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 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: L 1 → E , a fact well‐known for Banach spaces. We also study the relationship between Radon–Nikodym property and the characterization of the dual of L 1 {E} as the space L ∞ { E ′ b }.