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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 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,A 1 ,A 2 ,A 3 , are nonempty closed convex subsets of X with A=τ m -limA n , then A 0 =τ M -limA n 0 ;

(2) τ M is a Hausdorff topology on the nonempty closed convex subsets of X;

(3) the arg min multifunction f{xX: f(x)=inf X f} on the proper lower semicontinuous convex functions on X, equipped with τ M , has a closed graph.

MSC:
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