  We introduce a new class $\mathcal{FV}(\Omega,E)$ of spaces of weighted
functions on a set $\Omega$ with values in a locally convex Hausdorff space $E$
which covers many classical spaces of vector-valued functions like continuous,
smooth, holomorphic or harmonic functions. Then we exploit the construction of
$\mathcal{FV}(\Omega,E)$ to derive sufficient conditions such that
$\mathcal{FV}(\Omega,E)$ can be linearised, i.e. that $\mathcal{FV}(\Omega,E)$
is topologically isomorphic to the $\varepsilon$-product
$\mathcal{FV}(\Omega)\varepsilon E$ where
$\mathcal{FV}(\Omega):=\mathcal{FV}(\Omega,\mathbb{K})$ and $\mathbb{K}$ is the
scalar field of $E$.
