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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

Kung Fu Ng
Dept. of Mathematics, Chinese University of Hong Kong, Hong Kong, P. R. China

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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