  We prove that the ring $\Aff{\R}{M}$ of all polynomials defined on a real
algebraic variety $M\subset\R^n$ is dense in the Hilbert space
$L^2(M,e^{-|x|^2}\de\mu)$, where $\de\mu$ denotes the volume form of $M$ and
$\de\nu=e^{-|x|^2}\de\mu$ the Gaussian measure on $M$.
