On the Young-Fenchel Transform for Convex Functions

Abstract

Let $\Gamma (X)$ be the proper lower semicontinuous convex functions on a reflexive Banach space $X$. We exhibit a simple Vietoris-type topology on $\Gamma (X)$, compatible with Mosco convergence of sequences of functions, with respect to which the Young-Fenchel transform (conjugate operator) from $\Gamma (X)$ to $\Gamma ({X^*})$ 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.