Cameron-Martin space of a Gaussian measure ($E_{\mu }=\mathrm{Ran}{\Sigma }^{1/2}$)

For a centered Gaussian measure on a real Hilbert space with injective covariance operator of a Gaussian measure $\Sigma$, its Cameron-Martin space is $E_{\mu }=\mathrm{Ran}{\Sigma }^{1/2}$, with inner product $⟨h,g{⟩}_{E_{\mu }}=⟨{\Sigma }^{-1/2}h,{\Sigma }^{-1/2}g⟩$. In an orthonormal basis satisfying $\Sigma e_j={\lambda }_j{e}_j$, it consists of exactly the vectors $h$ for which ${\sum }_j{h}_j^2/{\lambda }_j<\infty$. It is complete in this stronger norm even if its range is not closed in the ambient norm. Its importance is that it describes precisely the translations that preserve the null sets of the Gaussian measure, as in the Cameron-Martin theorem for a Gaussian measure. For generalized Gaussian white noise over $L^2$, the corresponding Cameron-Martin space is L2 space itself.

Let $\mathcal{G}$ be the first Gaussian chaos of a centered Gaussian random variable in a Banach space. By Fernique's theorem, $Sg=\mathbb{E}[Xg]$ is a Bochner integral in $B$. The map $S$ is injective, because $Sg=0$ implies $\mathbb{E}[g\ell (X)]=0$ for every continuous linear functional $\ell$, hence $g=0$. Its range becomes a Hilbert space under $⟨Sg,Sv{⟩}_H=\mathbb{E}[gv]$. The embedding in $B$ is continuous by Cauchy-Schwarz inequality. Its reproducing property is $⟨h,S\ell (X){⟩}_H=\ell (h)$. Thus this is the Reproducing kernel Hilbert space of the random variable, including degenerate laws. For $h=Sg$, the scalar coordinate $\hat{h}(X)=g$ has variance $\Vert h{\Vert }_H^2$; it is generally a measurable linear coordinate rather than a continuous functional on $B$.