Approximation by DC functions and application to representation of a normed semigroup (Q2922467)

The present article presents an approximation result in the spirit of \textit{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 \(X\).