Existence and uniqueness of greedy bases in Banach spaces

TL;DR

This paper investigates the conditions under which greedy bases exist and are unique in Banach spaces, providing new examples of spaces with unique greedy bases and identifying spaces where such bases do not exist.

Contribution

It demonstrates the non-existence of greedy bases in certain Nakano and Orlicz spaces and introduces the first known examples of non-trivial spaces with a unique greedy basis.

Findings

Non-existence of greedy bases in some Nakano and Orlicz spaces

First examples of non-trivial spaces with a unique greedy basis

Identification of spaces where greedy bases do not exist

Abstract

Our aim is to investigate the properties of existence and uniqueness of greedy bases in Banach spaces. We show the non-existence of greedy basis in some Nakano spaces and Orlicz sequence spaces and produce the first-known examples of non-trivial spaces (i.e., different from $c_0$, $\ell_1$, and $\ell_2$) with a unique greedy basis.