I am reviewing materials on reproducing kernel Hilbert space (RKHS) and I've found various versions of Mercer's theorem:
About the positive-definiteness conditions.
In the Wikipedia pages on RKHS above or on the theorem itself, the condition is
$$\sum_{i,j =1}^n c_i c_j K(x_i, x_j) \ge 0 , \forall n \in \mathbb{N},\ x_1, \dots, x_n \in X,\ \mbox{and}\ c_1, \dots, c_n \in \mathbb{R},$$
while in some other materials (for example this, page 16), the condition becomes an integral:
In T. Hofmann et. al there is a short comment on this:
So according to the text they are different. Thus my first question is, what is the connection between these two conditions?
About the function space of $K$.
Again in the Wikipedia pages on Mercer's theorem, it mentions that $K$ is a Hilbert–Schmidt integral operator, which is like saying $K\in\mathcal{L}^2(\mathcal{X}^2)$ if I understand correctly. But in the attached slide above, it requires $K\in\mathcal{L}^\infty(\mathcal{X}^2)$, and gives $(\lambda_j)\in\ell_1$. This conclusion seems not covered by the spectral theory of compact and self-adjoint operators on Hilbert space, which only states a weaker result $\lambda_j\to 0$. It looks to me that $K\in\mathcal{L}^\infty(\mathcal{X}^2)$ helps give $(\lambda_j)\in\ell_1$. So my second question is, what is the general requirement of the function space for $K$, and how will the choice of spaces affect the properties of $(\lambda_j)$?