Let ${\mathcal S}'$ be the set of all distributions. Denote by ${\mathcal P}$ the set of all polynomials, which is embedded into ${\mathcal S}'$ as a closed subspace. Equip ${\mathcal S'}/{\mathcal P}$ with the quotient topology.

It is known that ${\mathcal S}'/{\mathcal P}$ is algebraically isomorphic to the dual of ${\mathcal S}_{\infty}$, where ${\mathcal S}_\infty$ is the set of all $f \in {\mathcal S}$ for which ${\mathcal F}f=O(|\xi|^m)$ for all $m \in {\mathcal N}$. My question is as follows: If ${\mathcal S}_\infty'$, the dual of ${\mathcal S}_\infty$, is equipped with the weak topology, then can we say that ${\mathcal S}_\infty'$ and ${\mathcal S}'/{\mathcal P}$ is topologicall isomorphic ?