Abstract

Let Ω\Omega be a nonempty compact set of a locally convex space LL, and let C(Ω)C(\Omega) be the Banach space of all real-valued continuous functions on Ω\Omega endowed with the sup⁡\sup-norm. In this paper, we show first that for every f∈C(Ω)f\in C(\Omega), and for every ε>0\varepsilon>0, there are continuous affine functions (gi)i=1m,(hj)j=1n(g_i)_{i=1}^m, (h_j)_{j=1}^n on LL for some m,n∈Nm,n\in\mathbb{N} such that ∣f(ω)−[(g1∨g2∨⋯∨gm)−(h1∨h2∨⋯∨hn)](ω)∣<ε|f(\omega)-[(g_1\vee g_2\vee\cdots\vee{g_m})-(h_1\vee h_2\vee \cdots\vee{h_n})](\omega)|<\varepsilon uniformly for ω∈Ω\omega\in\Omega. We prove then that if Ω=BX∗\Omega=B_{X^*}, the closed unit ball of X∗X^* of a Banach space XX endowed with the w∗w^*-topology, then C(Ω)∗C(\Omega)^* is just the dual of the normed semigroup b(X)(X) generated closed balls in XX

Author information

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Lixin Cheng

School of Mathematical Sciences, Xiamen University, Xiamen 361005, China

lxcheng@xmu.edu.cn

Yu Zhou

School of Fundamental Studies, Shanghai University of Engineering Science, Shanghai 201620, China

roczhou_fly@126.com

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Suggested citation

L. Cheng, Y. Zhou. “Approximation by DC Functions and Application to Representation of a Normed Semigroup.” Journal of Convex Analysis 21 (2014), No. 3, 651–661. https://doi.org/10.68381/jca21034

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Published by Heldermann Verlag, 2014. Rights now held by Banach Press.