Asplund sets, differentiability and subdifferentiability of functions in Banach spaces (Q855460)

Preiss' celebrated differentiability theorem and the Borwein--Preiss variational principle apply (in the Fréchet case) to functions which are defined on Asplund spaces. The purpose of the present work is to investigate differentiability of functions defined on general Banach spaces, but along directions which belong to a given Asplund set \(M\). It is shown that Stegall's factorization theorem allows indeed to carry over the classical results to this frame, and for instance to extend a result of Giles and Sciffer by showing that Lipschitz functions are densely \(M\)-differentiable when \(M\) is an Asplund set, and a result of Fabian by showing dense \(M\)-subdifferentiability of lower semi-continuous functions.