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

Abstract

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

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