On Banach spaces invariant under differentiation (Q2717507)

Let \({\mathcal D}(\Omega)\) denote the space of usual test functions on an open set \(\Omega\subset \mathbb{R}^n\) and \({\mathcal D}_\sigma'(\Omega)\) the space of distributions with its weak topology.NEWLINENEWLINENEWLINEIn this paper the authors give a short and elegant proof of the fact that there is no Banach space \(E\) such that NEWLINE\[NEWLINE{\mathcal D}(\Omega)\subset E\subset{\mathcal D}_\sigma'(\Omega)NEWLINE\]NEWLINE with all partial differentiations acting continuously in \(E\). They also show that a Banach space \(E\) with all partial differentiations acting continuously on it and contained in \({\mathcal D}_\sigma'(\Omega)\) is already contained in a canonical weighted space of entire functions.