The approximation property for weighted spaces of differentiable functions

Abstract

We study spaces $\mathcal{CV}^{k}(\Omega,E)$ of $k$-times continuously partially differentiable functions on an open set $\Omega\subset\mathbb{R}^{d}$ with values in a locally convex Hausdorff space $E$. The space $\mathcal{CV}^{k}(\Omega,E)$ is given a weighted topology generated by a family of weights $\mathcal{V}^{k}$. For the space $\mathcal{CV}^{k}(\Omega,E)$ and its subspace $\mathcal{CV}^{k}_{0}(\Omega,E)$ of functions that vanish at infinity in the weighted topology we try to answer the question whether their elements can be approximated by functions with values in a finite dimensional subspace. We derive sufficient conditions for an affirmative answer to this question using the theory of tensor products.