Centered Borel Gaussian random element on a separable Banach space

Let $(\mathrm{\Omega },\mathcal{F},P)$ be a probability space and let $B$ be a real separable Banach space with its norm Borel sigma-algebra. A map $W:\mathrm{\Omega }\to B$ is a centered Borel Gaussian random element if it is Borel-measurable and, for every $\mathrm{\ell }\in B^{\ast }$, the real random variable $\mathrm{\ell }(W)$ has a centered (possibly zero-variance) normal law. Its law is the Borel probability $\mu =P\circ W^{-1}$. When the word tight is used, it means that for every $\delta >0$ there is a norm-compact $K\subseteq B$ with $\mu (K)>1-\delta$.

This definition concerns a random element in $B$. A Gaussian process on an arbitrary index set with only a cylinder-space law does not satisfy it unless a Borel $B$-valued realization is separately constructed.