Let X be a Banach space. We study the circumstances under which there exists an uncountable set 𝓐 ⊂ X of unit vectors such that ||x-y|| > 1 for any distinct x,y ∈ 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|| ≥ slant 1 + ε for some ε > 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).
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm8353-2-2016,
author = {Tomasz Kania and Tomasz Kochanek},
title = {Uncountable sets of unit vectors that are separated by more than 1},
journal = {Studia Mathematica},
volume = {233},
year = {2016},
pages = {19-44},
zbl = {06575021},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm8353-2-2016}
}
Tomasz Kania; Tomasz Kochanek. Uncountable sets of unit vectors that are separated by more than 1. Studia Mathematica, Tome 233 (2016) pp. 19-44. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm8353-2-2016/