Equilateral Sets in Banach Spaces of th form C(K)

TL;DR

This paper investigates the existence and size of uncountable equilateral sets in the unit ball of Banach spaces C(K) for various compact non-metrizable spaces, revealing both their abundance and limitations.

Contribution

It demonstrates that for most non-metrizable compact spaces, C(K) contains uncountable 2-equilateral sets, and provides examples where maximal equilateral sets are countable.

Findings

Uncountable 2-equilateral sets exist in C(K) for most non-metrizable spaces.

Examples show some spaces have only countable maximal equilateral sets.

The size of maximal equilateral sets varies depending on the space.

Abstract

We show that for "most" compact non metrizable spaces, the unit ball of the Banach space C(K) contains an uncountable 2-equilateral set. We also give examples of compact non metrizable spaces K such that the minimum cardinality of a maximal equilateral set in C(K) is countable.