My Japanese book «Theory of Besov spaces, including a remark on the space $S'$ over $P$»

Let ${\mathcal S}'$ denote the set of all Schwartz distributions and ${\mathcal P}$ the set of all polynomials. If we define ${\mathcal S}_\infty$ to be the set of all $f \in {\mathcal S}$ such that $\int_{{\mathbb R}^n}x^\alpha f(x)\,dx=0$ for all $\alpha$, we can consider the dual space ${\mathcal S}_\infty'$.
We know that ${\mathcal S}_\infty'$ is isomorphic to ${\mathcal S}'/{\mathcal P}$ as linear spaces. But it seems to me that this is true topologically. In my Japanese book, I wrote a proof but I have commited the mistake. But recently I modified the proof. My result is as follows.
Theorem. Equip ${\mathcal S}'$ and ${\mathcal S}'_\infty$ with the weak star topology. Then the restriction mapping from ${\mathcal S}'$ to ${\mathcal S}_\infty'$ is open.