On the global shape of convex functions on locally convex spaces

Abstract

In the recent paper [1] D Azagra studies the global shape of continuous convex functions defined on a Banach space X. More precisely, when X is separable, it is shown that for every continuous convex function f : X → R there exist a unique closed linear subspace Y of X, a continuous function h : X/Y → R with the property that lim t→∞ h(u+tv) = ∞ for all u, v ∈ X/Y , v = 0, and x * ∈ X * such that f = h • π + x * , where π : X → X/Y is the natural projection. Our aim is to characterize those proper lower semicontinuous convex functions defined on a locally convex space which have the above representation. In particular, we show that the continuity of the function f and the completeness of X can be removed from the hypothesis of Azagra's theorem.