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

Volume 232 / 2016

Tomasz Kania, Tomasz Kochanek Studia Mathematica 232 (2016), 19-44 MSC: Primary 46B20, 46B04; Secondary 46E15, 46B26. DOI: 10.4064/sm8353-2-2016 Published online: 2 March 2016


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).


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

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