Suppose you have Banach spaces $\mathcal B_\alpha$ where $\alpha$ is in some index set $I$. Let $\mu_\alpha$ be Gaussian measures on $\mathcal B_\alpha$ with Cameron-Martin spaces $\mathcal H_{\mu_\alpha}$.
Is it then true that the product space of all the Banach spaces with the product measure (independent components), call it $\mathcal B$, has Cameron-Martin space $\mathcal H_\mu=\{\mathbf v=\{v_\alpha\}_{\alpha\in I}:\sum_{\alpha\in I}\|v_\alpha\|_{\mu_\alpha}^2<\infty\}$?
I believe I read this somewhere but I can't find a reference.
Best Answer
At least if $I$ is countable this should be true, but I do not have a reference:
Let $H$ be the CM space of the Frechet space $\mathcal{B} := \prod_{\alpha \in I} \mathcal{B}_{\alpha}$ with the product topology and the product measure. Recall that the Hilbert space reproducing kernel of $(\mathcal{B}, \mu)$ is isometrically isomorphic ($\Vert \cdot \Vert_{L^2(\mathcal{B}^{\ast}, \mu)} = \Vert \cdot \Vert_H$) to the CM space $H$ and is defined as the $L^2$ closure of $j(\mathcal{B}^{\ast})$ where $j$ is the canonical inclusion $\mathcal{B}^{\ast} \rightarrow L^2(\mathcal{B}, \mu)$.
In our case, the dual to $\mathcal{B}$ can be identified with the sequences in $\prod_{\alpha \in I} \mathcal{B}_{\alpha}^{\ast}$ with finite support and the pairing $\mathcal{B}^{\ast} \times \mathcal{B} \rightarrow \mathbb{R}$ given by
\begin{equation} v(x) = \sum_{\alpha \in I} v_{\alpha} (x_{\alpha}), ~~ v_{\alpha} \in \mathcal{B}_{\alpha}^{\ast}, x_{\alpha} \in \mathcal{B}_{\alpha} . \end{equation}
where the sum is, of course, finite.
Restricted to $j(\mathcal{B}^{\ast})$, the $L^2$-norm is
\begin{equation} \Vert v \Vert_H = \Vert j(v) \Vert_{L^2} = \int \sum_{\alpha, \beta \in I} v_{\alpha} (x_{\alpha}) v_{\beta} (x_{\beta}) d \mu(x) = \sum_{\alpha \in I} \int v_{\alpha} (x_{\alpha})^2 d \mu(x) = \sum_{\alpha \in I} \Vert v_{\alpha} \Vert_{H_{\alpha}}^2 \end{equation}
where we used the product structure of $\mu$. Hence indeed
\begin{equation} H = \{\mathbf v=\{v_\alpha\}_{\alpha\in I}:\sum_{\alpha\in I}\|v_\alpha\|_{\mu_\alpha}^2<\infty\} . \end{equation}