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

TL;DR

The paper introduces a new class of weighted vector-valued function spaces that unify many classical function spaces and provides conditions for their linearization via the $ ext{ extepsilon}$-product, enhancing understanding of their structure.

Contribution

It defines the space $ ext{ extcal{FV}}( ext{ extOmega},E)$ and establishes conditions under which it is topologically isomorphic to the $ ext{ extepsilon}$-product $ ext{ extcal{FV}}( ext{ extOmega}) ext{ extepsilon} E$, generalizing classical spaces.

Findings

Unified many classical vector-valued function spaces.

Derived conditions for linearization via $ ext{ extepsilon}$-product.

Established topological isomorphism results.

Abstract

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 $E$.