Overcomplete sets in non-separable Banach spaces

TL;DR

This paper introduces the concept of overcomplete sets in non-separable Banach spaces, extending classical overcomplete sequences, and investigates their existence, non-existence, and properties across various Banach spaces.

Contribution

It generalizes the notion of overcomplete sequences to overcomplete sets in non-separable Banach spaces and analyzes their fundamental properties and existence conditions.

Findings

Existence of overcomplete sets in certain non-separable Banach spaces

Non-existence results for overcomplete sets in other classes of Banach spaces

Properties of overcomplete sequences are partially retained in overcomplete sets

Abstract

We introduce and study the notion of overcomplete set in a Banach space, that subsumes and extends the classical concept of overcomplete sequence in a (separable) Banach space. We give existence and non-existence results of overcomplete sets for a wide class of (non-separable) Banach spaces and we study to which extent properties of overcomplete sequences are retained by every overcomplete set.