An observation of the subspaces of S′

Abstract

The spaces S′/P equipped with the quotient topology and S′∞ equipped with the weak-* topology are known to be homeomorphic, where 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 [5], we give an alternative proof of this fact and then we extend this proposition so that we can include some related function spaces.