1.
It is well known that the Hermite functions:
$h_n(x) = a_n \ H_n(x) \ e^{-\frac{1}{2}x^2}$, with
$a_n:= \text{normalization factor}$
A
$H_n:= n \text{th Hermite polynomial}$
forms a complete orthonomal system on the Hilbert space $L^2(\mathbb{R})$.
2.
I also know that if $H_1$ and $H_2$ are Hilbert spaces and $(\phi_n)$ $(\psi_n)$ are complete orthonomal systems on $H_1$ and $H_2$, then $(\phi_n \otimes \psi_m); n,m \in \mathbb{N}$ forms a complete orthonomal system on $H_1 \otimes H_2$.
3.
In my quantum pyhsics class we use that $(h_n \otimes h_m) = h_n(x)h_m(y)$ forms a complete orthonomal system on $L^2(\mathbb{R}^2)$, since as stated above the Hermite functions themself are an orthonomal system on $L^2(\mathbb{R})$.
My question is: Since we can't just use that $L^2(\mathbb{R}) \otimes L^2(\mathbb{R}) = L^2(\mathbb{R}^2)$, is there a formal way to prove statement 3.?