Large separated sets of unit vectors in Banach spaces of continuous functions

TL;DR

This paper investigates the existence of large, well-separated sets of unit vectors in Banach spaces of continuous functions, establishing conditions under which such sets are equilateral or have large cardinality.

Contribution

It proves that nonseparable (K) spaces with density at most continuum contain large separated sets, and for certain classes, these sets can be made 2-equilateral.

Findings

Existence of large separated sets in (K) spaces with density at most continuum.

Construction of 2-equilateral sets in specific (K) spaces.

Results extend understanding of geometric structures in Banach spaces.

Abstract

The paper concerns the problem whether a nonseparable $\C(K)$ space must contain a set of unit vectors whose cardinality equals to the density of $\C(K)$ such that the distances between every two distinct vectors are always greater than one. We prove that this is the case if the density is at most continuum and we prove that for several classes of $\C(K)$ spaces (of arbitrary density) it is even possible to find such a set which is $2$-equilateral; that is, the distance between every two distinct vectors is exactly 2.