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Approximation by DC functions and application to representation of a normed semigroup. (English) Zbl 1315.46050
The present article presents an approximation result in the spirit of M. Cepedello Boiso [Isr. J. Math. 106, 269–284 (1998; Zbl 0920.46010)], namely, that each continuous function defined on a compact subset $$\Omega$$ of a locally convex space $$L$$ can be approximated by DC functions. Indeed, the subspace of the so-called DAPS-functions (difference of affine polyhedral support functions) is a lattice in $$C(\Omega )$$ and the Kakutani-Krein approximation theorem applies. It is finally stated that, in the case when $$\Omega$$ is the $$w^{\ast }$$-compact set $$B_{X^{\ast }}$$ of the locally convex space $$(X^*,w^{\ast })$$, where $$X$$ is a Banach space, then $$C(\Omega )^{\ast}$$ is the dual of the normed semigroup $$b(X)$$ generated by the closed balls in $$X$$.

##### MSC:
 46G05 Derivatives of functions in infinite-dimensional spaces 49J52 Nonsmooth analysis 54C30 Real-valued functions in general topology
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