Weierstrass Approximation Theorem for a Product Space?

Question

I am faced with the following problem:

Let $X$ and $Y$ be compact Hausdorff spaces and $f$ belong to $C(X \times Y)$. Show that for each $\epsilon > 0$, there are functions $f_{1}, f_{2}, \cdots , f_{n}$ in $C(X)$ and functions $g_{1}, g_{2}, \cdots, g_{n}$ in $C(Y)$ such that $\displaystyle \left \vert f(x,y) - \sum_{k=1}^{n}f_{k}(x)\cdot g_{k}(y) \right \vert < \epsilon$ for all $ (x,y) \in X \times Y$.

The only idea that I have is that this is kind of like a Weierstrass Approximation Theorem for the product space $X \times Y$, but I am at a loss for how to prove it.

Any suggestions on how to proceed would be most welcome. Also, please be willing to answer follow-up questions, because I'm the kind of person who has them.

Thanks.

Answer 1

Stone-Weierstrass theorem. Functions of the given form are an algebra ...