Prove that if f in $C(X \times Y)$ then there exists functions.

Question

Let $X$ and $Y$ be compact metric spaces. I am trying to prove that if $f \in C(X \times Y)$ and $\varepsilon > 0 $, then there exist functions $g_1, g_2,...,g_n \in C(X)$ and $h_1,...h_n \in C(Y)$ so that $|f(x,y) - \sum_{k=1}^n g_k(x)h_k(y) | &lt \varepsilon $ for all $(x,y) \in X \times Y$. I need help. Thank you.

Answer 1

The space of functions of the form $(x,y)\mapsto \sum_{i=1}^nf_i(x)g_i(y)$ (for some $n\geq 1$ and some $f_1,\ldots,f_n\in C(X)$, $g_1,\ldots,g_n\in C(Y)$) is a Banach algebra which separates the points of $X\times Y$. Your statement is thus a direct application of the Stone-Weierstrass theorem.