I would like to get a confirmation if I understand well Mercer's theorem. I'm following the Wikipedia article about RKHS (https://en.wikipedia.org/wiki/Reproducing_kernel_Hilbert_space).
Let $X$ a compact subset of $\mathbb{R}^{d}$. You take a kernel $K:X \times X\to\mathbb{R}$ continuous, symmetric and positive semi-definite. Let $T_K$ the integral operator associated with $K$ defined in $L_2(X)$. We know from the spectral theorem that there is at most a countable decreasing sequence of eigenvalues of $T_K$ such that $\sigma_i \geq0 $ where $\lim\limits_{i\to+\infty}\sigma_i = 0$ with the eigenvectors associated. The eigenvectors $\{\phi_i\}$ form an orthonormal basis of $L_2(X)$. They are claming that as $T_K$ is positive, $\sigma_i > 0$. Does it mean that it is true only if $T_K$ is positive define ?