An observation of the subspaces of ${\mathcal S}'$

The spaces ${\mathcal S}'/{\mathcal P}$ equipped with the quotient topology and ${\mathcal S}'_\infty$ equipped with the weak-* topology are known to be homeomorphic, where ${\mathcal P}$ denotes the set of all polynomials. The proof is a combination of the fact in the textbook by Treves and the well-known bipolar theorem. In this paper by extending slightly the idea employed in \cite{NNS15}, we give an alternative proof of this fact and then we extend this proposition so that we can include some related function spaces.