The approximation property for weighted spaces of differentiable functions

We study spaces ${\cal CV}^{k}(\Omega,E)$ 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 ${\cal CV}^{k}(\Omega,E)$ is given a weighted topology generated by a family of weights ${\cal V}^{k}$. For the space ${\cal CV}^{k}(\Omega,E)$ and its subspace ${\cal CV}^{k}_{0}(\Omega,E)$ 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.