Let $\{e_k\}_{k\in \mathbb{N}}$ be a orthonormal basis for $L^2(\mathbb{R}^n,\mathbb{F})$ with $\mathbb{F}=\mathbb{R}$ or $\mathbb{C}$. Prove that the collection \begin{equation} \{e_{k,j}\}_{k,j\in \mathbb{N}} \qquad \mbox{where} \qquad e_{k,j}(x,y):=e_k(x)e_j(y) \end{equation} gives an orthonormal basis for $L^2(\mathbb{R}^{2n},\mathbb{F})$.
My idea:
I know that \begin{equation} \int_{\mathbb{R}^n} e_k(x)\overline{e_j(x)}dx=\delta_{k,j} \end{equation} so I can write: \begin{equation} \int_{\mathbb{R}^{2n}} e_{k,j}(x,y)\overline{e_{h,l}(x,y)}d(x,y)=\int_{\mathbb{R}^{n}}e_{k}(x)\overline{e_h(x)}(\int_{\mathbb{R}^{n}} e_j(y)\overline{e_{h}(y)}dy)dx \end{equation} that is equal to $1$ if $k=h$ and $j=l$, equal to $0$ otherwise. Hence $\{e_{k,j}\}_{k,j\in \mathbb{N}}$ is a orthonormal system of $L^2(\mathbb{R}^{2n},\mathbb{F})$.
Now, to prove the completeness I use the maximality condition of $\{e_{k\}_{k}\in \mathbb{N}}$: \begin{equation} 0=\int_{\mathbb{R}^{2n}} f(x,y)\overline{e_{k,j}(x,y)}d(x,y)=\int_{\mathbb{R}^{n}}F_k(y)\overline{e_j(y)}dy \end{equation} so $F_k=0$ for the maximality condition of $\{e_{k\}_{k}\in \mathbb{N}}$, that implies $f=0$. I can conclude that $\{e_{k,j}\}_{k,j\in \mathbb{N}}$ is an orthonormal basis for $L^2(\mathbb{R}^{2n},\mathbb{F})$.