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Let $X$ and $Y$ be compact metric spaces. Let $$ F= \Bigl\{\sum_{i=1}^n A_i f_i(x) g_i(y): f_i\in C(X,\mathbb{R}),g_i\in C(Y,\mathbb{R}), 1\le i\le n \Bigr\}. $$ Prove that $F$ is dense in $C(X\times Y,\mathbb{R})$.

Please, I can't figure it out.

I will be thankful for any help.

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    $\begingroup$ this is difficult to read, I suggest that you edit this post to make it more legible. $\endgroup$ – Rustyn Apr 18 '14 at 9:41
  • $\begingroup$ what about now ? this is the first question for me and i have so much trouble to make it. $\endgroup$ – user143991 Apr 18 '14 at 20:18
  • $\begingroup$ You probably meant to say that $A_i\in {\bf Q}$ or $A_i\in{\bf R}$. $\endgroup$ – tomasz Apr 19 '14 at 13:08
  • $\begingroup$ Also, this question (which is on top of the related questions on the right hand side!) is kind of a generalisation of your question, and contains a possible hint for you. math.stackexchange.com/q/63416/30222 $\endgroup$ – tomasz Apr 19 '14 at 13:12
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Use the Stone-Weierstrass theorem.

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  • $\begingroup$ can you help me step by step.. i have problem with this chapter $\endgroup$ – user143991 Apr 20 '14 at 11:14
  • $\begingroup$ Compact+Hausdorff $\implies$ normal $\implies$ $C(X,\mathbb{R})$ separates points in $X$, $C(Y,\mathbb{R})$ separates points in $Y$. Take $(x_1,y_1),(x_2,y_2)\in X\times Y$ with $(x_1,y_1)\ne(x_2,y_2)$. Suppose wlog that $x_1\ne x_2$. Now, construct a separating function $\in F$. $\endgroup$ – Martín-Blas Pérez Pinilla Apr 20 '14 at 13:38

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