On the Young-Fenchel Transform for Convex Functions

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