Smooth Banach spaces, weak asplund spaces and monotone or usco mappings
- 111 Downloads
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
Unable to display preview. Download preview PDF.
- [B-F-K]J. Borwein, S. Fitzpatrick and P. Kenderov,Minimal convex uscos and monotone operators on small sets, preprint.Google Scholar
- [St1]C. Stegall,A class of topological spaces and differentiation of functions on Banach spaces, Proc. Conf. on Vector Measures and Integral Representations of Operators, Vorlesungen aus dem Fachbereich Math., Heft 10 (W. Ruess, ed.), Univ. Essen, 1983.Google Scholar
- [St2]C. Stegall,More Gateaux differentiability spaces, Proc. Conf. Banach Spaces, Univ. Missouri, 1984 (N. Kalton and E. Saab, eds.), Lecture Notes in Math., Vol. 1166, Springer-Verlag, Berlin, 1985, pp. 158–168.Google Scholar
- [V-V]A. Verona and M. E. Verona,Locally efficient monotone operators, Proc. Am. Math. Soc., to appear.Google Scholar