Extension of vector-valued functions and sequence space representation

TL;DR

This paper develops a unified framework for extending scalar-valued functions to vector-valued functions in locally convex spaces, using operator representations and sequence space methods, generalizing previous results.

Contribution

It introduces a unified approach for extending functions with values in locally convex spaces and derives a sequence space representation for these vector-valued function spaces.

Findings

Provides a general extension method for vector-valued functions

Derives a sequence space representation of vector-valued function spaces

Extends previous results by Bonet, Frerick, Gramsch, and Jordá

Abstract

We give a unified approach to handle the problem of extending functions with values in a locally convex Hausdorff space $E$ over a field $\mathbb{K}$, which have weak extensions in a space $\mathcal{F}(\Omega,\mathbb{K})$ of scalar-valued functions on a set $\Omega$, to functions in a vector-valued counterpart $\mathcal{F}(\Omega,E)$ of $\mathcal{F}(\Omega,\mathbb{K})$. The results obtained base upon a representation of vector-valued functions as linear continuous operators and extend results of Bonet, Frerick, Gramsch and Jord\'{a}. In particular, we apply them to obtain a sequence space representation of $\mathcal{F}(\Omega,E)$ from a known representation of $\mathcal{F}(\Omega,\mathbb{K})$.