Topological duals of Banach function spaces

TL;DR

This paper characterizes the topological duals of Banach function spaces, extending classical duality results to more general spaces including Musielak-Orlicz and convex risk measure spaces, with special focus on rearrangement invariant cases.

Contribution

It provides a unified description of duals for a broad class of Banach function spaces, including new results for Musielak-Orlicz and risk measure spaces.

Findings

Duals are identified with a sum of K"othe dual, finitely additive measures, and the annihilator of L^8.

In rearrangement invariant spaces, the finitely additive component vanishes.

New duality results for Lebesgue, Orlicz, Lorentz-Orlicz, and risk measure spaces.

Abstract

This paper studies topological duals of Banach function spaces (BFS). We assume a finite measure but our arguments extend to general locally convex function spaces whose topology is generated by seminorms that satisfy the usual BFS axioms. The dual is identified with the direct sum of another space of random variables (K\"othe dual), a space of purely finitely additive measures and the annihilator of $L^\infty$. In the special case of rearrangement invariant spaces, the second component in the dual vanishes and we obtain various classical as well as new duality results e.g. on Lebesgue, Orlicz, Lorentz-Orlicz spaces and spaces of finite moments. Beyond rearrangement invariant spaces, we find the topological duals of Musielak-Orlicz spaces and those associated with general convex risk measures.