  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.
