Smooth Banach spaces, weak asplund spaces and monotone or usco mappings

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 denseGδ subset ofE at every point of whichf is Gateaux differentiable. More generally, for any maximal monotone operatorT on such a space, there exists a denseGδ 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.

[1]  J. Rainwater.  Yet more on the differentiability of convex functions , 1988 .

[2]  R. Phelps.  Gateaux differentiability spaces , 1989 .

[3]  E. Asplund.  Fréchet differentiability of convex functions , 1968 .

[4]  M. Day,et al.  Normed Linear Spaces , 1960 .

[5]  Eduardo H. Zarantonello,et al.  Dense single-valuedness of monotone operators , 1973 .

[6]  J. Borwein,et al.  A smooth variational principle with applications to subdifferentiability and to differentiability of convex functions , 1987 .

[7]  Dominikus Noll,et al.  Generic Gâteaux-differentiability of convex functions on small sets , 1990 .

[8]  Richard Haydon,et al.  A Counterexample to Several Questions about Scattered Compact Spaces , 1990 .

[9]  I. Ekeland,et al.  Generic Fréchet-differentiability and perturbed optimization problems in Banach spaces , 1976 .

[10]  S. Mazur.  Über konvexe Mengen in linearen normierten Räumen , 1933 .

[11]  Jonathan M. Borwein,et al.  Minimal Convex Uscos and Monotone Operators on Small Sets , 1991, Canadian Journal of Mathematics.

[12]  C. Stegall.  More gateaux differentiability spaces , 1985 .

[13]  David Preiss,et al.  Differentiability of Lipschitz functions on Banach spaces , 1990 .

[14]  John C. Oxtoby.  7. THE BANACH-MAZUR GAME AND BANACH CATEGORY THEOREM , 1958 .

[15]  Jens Peter Reus Christensen.  Theorems of Namioka and R. E. Johnson type for upper semicontinuous and compact valued set-valued mappings , 1982 .

[16]  Locally efficient monotone operators , 1990 .

[17]  P. Müller.  M. Day, Normed Linear Spaces. Second printing corrected. VIII + 139 S. Berlin/Göttingen/Heidelberg 1962. Springer-Verlag. Preis brosch. DM 17,80 , 1963 .

[18]  Renormages de Quelquesℒ(K) , 1986 .

[19]  R. Phelps,et al.  Gateaux Differentiability of Convex Functions on Banach Spaces , 1979 .

[20]  R. Phelps.  Convex Functions, Monotone Operators and Differentiability , 1989 .