Theorem 1.5.1 ([Theorem 8.3.1, Str24]).label Let $H$ be an infinite-dimensional, separable Hilbert space over $\real$, $E$ be a Banach space with $H$ continuously embedded as a dense subspace.
Let $\seq{h_n}\subset H$ be an orthonormal basis and $\seq{X_n}$ be mutually independent $\gamma_{0, 1}$-distributed random variables on a probability space $(\Omega, \cf, \bp)$. If
- (1)
$\sum_{n = 1}^{\infty} X_{n} h_{n}$ converges in $E$ almost surely.
- (2)
$S: \Omega \to E$ is given by
\[S(\omega) = \begin{cases}\sum_{n = 1}^{\infty} X_{n}(\omega) h_{n}&\text{The series converges in }$E$ \\ 0&\text{Otherwise}\end{cases}\] - (3)
$\wien = S_{*} \bp$.
then $(H, E, \wien)$ is an abstract Wiener space.
Proof. Let $S_{n} = \sum_{k = 1}^{n} X_{k}h_{k}$, then
\[\wh \wien(x^{*}) = \limv{n}\ev^{\bp}\braks{e^{\dpb{S_n, x^*}{E}}}= \limv_{n}\prod_{k = 1}^{n} e^{-\dpb{\iota^*x^*, h_k}{H}^2/2}= e^{-\norm{\iota^*x^*}_H^2/2}\]
$\square$