Series representations in spaces of vector-valued functions via Schauder decompositions

TL;DR

This paper investigates conditions under which series representations of scalar-valued functions can be extended to vector-valued functions in locally convex spaces, using Schauder decompositions to generalize classical power series results.

Contribution

It provides sufficient conditions based on Schauder decompositions for lifting series representations from scalar to vector-valued function spaces.

Findings

Sufficient conditions for series representation lifting are established.

Applicable to many classical function spaces with an equicontinuous Schauder basis.

Extends classical power series results to vector-valued functions.

Abstract

It is a classical result that every $\mathbb{C}$-valued holomorphic function has a local power series representation. This even remains true for holomorphic functions with values in a locally complete locally convex Hausdorff space $E$ over $\mathbb{C}$. Motivated by this example we try to answer the following question. Let $E$ be a locally convex Hausdorff space over a field $\mathbb{K}$, $\mathcal{F}(\Omega)$ be a locally convex Hausdorff space of $\mathbb{K}$-valued functions on a set $\Omega$ and $\mathcal{F}(\Omega,E)$ be an $E$-valued counterpart of $\mathcal{F}(\Omega)$ (where the term $E$-valued counterpart needs clarification itself). For which spaces is it possible to lift series representations of elements of $\mathcal{F}(\Omega)$ to elements of $\mathcal{F}(\Omega,E)$? We derive sufficient conditions for the answer to be affirmative using Schauder decompositions which are applicable for many classical spaces of functions $\mathcal{F}(\Omega)$ having an equicontinuous Schauder basis.