Mosco convergence and reflexivity

In this note we aim to show conclusively that Mosco convergence of convex sets and functions and the associated Mosco topology τ_M 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 X to be reflexive: (1) whenever A , A₁, A₂, A₃, ... are nonempty closed convex subsets of X with A = τ_M — lim A_n , then A° = τ_M — lim A°/_n ; (2) τ_M is a Hausdorff topology on the nonempty closed convex subsets of X ; (3) the arg min multifunction ∫ ⇉ {x ∈ X : ∫(x) = inf_x ∫} on the proper lower semicontinuous convex functions on X , equipped with τ_M , has closed graph.