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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 $\tau\sb M$ 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 $A,A\sb 1,A\sb 2,A\sb 3,\dots$ are nonempty closed convex subsets of $X$ with $A=\tau\sb m-\lim A\sb n$, then $A\sp 0=\tau\sb M- \lim A\sb n\sp 0$; (2) $\tau\sb M$ is a Hausdorff topology on the nonempty closed convex subsets of $X$; (3) the arg min multifunction $f\rightrightarrows\{x\in X$: $f(x)=\inf\sb X f\}$ on the proper lower semicontinuous convex functions on $X$, equipped with $\tau\sb M$, has a closed graph.

46B10Duality and reflexivity in normed spaces
46B20Geometry and structure of normed linear spaces
49J45Optimal control problems involving semicontinuity and convergence; relaxation
54B20Hyperspaces (general topology)
54C60Set-valued maps (general topology)
90C25Convex programming
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