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Spaces of Vector‐Valued Integrable Functions and Localization of Bounded Subsets
Author(s) -
Florencio Miguel,
Mayoral Fernando,
Paúl Pedro J.
Publication year - 1995
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.19951740109
Subject(s) - mathematics , bounded function , integrable system , pure mathematics , vector (molecular biology) , vector valued function , mathematical analysis , recombinant dna , biochemistry , chemistry , gene
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 }.