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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Mosco convergence and reflexivity


Authors: Gerald Beer and Jonathan M. Borwein
Journal: Proc. Amer. Math. Soc. 109 (1990), 427-436
MSC: Primary 46B10; Secondary 46B20, 49J45, 54B20, 90C25
MathSciNet review: 1012924
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Abstract: In this note 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}, \ldots $ are nonempty closed convex subsets of $ X$ with $ A = {\tau _M} - \lim {A_n}$, then $ {A^ \circ } = {\tau _M} - \lim A_n^ \circ $; (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(x) = \inf {}_Xf\} $ on the proper lower semicontinuous convex functions on $ X$, equipped with $ {\tau _M}$, has closed graph.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1990-1012924-9
PII: S 0002-9939(1990)1012924-9
Keywords: Mosco convergence, polar convex set, conjugate convex function, arg min multifunction, reflexivity
Article copyright: © Copyright 1990 American Mathematical Society