On the uniform density of C(X) ⊗ C(Y) in C(X × Y) ✩

We prove that if X and Y are compact Hausdorff spaces, then every f ∈ C(X × Y)+, i.e. f(x, y) ≥ 0 for all (x, y) ∈ X × Y, can be approximated uniformly from below and above by elements of the form , where fi ∈ C(X)+ and gi ∈ C(Y)+ for i = 1, 2, …, n. The proof uses only elementary topology. We use this result, in conjuction with Kakutani's M-spaces representation theorem, to obtain an alternative proof for a known property of Fremlin's Riesz space tensor product of Archimedean Riesz spaces.