Weak upper topologies and duality for cones

In functional analysis it is well known that every linear functional definedon the dual of a locally convex vector space which is continuous for the weaktopology is the evaluation at a uniquely determined point of the given vectorspace. M. Schroeder and A. Simpson have obtained a similar result for lowersemicontinuous linear functionals on the cone of all Scott-continuousvaluations on a topological space endowed with the weak upper topology, anasymmetric version of the weak topology. This result has given rise to severalproofs, originally by the Schroeder and Simpson themselves and, more recently,by the author of these Notes and by J. Goubault-Larrecq. The proofs developedfrom very technical arguments to more and more conceptual ones. The presentNote continues on this line, presenting a conceptual approach inspired byclassical functional analysis which may prove useful in other situations