Let $\Omega$ be some domain. Let $k(x,y): \Omega^2 \to \mathbb{R}$ be a kernel on the domain and $H$ to be the RKHS induced by $k$. Here, I want to consider the projection from $L_2(\Omega)$ to $H$. For any given $f \in L_2(\Omega)$, can we define
$$f_{\mathrm{proj}} = \arg \min_{\tilde f \in H} || f - \tilde f ||^2_{L_2(\Omega)}$$
where $|| f ||_{L_2(\Omega)} = \sqrt{E[f^2(x)]}$ is the norm in $L_2(\Omega)$?
Also, is it true that residual is orthogonal to $H$? i.e. given $f_{\mathrm{proj}}$,
$$\int_\Omega (f(x) - f_{\mathrm{proj}}(x))\tilde f(x) P(x) \mathrm{d}x = 0$$
for all $\tilde f \in H$? I can easily see this is true when $k$ is a polynomial kernel, but can we generalize this result to an infinite dimensional kernel (such as a Gaussian kernel)? I appreciate your help!