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Mosco convergence and reflexivity. (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 $f⇉\left\{x\in X$: $f\left(x\right)={inf}_{X}f\right\}$ on the proper lower semicontinuous convex functions on $X$, equipped with ${\tau }_{M}$, has a closed graph.

##### MSC:
 46B10 Duality and reflexivity in normed spaces 46B20 Geometry and structure of normed linear spaces 49J45 Optimal control problems involving semicontinuity and convergence; relaxation 54B20 Hyperspaces (general topology) 54C60 Set-valued maps (general topology) 90C25 Convex programming