
Journal of Convex Analysis 19 (2012), No. 1, 201212 Copyright Heldermann Verlag 2012 On Approximation by ΔConvex Polyhedron Support Functions and the Dual of cc(X) and wcc(X) Lixin Cheng School of Mathematical Sciences, Xiamen University, Xiamen 361005, P. R. China lxcheng@xmu.edu.cn Yu Zhou School of Mathematical Sciences, Xiamen University, Xiamen 361005, P. R. China roczhoufly@126.com [Abstractpdf] 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 $wcc(X)$. Keywords: Weierstrass theorem, function approximation, weakly continuous function, weakly compact set, normed semigroup, Deltaconvex polyhedron support function. MSC: 41A10, 41A30, 41A65, 46A20; 46B20, 46E05, 46J10 [ Fulltextpdf (166 KB)] for subscribers only. 