Approximating continuous functions from $K\times L$ into $[0,1]$

Question

Let $K$ and $L$ be compact Hausdorff spaces, let $f:K\times L\to [0,1]$ be continuous and let $\varepsilon>0$. Can we find continuous $g_{1},...,g_{n}:K\to[0,+\infty)$ and $h_{1},...,h_{n}:L\to[0,+\infty)$ such that $\left\|f-\sum\limits_{i=1}^{n}g_{i}h_{i}\right\|<\varepsilon$?

If there was no requirement of positivity of $g_{i}$'s and $h_{i}$'s, the result would follow immediately from Stone-Weierstrass theorem.

It can also be deduced from some properties of tensor products of vector lattices, but the proofs of those properties are rather difficult, and so we are looking for a direct proof.

Answer 1

The following provides a construction of $g_i, h_i$ satisfying the required conditions (the ranges of these functions can even be in $[0, 1]$):

Claim: For each $k \in K$, there exists an open neighborhood $U_k$ of $k$ s.t. $|f(k', l) - f(k, l)| < \epsilon$ for all $k' \in U_k, l \in L$.

Proof of claim: For each $l \in L$, there exists an open neighborhood $V$ of $(k, l)$ s.t. $|f(v) - f(k, l)| < \epsilon/2$ for all $v \in V$. By the construction of the product topology, we may assume $V = K_l \times L_l$ where $K_l$ is an open neighborhood of $k$ and $L_l$ is an open neighborhood of $l$. Then $\{L_l\}$ is an open cover of $L$ so we may take a finite subcover $L_{l_1}, \cdots, L_{l_m}$. Let $U_k = \cap_{i = 1}^m K_{l_i}$. It is then easy to verify that this satisfies the requirements of the claim.

Now, all such $\{U_k\}$ is an open cover of $K$, so we may again take a finite subcover $U_{k_1}, \cdots, U_{k_n}$. Let $\{g_i\}$ be a partition of unity subordinate to $\{U_{k_i}\}$ and $h_i$ be defined by $h_i(l) = f(k_i, l)$. It is easy to prove they satisfy the required conditions.