Theorem 2 (Stone)

Let X be a compact space, and let H be a vector subspace of $\mathscr{C}(X; \mathbf{R})$ such that 1) the constant functions belong to H; 2) if $u \in H$, then $|u| \in H$; 3) H separates the points of X. Then every continuous real-valued function on X can be uniformly approximated by functions of H.

It is enough to show that H satisfies the conditions of the Corollary to Proposition 2. By hypothesis, if $u \in H$ and $v \in H$, we have

and

On the other hand, let x and y be any two distinct points of X, and let α, β be any two real numbers. By hypothesis, there is a function $h \in H$ such that $h(x) \neq h(y)$: say $h(x) = \gamma$ and $h(y) = \delta$. Since the constants belong to H, the function

belongs to H and is such that $g(x) = \alpha$ and $g(y) = \beta$.