Assume we have a Gaussian measure $\mu$ supported on a Banach space $X$. Can we always find a Hilbert space $H$ embedded in $X$ sch that $\mu$ is also supported on $H$?



Gaussian measure on Banach space
No, not necessarily. It is shown in Examples 3.6.6 and 3.6.7 of Bogachev's *Gaussian Measures* that if $X = C([0,1])$ and $\mu$ is classical Wiener measure, then for any Hilbert space $H$ embedded in $X$, we have $\mu(H) =0$, so that $\mu$ is not supported on $H$.


