Some results on Hahn–Banach-type theorems for continuous D-cones

Abstract

Like the extended non-negative reals R ̄ + equipped with the Scott topology, there are other real topological cones such that the specialisation order yields a directed complete partially ordered set (dcpo). We will call them d-cones. Further examples are the extended probabilistic powerdomain, the set of all lower semicontinuous functions f : X → R ̄ + for any topological space X and arbitrary products of given d-cones. The dual cone C ∗ for a given d-cone C consists of all linear continuous functions Λ : C → R ̄ + . With respect to the pointwise order, addition and scalar multiplication the dual cone becomes also a d-cone. We are interested in obtaining results with our concept of d-cones that are comparable to Hahn–Banach-type theorems in functional analysis. Indeed, we can prove an Extension Theorem and a Separation Theorem for the continuous d-cones. In particular, the second implies that the elements of the dual cone C ∗ separate the points of C. As a consequence of the Extension Theorem, we obtain a Sum Theorem for continuous d-cones. We will give some sufficient conditions when the previous examples of d-cones are continuous and have an additive way-below relation.