Let $E=C[0,1]$, space of all real-valued continuous functions on $[0,1]$, $\mathcal{E}$ be its Borel $\sigma$-algebra and $\mu$ a Gaussian measure on $E$. Let $E^*$ be a space of all continuous linear functions on $E$. Define map $R$ on $E^*$ by $x^* \mapsto R(x^*)=\int_E\langle x^*,x\rangle x\;\mu(dx)=\int_E x^*(x)\; x\;\mu(dx)$ And let $H_\mu$ be the completion of $R(E^*)$ with respect a norm induced by an inner product defined as $\langle Rx^*,Ry^* \rangle=\int_Ex^*(x)y^*(y)\;\mu(dx)$.
$H_\mu$ stands for Reproducing Kernel Hilbert Space and it is dense$^1$ in $E$ if topological support$^2$ of $\mu$ is the whole space $E$. Why?
I think I understand the construction well enough, but the statement is somewhat unexpected.
$^1$ $i(H_\mu)$ to be precise, $i$ for inclusion from $H_\mu$ to $E$.
$^2$ topological support is the smallest closed set $F$ such that $\mu(F) = 1$.
Edit This is page 84 of Interest Rate Models: an Infinite Dimensional Stochastic Analysis Perspective. Read online on Springer: http://www.springerlink.com/content/978-3-540-27065-2#section=411613&page=84 (the statement is on page 88)