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

Abstract

We study the structure of bounded sets in the space L ' ( 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' -. 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'{E} as the space LODIE;}.

Testing on null sequences is enough for Bochner integrability

“…We refer the reader to the books by Jarchow [5], Kothe [6], Perez Carreras and Bonet [7] or Robertson and Robertson [8] for the terminology about locally convex spaces and to the monographs by Bourbaki [2], Diestel and Uhl [3], Schwartz [9] or Thomas [10] for the properties of measurable functions. Our paper [4]…”

Dieudonné—köthe Duality for Vector—valued Function Spaces: Localization of Bounded Sets and Barrelledness

“…We think that measurability in the sense of Lusin, used e.g. in [3], [4], [10], [11], [12], [13], [41], [42], and [43], is the most appropriate for the case of a general locally convex space. If E is a Fréchet space, it coincides with the usual notion of strongly measurable function as the a.e.…”

“…Another useful fact is that if a measurable function f is localized in E D for some absolutely convex and closed set D, then the scalar function t → p D (f (t)) is also measurable. We shall make frequent use of the following result from [12].…”

“…Dieudonné [9] extended this theory to spaces of measurable functions by replacing the in the definition of Köthe and Toeplitz with a suitable . More recent developments include the study of the vector-valued cases: Köthe spaces of sequences from a locally convex space [5], [14], [17], [18], [33], [35], L p spaces with values in a Banach space [8] or a locally convex space [10], [11], [12], [13], [23], and Dieudonné-Köthe spaces of measurable functions with values in a normed space [15], [30], [32].…”

“…Hence in our case we need a theory of absolute integration of functions with values in a locally convex space analogous to the theory of Bochner integral for Banach spaces. We shall use the theories of summable functions and absolutely p-integrable functions developed by Thomas [41], [43] and continued in our previous paper [12].…”