I have recently been trying to gain some intuition about Mercer's Theorem:
$$K(x,y)=\sum_{i=1}^{\infty}\lambda_ie_i(x)e_i(y)$$
According to this video, each symmetric, positive semi-definite kernel constitutes an inner product in some associated (Reproducing Kernel) Hilbert space $\mathcal{H}$. This is relatively clear to me, as the equation above can be easily interpreted as an inner product between
$$k(\cdot,x)=[\sqrt{\lambda_1}e_1(x),...,\sqrt{\lambda_\infty}e_\infty(x)]$$ $$k(\cdot,y)=[\sqrt{\lambda_1}e_1(y),...,\sqrt{\lambda_\infty}e_\infty(y)]$$
What has me stumped, however, is on what basis exactly this associated Hilbert space is formed. Since we have eigenfunctions $e_i$ and eigenvalues $\lambda_i$, we must have decomposited some linear operator. According to Wikipedia, this operator was the Hilbert-Schmidt integral operator defined on some functions $\phi$.
$$[T_K\phi](x)=\int_a^bK(x,s)\phi(s)ds$$
But why exactly this integral operator and not something else? From where to where are these functions $\phi$ mapping? Are they functions defined in the Hilbert space? Why are they absent from both Mercer's theorem and the RKHS? The derivation just seems rather arbitrary to me. Is there something I have misunderstood?
