w-Frchet可微性质和Radon-Nikod(?)m性质以及w-Asplund空间

The w-Frchet Differentiability Property, the Radon-Nikod(?)m Property and w-Asplund Spaces

我们称定义在一个Banach空间的对偶空间上的广义实值w*-下半连续凸函数f具有w*-Frechet可微性质(w*-FDP),如果对于该对偶空间上的每个w*-下半连续的广义实值凸函数g,只要g≤f,就有g在intdom g的某个稠密的Gδ-子集上处处Frechet可微.本文用集合的Radon-Nikodym性质刻划了该种函数的特征.作为它的一个直接推论,给出了局部化的Collier定理.

We say an extended real-valued w*-lower semi-continuous convex function f on a dual Banach space E* has w*-Frechet differentiablity property (w*-FDP), if for every w*-lower semi-continuous proper convex function g on E*, g ≤f, implies that g is Frechet differentiable at each point of a dense Gδ-subset of the interior of dom f (the effective domain of f). This paper characterizes the w*-FDP of the functions f by the Radon- Nikodym property of subsets in the pre-dual E, and as a direct consequence, it gives a localized Collier's theorem.