I currently try to prove the following:
Let $X,Y$ be compact metric spaces. $A = \{(x,y) \rightarrow \sum_{i=1}^{n} f_i(x)g_i(y) \ | \ f_i \in C(X,\mathbb{R})$ and $g_i \in C(Y,\mathbb{R}), 1 \leqslant i \leqslant n \}$.
I want to prove that
(i) $A$ is an algebra and
(ii) $A$ is uniformly dense in $C(X \times Y, \mathbb{R})$.
Regarding (ii) I think I will just show that (given $A$ is an algebra) the constant functions are in $A$ and that $A$ separates points which I think is straight forward. However, I have difficulties to show that $A$ is indeed an algebra. Could someone show me how to prove this rigorously?