Approximation by DC Functions and Application to Representation of a Normed Semigroup

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