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