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

TL;DR

This paper provides an alternative proof for the homeomorphism between certain quotient spaces of tempered distributions and extends the result to include related function spaces, enhancing understanding of their topological structures.

Contribution

It offers a new proof of a known topological equivalence and extends the result to additional function spaces, broadening the theoretical framework.

Findings

Alternative proof of the homeomorphism between ${ m S}'/{ m P}$ and ${ m S}'_inity$

Extension of the homeomorphism to related function spaces

Enhanced understanding of the topological structure of distribution spaces

Abstract

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.