Uncountable sets of unit vectors that are separated by more than 1

TL;DR

This paper investigates conditions under which Banach spaces contain large uncountable sets of unit vectors separated by more than 1, revealing new existence results in various classes of Banach spaces.

Contribution

It establishes the existence of uncountable separated sets in non-separable, quasi-reflexive, and super-reflexive Banach spaces, and extends results to certain function spaces, solving an open problem.

Findings

Existence of uncountable separated sets in non-separable, quasi-reflexive Banach spaces.

Enhanced separation in super-reflexive spaces with a uniform gap.

Construction of large separated sets in $C(K)$ spaces for non-metrisable compact spaces.

Abstract

Let $X$ be a Banach space. We study the circumstances under which there exists an uncountable set $\mathcal A\subset X$ of unit vectors such that $\|x-y\|>1$ for distinct $x,y\in \mathcal A$. We prove that such a set exists if $X$ is quasi-reflexive and non-separable; if $X$ is additionally super-reflexive then one can have $\|x-y\|\geqslant 1+\varepsilon$ for some $\varepsilon>0$ that depends only on $X$. If $K$ is a non-metrisable compact, Hausdorff space, then the unit sphere of $X=C(K)$ also contains such a subset; if moreover $K$ is perfectly normal, then one can find such a set with cardinality equal to the density of $X$; this solves a problem left open by S. K. Mercourakis and G. Vassiliadis.