Example 2

Let $X$ be a Hausdorff topological space, $\mu$ a positive measure of norm 1 on $X$ and $E$ a hilbertian subspace of the real hilbertian space $L^2_{\mathbf{R}}(X, \mu)$. We say that $E$ is a gaussian space if the following equivalent conditions are satisfied :

a) for all $f \in E$, we have $\int_X e^{i f} d\mu = \exp(-\| f \|^{2}/2) ;$

b) for all $f \in E$ with norm 1, the image of the measure $\mu$ under $f$ is the measure

on $\mathbf{R}$.

Suppose $E$ is a gaussian space. Let $f_1, ..., f_n$ be functions whose classes $f_i$ belong to $E$. We define a function : $f_1 ... f_n$ : on X (called « Wick’s product » of $f_1, ..., f_n$) by the formula

where $I_p$ is the set of permutations $\sigma$ of $\{1, 2, ..., n\}$ such that we have

Then there exists an isomorphism $\phi$ from $\hat{S}(E)$ onto a hilbertian subspace of $L^2_R(X, \mu)$ which transforms the product $f_1 ... f_n$ of $f_1, ..., f_n$, calculated in $\hat{S}(E)$, into $(:f_1 ... f_n:)$. Suppose that X is a Souslin space and that there exists a countable family $(f_n)$ of functions whose classes belong to E and which separate the points of X. Then $\phi$ is an isomorphism from $\hat{S}(E)$ onto $L^2_R(X, \mu)$. \*