*(English)*Zbl 0763.46006

Summary: We aim to show conclusively that Mosco convergence of convex sets and functions and the associated Mosco topology ${\tau}_{M}$ 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,{A}_{1},{A}_{2},{A}_{3},\cdots $ are nonempty closed convex subsets of $X$ with $A={\tau}_{m}-lim{A}_{n}$, then ${A}^{0}={\tau}_{M}-lim{A}_{n}^{0}$;

(2) ${\tau}_{M}$ is a Hausdorff topology on the nonempty closed convex subsets of $X$;

(3) the arg min multifunction $f\rightrightarrows \{x\in X$: $f\left(x\right)={inf}_{X}f\}$ on the proper lower semicontinuous convex functions on $X$, equipped with ${\tau}_{M}$, has a closed graph.