Exercise 5, § 4

Let X be a compact space and let H be a set of continuous real-valued functions on X. Let R be the equivalence relation “u(x) = u(y) for all u ∈ H”. A continuous real-valued function f on X can then be uniformly approximated by polynomials (resp. polynomials with no constant term) in the functions of $H$ if and only if $f$ is constant on each equivalence class mod $R$ [resp. if and only if $f$ is constant on each equivalence class mod $R$ and vanishes at all points $x$ such that $u(x) = 0$ for all $u \in H$].