On the Young-Fenchel transform for convex functions

Abstract

Let Γ ( X ) \Gamma (X) be the proper lower semicontinuous convex functions on a reflexive Banach space X X . We exhibit a simple Vietoris-type topology on Γ ( X ) \Gamma (X) , compatible with Mosco convergence of sequences of functions, with respect to which the Young-Fenchel transform (conjugate operator) from Γ ( X ) \Gamma (X) to Γ ( X ∗ ) \Gamma ({X^*}) is a homeomorphism. Our entirely geometric proof of the bicontinuity of the transform halves the length of Mosco’s proof of sequential bicontinuity, and produces a stronger result for nonseparable spaces.