Smooth Banach spaces, weak asplund spaces and monotone or usco mappings
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It is shown that if a real Banach spaceE admits an equivalent Gateaux differentiable norm, then for every continuous convex functionf onE there exists a denseG δ subset ofE at every point of whichf is Gateaux differentiable. More generally, for any maximal monotone operatorT on such a space, there exists a denseG δ subset (in the interior of its essential domain) at every point of whichT is single-valued. The same techniques yield results about stronger forms of differentiability and about generically continuous selections for certain upper-semicontinuous compact-set-valued maps.
KeywordsBanach Space Monotone Operator Hausdorff Space Maximal Monotone Operator Winning Strategy
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