Let \(R\) be a normed linear space, \(\{\phi_ i: i-1,2,\dots,n\}\) be \(n\) linearly independent elements of \(R\), \(\Phi_ n=\text{span}\{\phi_ 1,\dots,\phi_ n\}\) be the \(n\)-dimensional subspace of \(R\) (called the set of generalized polynomials) and \(K\) be an arbitrary convex subset of \(\Phi_ n\). This paper gives a general characterization for a best approximation from \(K\) in form of `zero in the convex hull'. Applying this characterization to the uniform approximation by generalized polynomials with restricted ranges, an alternation characterization is obtained. The results proved in this paper contain the special cases of interpolatory approximation, positive approximation, copositive approximation, and the classical characterizations in forms of convex hull and alternation in approximation without restriction. Main result: Assume that \(\Phi_ n\subset R\), the convex set \(K\subset \Phi_ n\), \(f\in R| K\). If \(p\in K\) is not a best approximation to \(f\) from \(\Phi_ n\), or the best-approximation to \(f\) from \(\Phi_ n\) is unique, then \(p\) is a best approximation to \(f\) from \(K\) iff there exists a non-vanishing vector \(g\in K^*\) for which \(0\in h(\{g\}\cup K^*_ p\), which means \(0\in h(K^*_ p)\) if \(K^*=\phi\). Here \[ K^*=h\{g\in\Phi_ n: g\neq 0,\;(g,q-p)\leq 0\;\forall q\in K\}, \] \[ K^*_ p=h\{g\in\Phi_ n: g\neq 0,\;(g,q-p)\leq 0\;\forall q{\i}K_ p\}, \] where \(K_ p=\{q\in\Phi_ n: \| f-q\|\leq\| f-p\|\}\) and \(h(Q)\) denotes the convex hull of the set \(Q\) of \(\Phi_ n\) and \((\cdot,\cdot)\) denotes inner product in \(\Phi_ n\).