Let $\mathcal{H}$ be a Hilbert space with a reproducing kernel $K:X \times X \rightarrow \mathbb{R}$. For any finite sequence $x_1,...,x_n$ of distinct points in $X$ we define Gramm matrix as $$ M(x_1,...,x_n) := \begin{pmatrix} K(x_1,x_1) & \cdots & K(x_1,x_n) \\ \vdots & \ddots & \vdots \\ K(x_n,x_1) & \cdots & K(x_n,x_n) \end{pmatrix} .$$
In reading one the articles from this field the author notes that the invertibility of matrix $M$ is safely assumed due to Mercer's theorem. When looking up the theorem I was not able to conclude this by myself. Can somebody point out to me, how exactly does the invertibility follows from the theorem?