within the context of vector-valued sequences, to spaces of vector-valued functions. We prove that this extension, that at first sight looks more restrictive, coincides with the original property (B) for quasicomplete spaces. Then we show that when dealing with a locally convex space, property (B) provides the link to prove the equivalence between Radon–Nikodym property (the existence of a density function for certain vector measures) and the integral representation of continuous linear operators T: L1 → E, a fact well-known for Banach spaces. We also study the relationship between Radon–Nikodym property and the characterization of the dual of L1{E} as the space L∞{E′b}.