I am reading Nourdin and Peccati’s textbook (Normal Approximations with Malliavin Calculus From Stein’s Method to Universality). My question is about Lemma 1.1.6. Which says
Lemma 1.1.6: The operator $D^{p}\colon \mathcal{S}\subset L^{q}(\gamma)\to L^{q}(\gamma)$ is closable for every $q\in[1,\infty)$ and every integer $p\geq 1$.
Here $\mathcal{S}$ is the space of $C^{\infty}(\mathbb{R},\mathbb{R})$ such that $f$ and all its derivatives have at most polynomial growth.
In case someone doesn’t remember, here is the definition of closable. A linear operator $A\colon D(A)\subset \mathcal{B}\to \mathcal{B}$ where $\mathcal{B}$ is a Banach space is closable if it has a closed extension. The following characterization might also be useful:
The operator is closable iff for every sequence $\{f_{n}\}_{n=1}^{\infty}\subset D(A)$ such that $f_{n}\to 0$ and $Af_{n}\to g$, then we have $g=0$.
My question is the following. How does one prove this lemma when $q=1$? When $q\geq 1$, Holder’s inequality helps. But if $q=1$ the same trick doesn’t apply! The authors say it needs some specific argument which uses the $(L^{1}(\gamma))’=L^{\infty}(\gamma)$. Here $\gamma$ is the Gaussian measure on $\mathbb{R}$, given by $ \gamma(A)=\int_{A}\frac{1}{\sqrt{2 \pi}}\mathrm{e}^{-x^2/2}dx$.