Example 1

Let $d \geq 1$ be an integer and $\omega$ a positive function on $\mathbf{R}^d$, locally integrable with respect to the Lebesgue measure $\mu$. Let $E$ be the hilbertian space $L^2(\mathbf{R}^d, \omega.\mu)$, and let $S = S(E)$. Then $S$ can be identified with the space of all sequences $f = (f_n)_{n \geq 0}$, where each $f_n$ is a function on $(\mathbf{R}^d)^n$ which is measurable with respect to the Lebesgue measure $\mu \otimes ... \otimes \mu$ ($n$ factors) and invariant under the permutations of the $n$ factors in $(\mathbf{R}^d)^n$, and is such that

is finite. The norm $\| f \|$ in $S$ is defined by formula (22). The hilbertian space $S$ defined above is called the symmetric Fock space corresponding to the weight $\omega$.