I'm a little bit confused and I know that the question is trivial but I can't find a good argument for the following assumption.
Lets say that we have a gaussian measure $\mu$ on a separable banach space $X$ with mean
$a_{\mu}(f)=\int_{X}f(x)\mu(dx)$ ,$f\in X^{*}$
and covariance
$Q_{\mu}(f,f)=\int_{X}(f(x)-a_{\mu}(f))^{2}\mu(dx)$,$f\in X^{*}$
My question is the following:
if we have that a functional $f\in X^{*}$ then $f-a_{\mu}(f)$ is going to be in $X^{*}$??
Because I have a mapping $j:X^{*}\rightarrow L^{2}(X,\mu)$ and I want to see if it's an inclusion map.