The approximation property for weighted spaces of differentiable functions

We study spaces ${\mathcal{C}\mathcal{V}}^k(\mathrm{\Omega },E)$ of $k$-times continuously partially differentiable functions on an open set $\mathrm{\Omega }\subset {\mathbb{R}}^d$ with values in a locally convex Hausdorff space $E$. The space ${\mathcal{C}\mathcal{V}}^k(\mathrm{\Omega },E)$ is given a weighted topology generated by a family of weights ${\mathcal{V}}^k$. For the space ${\mathcal{C}\mathcal{V}}^k(\mathrm{\Omega },E)$ and its subspace ${\mathcal{C}\mathcal{V}}_0^k(\mathrm{\Omega },E)$ 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.