I'm reading Hairer's notes on SPDEs here. On page 17 he has remark 3.33 where he considers $\Bbb R^{\Bbb R}$ with the product $\sigma$-algebra and product Gaussian measure. He claims the Cameron Martin space of this is $\{f\in \Bbb R^{\Bbb R}:\sum_{t\in \Bbb R}|f(t)|^2<\infty\}$.
I'm curious, in what sense is the infinite product of Gaussian measures a true honest Gaussian measure in sense of definition 3.2:
A Gaussian probability measure $\mu$ on a Banach space $\mathcal B$ is a Borel measure such that $\ell^\ast \mu$ is a real Gaussian probability measure on $\Bbb R$ for every linear functional $\ell:\mathcal B \to \Bbb R$
Here, $\Bbb R^{\Bbb R}$ isn't Banach. The definition of Cameron Martin space obviously involves the Banach structure so my question.
In what sense is the product Gaussian measure on $\Bbb R^{\Bbb R}$ actually a Gaussian measure? In what sense does it have a Cameron Martin space?