Real and Complex Linear Extensions for Locally Convex Cones

A locally convex cone is called linear over R (resp. C) if the scalar multiplication is extended to all real (resp. complex) numbers and a weakened version of the distributive law holds. We show that every locally convex cone may be embedded into a cone that is linear over R (resp. C). We investigate continuous linear functionals on these cones and introduce a new (modular) topology.