with $G$ in $\mathcal{S}$ and $h$ in $H$ belongs to ${\rm{dom}}(\delta)$ and that $$\label{eq:deltaprod}
\delta(G h)= G W(h) -\langle \nabla G, h\rangle_H,$$ see for example [@Nualartbook Relation (1.46)]. Note that for any $h$ in $H$, $\delta(h)=W(h)$.

Notice that in [@Sugita] the cylindrical space, that we will denote by $\mathcal P$ in the following, is the space of functionals $F$ of the form with $f$ a polynomial. More precisely let $\mathcal{P}$ be the set of polynomial cylindrical functionals, that is the set of random variables $F$ of the form $$\label{eq:cylindricalP}
F=f(W(h_1),\ldots,W(h_n)), \quad (h_1,\ldots,h_n) \in H^n, \; f \in \R^n[X], \text{ for some }n\geq 1,$$ where $\R^n[X]$ denotes the set of polynomials of degree less or equal to $n$. However, the closures of both $\mathcal S$ and $\mathcal P$ with respect to any $\|\cdot\|_{1,p}$ coincide, as any polynomial together with its derivative can be approximated in $L^p({\mathbb{R}}^n)$ (see Lemma \[lemma:tec1\] below).

\[lemma:tec1\] Let $G$ be in $\mathcal{P}$. There exists a sequence $(G^N)_{N\geq 1} \subset \mathcal{S}$ such that $\lim_{N\to+\infty} G^N = G$ in $\mathbb{D}^{1,r}$ for any $r \geq 1$.

Let $G:=f(W(h_1),\cdots,W(h_n))$ with $n\geq 1$, $h_i$ in $H$ and $f$ in ${\mathbb{R}}^n[X]$. Without loss of generality, we assume that the family $(h_1,\ldots,h_n)$ is orthonormal 