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Given $$ A=\left\{\sum^n_{k=0}f_k(x)g_k(y) : \ n \in \mathbb{Z}^+, \ f_k, g_k\in C[0,1]\right\}. $$

I am trying to use the Stone-Weierstrass Theorem to prove that $A$ is dense in $C([0,1]\times[0,1])$.

It is easy to see that $A$ is an algebra. I know $A$ vanishes nowhere, but I am confused about how to prove $A$ separates points. I know according to the Weierstrass Approximation Theorem, for all $\epsilon>0$ there exists a polynomial $p$ such that $|p(x,y)-f(x,y)|<\epsilon$, when $f\in C([0,1]\times[0,1])$, and I know polynomials separate points, and I have proved that all polynomials are in $A$.

My question is, how to prove $A$ separates points?

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Hint: Given two distinct points $(x_1,y_1),(x_2,y_2) \in [0,1]\times[0,1]$, try to give an example of a function of the form $F(x,y) = f(x)g(y)$ for which $$ F(x_1,y_1) \neq F(x_2,y_2). $$ You might find it helpful to separately handle the case where $x_1 \neq x_2$ and the case where $y_1 \neq y_2$. Note that our function $F$ is allowed to depend on the choice of points.

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  • $\begingroup$ Oh yes! I totally forgot that we can choose F depending on the choice of points. Thank you very much! $\endgroup$
    – kaaaTata
    Aug 4, 2020 at 20:23

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