Abstract
This paper studies topological duals of locally convex function spaces that are natural generalizations of Fréchet and Banach function spaces. The dual is identified with the direct sum of another function space, a space of purely finitely additive measures and the annihilator of L^infty . This allows for quick proofs of various classical as well as new duality results e.g. in Lebesgue, Musielak–Orlicz, Orlicz–Lorentz space and spaces associated with convex risk measures. Beyond Banach and Fréchet spaces, we obtain completeness and duality results in general paired spaces of random variables.
Highlights
Banach function spaces (BFS) provide a convenient set up for functional analysis in spaces of measurable functions
This paper studies topological duals of more general locally convex function spaces where the topology is generated by an arbitrary collection of seminorms satisfying the usual BFS axioms
Building on the classical result of Yosida and Hewitt [37,Section 2] on the dual of L∞, we identify the topological dual as the direct sum of another space of random variables (Köthe dual), a space of purely finitely additive measures and the annihilator
Summary
Banach function spaces (BFS) provide a convenient set up for functional analysis in spaces of measurable functions. Many well known properties of e.g. Lebesgue spaces and Orlicz spaces extend to BFS with minor modifications; see e.g. Extensions to Fréchet function spaces have been studied e.g. in [4]. This paper studies topological duals of more general locally convex function spaces where the topology is generated by an arbitrary collection of seminorms satisfying the usual BFS axioms. Building on the classical result of Yosida and Hewitt [37,Section 2] on the dual of L∞, we identify the topological dual as the direct sum of another space of random variables (Köthe dual), a space of purely finitely additive measures and the annihilator
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