Banach空间的p— Asplund 伴随空间

我们称一个定义在Banach空间E上的连续凸函数f具有Frechet可微性质（FDP），如果E上的每个实值凸函数g≤f均在E一个稠密的Gδ－子集上Frechet可微。本文主要证明了：对任何Banach空间E，均存在一个局部凸相容拓扑p使得1）（E，p)是Hausdorff局部凸空间；2） E上的每个范数连续具有FDP的凸函数均是p－连续的；3）每个p-连续的凸函数均具有FDP ；4）p等价某个范数拓扑当且仅不E是Asplund空间。

On the p-Asplund Space of a Banach Space

Recently, a sequence of articles studied the Frechet differentiability property of convex functions on general Banach spaces and even on topological linear spaces. Cheng et al introduced the notion of the FDP (Frechet differentiability property) of convex functions: an extended real valued proper convex function f on a Banach space E is said to have the FDP if every continuous convex function g with g≤f on E is Frechet differentiable on a dense Gδ subset of E. This paper mainly shows that all such continuous convex functions f on the space E are exactly all continuous convex functions on a locally convex space (E, τ) for some suitably locally convex topology τ, that the space (E, τ) is normable if and only if E is an Asplund space. It also presents a revised version of the main theorems of Cheng et al.