Level sets and continuity of conjugate convex functions

Abstract

A finite-valued convex function on a nonempty convex set C in F can always be extended to a proper convex function on F by assigning it the value + 0o outside of C. Let F and G be real vector spaces in duality with respect to a bilinear functional (x, y) for x e F and y E G (see [1, p. 48]). We shall henceforth assume F and G have each been supplied with a topology compatible with this duality [1, p. 67], so that each can be identified with the space of continuous linear functionals on the other. Unless explicit notice is given, all questions of closure, continuity and boundedness refer to these given topologies. The formulas