I’m currently trying to familiarize myself with the Stone-Weierstrass theorem and its applications. When browsing Wikipedia, I found the following:
If X and Y are two compact Hausdorff spaces and f : X × Y → R is a continuous function, then for every ε > 0 there exist n > 0 and continuous functions f1, ..., fn on X and continuous functions g1, ..., gn on Y such that || f − ∑ fi gi || < ε
I just can’t think of how I would prove that statement. Is there any textbook you know which provides a step by step solution or could you provide some guidance?
The said claim can be found here.