On Approximation by Δ-Convex Polyhedron Support Functions and the Dual of cc(X) and wcc(X)

The classical Weierstrass theorem states that every continuous function $f$ defined on a compact set $\Omega \subset\mathbb{R}^n$ can be uniformly approximated by polynomials. We show first that it is again valid if $\Omega$ is a compact Hausdorff metric space, i.e., it holds in the following sense: there exists a surjective isometry $T$ from a compact set $K_\Omega$ of a Banach sequence space $S$ to $\Omega$, such that for every $\varepsilon>0$ there is an $n$ variable polynomial $p$ satisfying $$|f(T(s))-p(s_1,s_2,\cdots,s_n)|<\varepsilon,\;\forall s=(s_j)\in K_{\Omega}.$$ We prove also that for any $weak$ ($w^*$, resp.) continuous positively homogenous function $f$ defined on a (dual, resp.) Banach space $X$ ($X^*$, resp.) then for all $\varepsilon>0$ and for every weakly compact set $K\subset X$( $w^*$ compact set $K\subset X^*$), there exist $\phi_i\in X^*$ ($X,$ resp.) for $i=1,2,\cdots, m,$ and $\psi_j\in X^*$ ($X,$ resp.) for $j=1,2, \cdots,n$ such that $$|f(x)-[(\phi_1\vee\phi_2\vee\cdots\vee\phi_m)(x)- (\psi_1\vee\psi_2\vee\cdots\vee\psi_n)(x)]|<\varepsilon$$ uniformly for $x\in K.$ Let $cc(X)$ ($wcc(X)$, reps.) be the norm semigroup consisting of all nonempty (weakly, resp.) compact convex sets of the space $X$. As its application, we give two representation theorems of the duals of $cc(X)$ and $wcc(X)$.