Journal of Convex Analysis 19 (2012), No. 1, 201--212
Copyright Heldermann Verlag 2012
On Approximation by Δ-Convex 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
[Abstract-pdf]
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, Delta-convex polyhedron support function.
MSC: 41A10, 41A30, 41A65, 46A20; 46B20, 46E05, 46J10
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