Let $X$ and $Y$ be compact metric spaces. I am trying to prove that if $f \in C(X \times Y)$ and $\varepsilon > 0 $, then there exist functions $g_1, g_2,...,g_n \in C(X)$ and $h_1,...h_n \in C(Y)$ so that $|f(x,y) - \sum_{k=1}^n g_k(x)h_k(y) | < \varepsilon $ for all $(x,y) \in X \times Y$. I need help. Thank you.
Prove that if f in $C(X \times Y)$ then there exists functions.
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functional-analysis
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0Is $T$ a typo for $Y$? If so, can you edit accordingly? Also, can you edit the title into something more informative? – 2012-04-20