Let $H$ be a separable Hilbert space and $W=(W(h))_{h\in H}$ be an isonormal Gaussian process on a probability space $(\Omega,\mathscr F,P)$, where $\mathscr F$ is generated by $W$.
Let $\mathscr S $ (resp. $\mathscr S_b$) be the space of all random variable $X$ of the form $$X=f(W(h_1),\cdots, W(h_n))$$ where $n=1,2,\cdots,\,h_1,\cdots,h_n \in H$ and $f\in C^\infty_p(\mathbb R^n)$ (resp. $f\in C_b^\infty(\mathbb R^n)$). Note that $\mathscr S$ and $\mathscr S_b$ are dense in $L^p(\Omega)$ for all $p\in [1,\infty)$. If $X=f(W(h_1),\cdots, W(h_n)) \in \mathscr S$, define the Malliavin derivative $DX$ as $$DX:= \sum_{j=1}^n \partial_jf(W(h_1),\cdots, W(h_n))h_j. $$
I want to prove the operator $D:\mathscr S\,(\subset L^p(\Omega)) \to L^p(\Omega;H) $ is closable for any $p\in[1,\infty) $, i.e. if $(X_n)_{n=1}^\infty$ is a sequence in $\mathscr S$ which converges to $0$ in $L^p(\Omega)$ and $(DX_n)_{n=1}^\infty$ converges to $\xi$ in $L^p(\Omega; H)$, then I want to conclude $\xi=0$. Under that assumption, we can prove that for all $Y \in \mathscr S_b$ and $h\in H$ $$E[Y\langle \tilde\xi,h\rangle_H]=0,$$ where $\tilde \xi := \exp(-|W(h)|^2)\xi$. If $1<p<\infty$, using the density of $\mathscr S_b$ in $L^q(\Omega)$ where $1/p +1/q =1 $, we can conclude $\langle \tilde\xi,h\rangle_H=0$ almost surely, which implies $\langle \xi,h\rangle_H=0$ almost surely, and since $h$ is arbitraly and $H$ is separable, it follows $\xi=0$ almost surely.
If $p=1$, how do we prove $\xi=0$ ?