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 generalizations of this elementary statement 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 function/(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 statements hold.Therefore we say that a Banach space is a strong di//erentiability space (SDS) if the following theorem holds for it.T~ o~ M. (Strong Differentiability Theorem.)Every continuous convex/unction is Frdchet di//erentiable on a dense G~ subset o/its domain o/continuity.If the following statement holds for a Banach space, we call it a weak di//erentiability space (WDS): T~E OR~ M. (Weak Differentiability Theorem.)Every continuous convex /unction is Gdteaux di//erentiable on a dense G~ subset o/ its domain o/ continuity.Some general remarks on these definitions are in order here.First, by a continuous convex function / on the Banach space E, we mean a function which is defined and convex on all of E, with values in ( -c~, co ], and finite valued and continuous at least at some point of E. Then the set of all points of E where / is finite valued and continuous is a non-empty, open, and convex subset of E which we call the domain o/ continuity of/.It is equal to the interior of dom/, the e//ective domain of/, defined by dom/= {x E E:/(x) < ~ }.