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

We introduce a new class $\mathcal{F}\mathcal{V}(\mathrm{\Omega },E)$ of spaces of weighted functions on a set $\mathrm{\Omega }$ 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 $\mathcal{F}\mathcal{V}(\mathrm{\Omega },E)$ to derive sufficient conditions such that $\mathcal{F}\mathcal{V}(\mathrm{\Omega },E)$ can be linearised, i.e. that $\mathcal{F}\mathcal{V}(\mathrm{\Omega },E)$ is topologically isomorphic to the $\epsilon$-product $\mathcal{F}\mathcal{V}(\mathrm{\Omega })\epsilon E$ where $\mathcal{F}\mathcal{V}(\mathrm{\Omega }):=\mathcal{F}\mathcal{V}(\mathrm{\Omega },\mathbb{K})$ and $\mathbb{K}$ is the scalar field of $E$.