Covariance operator and reproducing-kernel Hilbert space of a Banach Gaussian

Assume AC. Let $W$ be a centered Borel Gaussian random element in a real separable Banach space $B$, with law $\mu$. For $\mathrm{\ell }\in B^{\ast }$ define $C\mathrm{\ell }=E[\mathrm{\ell }(W)W]\in B,Q(\mathrm{\ell },m)=E[\mathrm{\ell }(W)m(W)](\mathrm{\ell },m\in B^{\ast })\mathrm{.}$ The Cameron–Martin reproducing-kernel Hilbert space $H$ is the completion of $B^{\ast }\mathrm{/}N$ in the inner product induced by $Q$, where $N=\{\mathrm{\ell }\in B^{\ast }:Q(\mathrm{\ell },\mathrm{\ell })=0\}$. The canonical map sends the class of $\mathrm{\ell }$ to $C\mathrm{\ell }$ and extends continuously to an injective map $i:H\to B$; we identify $H$ with its image in $B$ and use the completed Hilbert norm $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_H$. The preceding construction is well-defined by The Banach Gaussian covariance and RKHS construction is well-defined ↗.

The term RKHS here denotes this covariance-induced Hilbert space of shifts inside $B$; it is not the space of all continuous functions on an arbitrary index set.