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

Abstract

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 ∑ i = 1 n f i g i , where f i ∈ C(X) + and g i ∈ 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.