Weak upper topologies and duality for cones

TL;DR

This paper explores duality and topological structures in cones, extending classical functional analysis results to asymmetric topologies like the weak upper topology, with a focus on conceptual proofs.

Contribution

It introduces a conceptual approach to duality in cones using weak upper topologies, inspired by classical functional analysis, simplifying previous technical proofs.

Findings

Established a duality result for lower semicontinuous linear functionals on cones.

Provided a more conceptual proof method inspired by classical analysis.

Extended duality concepts to asymmetric topologies like the weak upper topology.

Abstract

In functional analysis it is well known that every linear functional defined on the dual of a locally convex vector space which is continuous for the weak topology is the evaluation at a uniquely determined point of the given vector space. M. Schroeder and A. Simpson have obtained a similar result for lower semicontinuous linear functionals on the cone of all Scott-continuous valuations on a topological space endowed with the weak upper topology, an asymmetric version of the weak topology. This result has given rise to several proofs, originally by the Schroeder and Simpson themselves and, more recently, by the author of these Notes and by J. Goubault-Larrecq. The proofs developed from very technical arguments to more and more conceptual ones. The present Note continues on this line, presenting a conceptual approach inspired by classical functional analysis which may prove useful in other situations.