I'm trying to prove that, given mutually different points $x_1,\dots,x_m$, the Gram matrix $G$ for the Gaussian kernel has $rank(G)=m$. If I can prove that the Gaussian kernel is strictly positive definite I could follow that all eigenvalues $\lambda_1,\dots,\lambda_m$ of $G$ are positive. Therefore, $det(G)=\Pi_{i=1}^m \lambda_i \neq 0$ which implies $rank(G)=m$. Is there an easy way to show that the Gaussian kernel is strictly positive definite? I've already searched on StackExchange but I couldn't find an explanation I could understand.
Thanks in advance!
EDIT: The definition of the Gram matrix is the following: Given a kernel $k$ and data $x_1,\dots,x_m \in X$, the $m \times m$ matrix $G=(g_{ij})$ with $g_{ij}=k(x_i,x_j)$ is called Gram matrix.