Weighted vector-valued functions and the $\varepsilon$-product

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 $E$.