Theorem 1.4.3.label Let $(H, E, \wien)$ be an abstract Wiener space, then
- (1)
There exists a unique linear isometry $\ci: H \to L^{2}(\wien; \real)$ such that $\ci(\iota^{*}x^{*}) = \dpb{\cdot, x^*}{E}$ for all $x^{*} \in E^{*}$.
- (2)
$\bracs{\ci(h): h \in H}$ is a Gaussian family.
The map in (1) is the Paley-Wiener map.
Proof. Let $x^{*} \in E^{*}$, then $\dpb{\cdot, x^*}{E}$ is a centred Gaussian random variable with variance $\norm{\iota^*x^*}_{H}^{2}$. By (3) of Lemma 1.3.1, $\iota^{*}(E^{*}) \subset H$ is dense. Thus $\ci$ extends uniquely into a continuous linear isometry.$\square$