Isomorphic classification of the spaces \(H_{\text{loc}}^k(\mathbb{R})\cap L^2(\mathbb{R})\) (Q1305328)

The authors study the Sobolev spaces \(H^k({\mathbb R}) \cap L^2({\mathbb R})\) from an isomorphic point of view. These are Fréchet spaces when equipped with the family of seminorms \(p_{n}(f)= \|f\|_{H^k[-n,n]} + \|f\|_{L^2({\mathbb R})}\). It is shown that \(H^k({\mathbb R}) \cap L^2({\mathbb R})\) is isomorphic to the Köthe sequence space of Moscatelli type \(\lambda_{2}(n^k)\). Consequently, \(H^k({\mathbb R}) \cap L^2({\mathbb R})\) and \(H^m({\mathbb R}) \cap L^2({\mathbb R})\) are isomorphic if and only if \(k=m\).