Summary: We aim to show conclusively that Mosco convergence of convex sets and functions and the associated Mosco topology are useful notions only in the reflexive setting. Specifically, we prove that each of the following conditions is necessary and sufficient for a Banach space to be reflexive:
(1) whenever are nonempty closed convex subsets of with , then ;
(2) is a Hausdorff topology on the nonempty closed convex subsets of ;
(3) the arg min multifunction : on the proper lower semicontinuous convex functions on , equipped with , has a closed graph.