Infima of convex functions

Abstract

Let Γ ( X ) \Gamma (X) be the lower semicontinuous, proper, convex functions on a real normed linear space X X . We produce a simple description of what is, essentially, the weakest topology on Γ ( X ) \Gamma (X) such that the value functional f → inf f f \to \inf f is continuous on Γ ( X ) \Gamma (X) . When X X is reflexive, convergence of a sequence in this topology is equivalent to Mosco plus pointwise convergence of the corresponding sequence of conjugate convex functions.