Let $h\_0, h\_1, \dots$ be the classical univariate Hermite polynomials, renormalized to have constant norm. Is
$$x\mapsto\prod\_{j=1}^n h\_{l\_j}(x\_j), \quad l\_j\in \mathbb N$$
a complete orthogonal system for $L^2(\mathbb R^n)$, as in the univariate case? The orthogonality is simple, but I still have some doubts on the completeness.



Multivariate Hermite Polynomials
In one dimension, you have
$$
h\_k(t)e^{-π t^2}=e^{π t^2}(\frac{d}{dt})^k\bigl(e^{-2π t^2}\bigr),
$$
and $h\_k$ is easily proven to be with degree $k$.
The completeness question amounts to proving
$L^2=\overline{\text{span}\{t^ke^{-π t^2}\}}=E$.
Taking $u\in E^\perp$, you obtain
$$
\int u(t)\sum\_k\frac{t^ki^k\xi ^k}{k!}e^{-π t^2} dt=0,
$$
and by injectivity of the Fourier transform $u(t)e^{-π t^2}=0$, so that $u=0$
(note that this argument works for $u$ tempered distribution).
In several variables, you get by tensorization 
$$
E=\overline{\text{span}\{x^\alpha e^{-π \vert x\vert^2}\}},
$$
and the same Fourier argument works.


