Is tensor product $L^\infty(X)\times L^\infty(Y)$ dense in $L^\infty(X\times Y)$ where $X, Y\subset \mathbb R^n$? and does any body know an example of a function $\psi \in L^\infty(X)\times L^\infty(Y)$ such that it can not be written as a series $\psi=\sum_ix_i(t)y_i(s)$ where $x_i(t)\in L^\infty(X)$ and $y_i\in L^\infty(Y)$.
I thank you in advance.