My professor told me that one can approximate a function $\varphi$ in $C_0^\infty(X \times Y)$ via
$$\varphi(x,y)=\lim_{n \to \infty} \sum_i \phi_i^n(x) \psi_i^n(y) $$
where $\phi_i^n$ is sequence in $C_0^\infty(X)$ and $\psi_i^n$ a sequence in $C_0^\infty(Y)$. You can imagine $X,Y$ sufficiently nice, in my case they where open and bounded subsets of $\mathbb{R}^n$, $n \in \mathbb{N}$.
A property like this is essential if one wants to prove a parabolic PDE in distributional sense and in one step separates the time and space test functions.
I asked for a proof of this statement and he told me that one can find proofs of this statement in books that are older than him and right now, he does not have any reference. Furthermore, I looked up some PDE books like Evans, Renardy, etc, but unfortunately I couldn't find such a statement.
So I would like to ask you to kindly tell me a source so I can look up the proof.
Thank you so much!