Conjugate convex functions and the epi-distance topology

Let Γ (X) denote the proper, lower semicontinuous, convex functions on a normed linear space, and let Γ ^ * (X^ * ) denote the proper, weak*-lower semicontinuous, convex functions on the dual X^ * of X. It is well-known that the Young-Fenchel transform (conjugate operator) is bicontinuous when X is reflexive and both Γ (X) and Γ ^ * (X^ * ) are equipped with the topology of Mosco convergence. We show that without reflexivity, the transform is bicontinuous, provided we equip both Γ (X) and Γ ^ * (X^ * ) with the (metrizable) epi-distance topology of Attouch and Wets. Convergence of a sequence of convex functions fₙ to f in this topology means uniform convergence on bounded subsets of the associated sequence of distance functional d( · ,epifₙ) to d( · ,epif).