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.

  • $\begingroup$ Apply the general version of Stone-Weierstrass theorem to the subalgebra $\{ \sum f_i g_i \vert f_i \in \mathcal{C}(X,\mathbb{R}), g_i \in \mathcal{C}(Y,\mathbb{R}) \}$ of $\mathcal{C}(X \times Y, \mathbb{R})$. $\endgroup$ – AlexL Oct 12 '18 at 3:11
  • $\begingroup$ I already tried but I think I’m making some conceptual mistakes. Do you probably know any material I could consult? $\endgroup$ – nonparametrix Oct 12 '18 at 3:25
  • $\begingroup$ I found this mast.queensu.ca/~speicher/Section14.pdf $\endgroup$ – AlexL Oct 12 '18 at 3:29
  • $\begingroup$ seems like this is not exactly what I was looking for, I was more looking for the specific statement I posted. but thank you for your effort AlexL $\endgroup$ – nonparametrix Oct 12 '18 at 4:04

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