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

• Autorzy: Tomasz Kania , Tomasz Kochanek
• Języki publikacji: EN

Abstrakty

EN

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).

Kategorie tematyczne

• 46E15: Banach spaces of continuous, differentiable or analytic functions
• 46B20: Geometry and structure of normed linear spaces
• 46B04: Isometric theory of Banach spaces
• 46B26: Nonseparable Banach spaces

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2016
• Tom: 232
• Numer: 1
• Strony: 19-44

Daty

wydano

2016

Twórcy

autor

Tomasz Kania

• Mathematics Institute, University of Warwick, Gibbet Hill Rd, Coventry, CV4 7AL, England

autor

Tomasz Kochanek

• Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-656 Warszawa, Poland
• Institute of Mathematics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland

Identyfikatory

DOI

10.4064/sm8353-2-2016