凸函数的方向可微性与一种广义Asplund空间

The Directional Differentiability of a Convex Function and a Kind of Generalized Asplund Spaus

讨论了Banach空间X上凸函数的方向可做性，引入了一种广义Asplund空间．若Y是X的子空间，且X的每一凸开集D上的连续凸函数在D的一个稠密Gδ集上沿（Y可微）Y的单位球一致可做，就称X关于Y是（弱）Aspund空间．我们得到当子空间Y是Asplund空间时，X关于Y是Asglund空间；当Y是光滑空间，弱紧生成的Banach空间或是可分空间时，X关于Y是弱Aspund空间．

In this paper, we discuss the directional differentiability of a convex function on a Banch space and introduce a kind of generalized Asplund spaces: If Y be a subspace of X and every continuous convex function on an open convex subset D of X is Y differentiable uniormly differentiable in the unit ball of Y on a dense Gδ subset of D, X is called a (weak) Asplund space with respect ot Y. We obtain that X is an As-plund space with respect Y if Y is an Asplund subspace of X and that X is a weak Aspund space with re-spect to Y if Y is a smooth subspace、 a weakly and compacdy generalized subspace of or a separable subspace of X.