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

• Authors: Regina Tix
• Venue: Theoretical Computer Science - Domains
• Year: 2001

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 &lgr; :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 suffcient conditions when the previous examples of d-cones are continuous and have an additive way-below relation