What is the difference between RKHS inner product and a "regular" inner product? I have a book (http://www.gaussianprocess.org/gpml/) that has in the notation section...
$$
\langle f, g \rangle_\mathcal{H} = \text{RKHS inner product} \\
\|f\|_\mathcal{H} = \text{RKHS norm}
$$
How are these different than the inner product of vectors $x^Tx$ and the L2 norm $\|x\|^2$ that uses the same notation without the $\mathcal{H}$?
 A: If $H$ is a real separable Hilbert space with a Gaussian measure $\mu$, then its covariance operator $S$ is positve, selfadjoint and of finite trace. Since the topological support of a Gaussian measure is a linear subvariety, we can consider the support of $\mu
$.
The topological support of $\mu$ is the closure of the linear span of the eigenvalues of $S$. The restriction makes $S$ injective on the support.
Now $\sqrt{S}$ exists, is positive, selfadjoint, Hilbert Schmidt and $\sqrt{S}(H)$ is dense in $H$. We define in $\sqrt{S}(H)$ the inner product
$$\langle \sqrt{S}x,\sqrt{S}y\rangle_0=\langle x,y\rangle_H.$$
which makes $\sqrt{S}H$ into a Hilbert space with $\sqrt{S} :H\to 
\sqrt{S}(H)$ an isometric isomorphism. The space $\sqrt{S}H$ is the reproducing kernel Hilbert space of $(H,\mu)$.
"How are these different?" This depends on the Hilbert space you're working with. If you're working with a non degenerate Gaussian measure in $\mathbb{R}^n$, then the RKHS is the entire space. If you're working with the Frechet space $\mathbb{R}^\mathbb{N}$, the space is $\ell^2$ with the usual $\ell^2$ norm.
