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Mosco convergence and reflexivity

1990; American Mathematical Society; Volume: 109; Issue: 2 Linguagem: Inglês

10.1090/s0002-9939-1990-1012924-9

1088-6826

Gerald Beer, Jonathan M. Borwein,

Advanced Banach Space Theory

In this note we aim to show conclusively that Mosco convergence of convex sets and functions and the associated Mosco topology τ M {\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 X to be reflexive: (1) whenever A , A 1 , A 2 , A 3 , … A,{A_1},{A_2},{A_3}, \ldots are nonempty closed convex subsets of X X with A = τ M − lim A n A = {\tau _M} - \lim {A_n} , then A ∘ = τ M − lim A n ∘ {A^ \circ } = {\tau _M} - \lim A_n^ \circ ; (2) τ M {\tau _M} is a Hausdorff topology on the nonempty closed convex subsets of X X ; (3) the arg min multifunction f ⇉ { x ∈ X : f ( x ) = inf X f } f \rightrightarrows \{ x \in X:f(x) = \inf {}_Xf\} on the proper lower semicontinuous convex functions on X X , equipped with τ M {\tau _M} , has closed graph.