Journal of Convex Analysis 26 (2019), No. 3, 761--772
Copyright Heldermann Verlag 2019
Differentiability of Convex Functions on a Locally Convex Topological Vector Space
Xi Yin Zheng
Dept. of Mathematics, Yunnan University, Kunming 650091, P. R. China
xyzheng@ynu.edu.cn
Kung Fu Ng
Dept. of Mathematics, Chinese University of Hong Kong, Hong Kong, P. R. China
kfng@math.cuhk.edu.hk
We introduce the notion of a smooth set in a locally convex topological vector space and extend Asplund's result on the strong differentiability space. We also establish Gateaux differentiability of a continuous convex function in a locally convex topological vector space. In particular, we extend Mazur's classical theorem on Gateaux differentiability from a separable Banach space to a separable locally convex topological vector space.
Keywords: Topological vector space, smooth set, uniform differentiability, Gateaux differentiability.
MSC: 52A41, 49J50, 46A55
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