= Cameron-Martin space of a Gaussian random variable in a Banach space
{c}
{title2=$H=S\mathcal G,\quad Sg=\mathbb E[Xg]$}
= Banach-space Gaussian RKHS
{c}
{synonym}
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 $\langle Sg,Sv\rangle_H=\mathbb E[gv]$. The embedding in $B$ is continuous by <Cauchy-Schwarz inequality>. Its <reproducing property> is $\langle h,S\ell(X)\rangle_H=\ell(h)$. Thus this is the <Reproducing-kernel Hilbert space> of the random variable, including degenerate laws. For $h=Sg$, the scalar coordinate $\widehat h(X)=g$ has <variance> $\|h\|_H^2$; it is generally a measurable linear coordinate rather than a continuous functional on $B$.
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