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

Tomasz Kania¹ ; Tomasz Kochanek²

¹ Mathematics Institute University of Warwick Gibbet Hill Rd Coventry, CV4 7AL, England
² Institute of Mathematics Polish Academy of Sciences Śniadeckich 8 00-656 Warszawa, Poland and Institute of Mathematics University of Warsaw Banacha 2 02-097 Warszawa, Poland

Studia Mathematica, Tome 232 (2016) no. 1, pp. 19-44

Résumé

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 $\|x-y\| \gt 1$ for any 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 $\varepsilon \gt 0$ 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 (2015).

Détail

DOI : 10.4064/sm8353-2-2016

Keywords: banach space study circumstances under which there exists uncountable set mathcal subset unit vectors x y distinct mathcal prove set exists quasi reflexive non separable additionally super reflexive have x y geqslant varepsilon varepsilon depends only non metrisable compact hausdorff space unit sphere contains subset moreover perfectly normal set cardinality equal density solves nbsp problem nbsp nbsp mercourakis nbsp vassiliadis