Exercise 4, § 4

Let X be a compact space and let H be a vector subspace of C(X; R) such that |u| ∈ H whenever u ∈ H. Then a continuous real-valued function f on X can be uniformly approximated by functions belonging to H if and only if, for each pair x, y of points of X, the function f satisfies every linear relation αf(x) = βf(y) where (α, β) are such that αβ ≥ 0 and αg(x) = βg(y) for every function g ∈ H. [Apply Proposition 2 of no. 1, observing that the image of H under the mapping u → (u(x) u(y)) is the whole plane R^2, or a line of equation αX = βY with αβ ≥ 0, or else consists of the single point (0, 0).]