# How to prove there is no finite-dimensional feature space for Gaussian RBF kernel?

How to prove that for the radial basis function $k(x, y) = \exp(-\frac{||x-y||^2)}{2\sigma^2})$ there is no finite-dimensional feature space $H$ such that for some $\Phi: \text{R}^n \to H$ we have $k(x, y) = \langle \Phi(x), \Phi(y)\rangle$?