Representations of the spaces ${\displaystyle C^{\infty }\left({ℝ}^N\right)\cap H^{k,p}\left({ℝ}^N\right)}$

We give a representation of the spaces ${\displaystyle C^{\infty }\left({ℝ}^N\right)\cap H^{k,p}\left({ℝ}^N\right)}$ as spaces of vector-valued sequences and use it to investigate their topological properties and isomorphic classification. In particular, it is proved that ${\displaystyle C^{\infty }\left({ℝ}^N\right)\cap H^{k,2}\left({ℝ}^N\right)}$ is isomorphic to the sequence space ${\displaystyle s^{\mathbb{N}}\cap l^2\left(l^2\right)}$, thereby showing that the isomorphy class does not depend on the dimension N if p=2.