 Solution

Suppose a feature map $\varphi :{\mathbb{R}}^d\to {\mathbb{R}}^p$ represented the Gaussian kernel. Choose $n>p$, a unit vector $e$, and points $x_i=iae$ for $i=0,\dots ,n-1$. Their kernel matrix isFor sufficiently large $a$, hence sufficiently small $r$, every row satisfiesThus $K$ is a symmetric strictly diagonally dominant matrix with positive diagonal and is therefore a positive-definite matrix, so $\mathrm{rank}K=n$.

On the other hand, if $\Phi$ is the $n\times p$ matrix whose $i$th row is $\varphi (x_i{)}^T$, then $K=\Phi {\Phi }^T$ and $\mathrm{rank}K\le p<n$, a contradiction. Hence every feature-space realization of the Gaussian kernel requires an infinite-dimensional vector space.