On the nuclearity of weighted spaces of smooth functions

Abstract

Nuclearity plays an important role for the Schwartz kernel theorem to hold and in transferring the surjectivity of a linear partial differential operator from scalar-valued to vector-valued functions via tensor product theory. In this paper we study weighted spaces $\mathcal{EV}(\Omega)$ of smooth functions on an open subset $\Omega\subset\mathbb{R}^{d}$ whose topology is given by a family of weights $\mathcal{V}$. We derive sufficient conditions on the weights which make $\mathcal{EV}(\Omega)$ a nuclear space.