# Two distribution spaces ${\mathcal S}'/{\mathcal P}$ and ${\mathcal S}_\infty'$

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 ?

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This question would benefit from some further explanation. For instance, when you say ${\cal F}f=O(|\xi|^m)$, you mean $\xi\to 0$ rather than $\xi\to\infty$, right? Otherwise, this does not seem to make sense. – Michael Renardy Aug 15 '13 at 16:26
Dear Professor Michael Renardy, I am sorry for my delay in reply. Yes, I meant $\xi \to 0$. – Yoshihiro Sawano Sep 3 '13 at 7:16