Weierstrass approximation and Stone--Weierstrass theorems

1. Let \(a<b\) and let \(f\colon[a,b]\to\CC\) be continuous. Then there exist polynomials \(P_n\) with complex coefficients such that \(P_n\to f\) uniformly on \([a,b]\). If \(f\) is real-valued, the \(P_n\) may be taken with real coefficients.

2. Let \(K\) be a compact metric space, let \(C(K)\) be the algebra of continuous functions \(K\to\CC\), and let \(\mathcal A\subseteq C(K)\) be a subalgebra, that is, a complex vector subspace closed under pointwise multiplication. Suppose that

• \(\mathcal A\) is self-adjoint: \(\bar g\in\mathcal A\) for every \(g\in\mathcal A\);

• \(\mathcal A\) separates points of \(K\): for all \(x\neq y\) in \(K\) there exists \(g\in\mathcal A\) with \(g(x)\neq g(y)\);

• \(\mathcal A\) vanishes at no point of \(K\): for every \(x\in K\) there exists \(g\in\mathcal A\) with \(g(x)\neq0\).

Then for every \(f\in C(K)\) there exist \(f_n\in\mathcal A\) such that \(f_n\to f\) uniformly on \(K\). [@Rud76].