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 $\mathcal{E}\mathcal{V}(\Omega )$ 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 $\mathcal{E}\mathcal{V}(\Omega )$ a nuclear space.