In stochastic equations in infinite dimensions written by Da Prato reproducing kernel Hilbert space is defined

Let $\mu$ be a symmetric Gaussian measures on a separable Banach space $E$. A linear subspace $H \subset E$ equipped with a Hilbert norm $|\cdot|_{H}$ is said to be a reproducing kernel Hilbert space for $\mu$ if H is complete, continuously embedded in $E$ and such that for arbitrary $\varphi \in E^{\star}$, the distribution of $\varphi$ is Gaussian $\mathcal{N}(0,|\varphi|^2_{H})$, where $$|\varphi|_{H}=\sup_{|h|\leq 1, h\in H}|\varphi(h)|$$

My question is

(1) How can this definition coincide with the general definition such as


(2) What is the advantage of this defination? Is there any convenience?


The two notions coincide in the particular case when $E = C(X)$ consists of the continuous functions from $X$ to $\mathbf{R}$ with $X$ playing the same role as in the Wikipedia article. (Say $X$ is a compact space to avoid technicalities.) In this case, if $H$ is defined as in Da Prato, since $H \subset C(X)$, the point evaluations $L_x$ are indeed continuous, so you are in the setting of the Wikipedia article. The "reproducing kernel" $K$ is then nothing but the covariance function of the measure $\mu$. In other words, the definition given by Da Prato is precisely the one guaranteeing that the reproducing kernel $K$ is given by $$ K(x,y) = \int_E f(x)f(y) \mu(df)\;. $$ It is true that the terminology is a bit confusing, many authors prefer to call $H$ the Cameron-Martin space of $\mu$ instead.


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