论文信息 - Fréchet differentiability of convex functions

Fréchet differentiability of convex functions

A continuous convex function of one real variable is differentiable, except perhaps at a countable subset of its interval of continuity. The present paper deals with generalizat ions of this e lementary s ta tement to convex functions which are defined on some Banach space E, and continuous in the norm topology, with "differentiable" replaced either by "Frdchet differentiable" or "Gateaux differentiable". Since for E = L ~ ( 0 , 1 ) the very norm funct ion/ (x) = Ilxll for x in E, which is convex and continuous on all of E, is nowhere even G~teaux differentiable (Mazur [13]), this amounts to a classification of the category of all Banach spaces depending upon whether certain differentiability s ta tements hold. Therefore we say tha t a Banach space is a strong di//erentiability space (SDS) if the following theorem holds for it.

E. Asplund

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