On the range of the Radon transform and its dual (Q759964)

The aim of the paper is to give an answer to the question: is a Radon transform a topological isomorphism on \({\mathcal D}\) and \({\mathcal E}'?\) It is proved that \(R:{\mathcal E}'({\mathbb{R}}^ n)\leftrightarrow {\mathcal E}'(S^{n- 1}\times {\mathbb{R}})\) is a topological isomorphism, but \(R:{\mathcal D}({\mathbb{R}}^ n)\to {\mathcal D}(S^{n-1}\times {\mathbb{R}})\) has no continuous inverse, the range of the Radon transform \(R({\mathcal D}({\mathbb{R}}^ n))\) is not topologically complemented in \({\mathcal D}(S^{n-1}\times {\mathbb{R}})\) and \(R({\mathcal D}({\mathbb{R}}^ n))\) is neither bornological nor barreled