Equilateral sets in infinite dimensional Banach spaces

TL;DR

This paper investigates the existence of infinite equilateral sets in Banach spaces, showing they exist under various conditions including containing $c_0$, having a bounded biorthogonal system, or being a space of continuous functions on certain compact spaces.

Contribution

It establishes new conditions under which Banach spaces contain large equilateral sets, including spaces with $c_0$, biorthogonal systems, and certain $C(K)$ spaces.

Findings

Banach spaces containing $c_0$ have infinite equilateral sets.

Spaces with bounded biorthogonal systems can be renormed to admit large equilateral sets.

Certain $C(K)$ spaces have uncountable equilateral sets under combinatorial conditions.

Abstract

We show that every Banach space $X$ containing an isomorphic copy of $c_0$ has an infinite equilateral set and also that if $X$ has a bounded biorthogonal system of size $\alpha$ then it can be renormed so as to admit an equilateral set of equal size. If $K$ is any compact non metrizable space, then under a certain combinatorial condition on $K$ the Banach space $C(K)$ has an uncountable equilateral set.