Lipschitz functions on Banach spaces which are actually on Asplund spaces

SINCE Namioka and Phelps, starting with Asplund’s pioneering work, introduced the no-tion of Asplund spaces (those are Banach spaces on which every continuous convex function isFrechet differentiable on a dense G_δ subset) and proved that the dual of an Asplund space hasthe Radon-Nikodym property (RNP), the study of differentiability properties of functions oninfinite dimensional spaces has continued widely and deeply (see, for instance, Phelps andGiles). The research attained a great achievment after Stegall’s theorem: If the dualspace E~* has the RNP, then E is an Asplund space. Because of the N-Ph-S theorem, we have