Mosco convergence and reflexivity

Beer, Gerald; Borwein, Jonathan M.

doi:10.1090/S0002-9939-1990-1012924-9

Mosco convergence and reflexivity
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by Gerald Beer and Jonathan M. Borwein

Proc. Amer. Math. Soc. 109 (1990), 427-436

DOI: https://doi.org/10.1090/S0002-9939-1990-1012924-9

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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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