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Large separated sets of unit vectors in Banach spaces of continuous functions

Volume 157 / 2019

Marek Cúth, Ondřej Kurka, Benjamin Vejnar Colloquium Mathematicum 157 (2019), 173-187 MSC: Primary 46B20, 46B04, 46E15; Secondary 54D30, 46B26. DOI: 10.4064/cm7648-1-2019 Published online: 8 April 2019

Abstract

The paper concerns the problem of whether a nonseparable $\mathcal {C}(K)$ space must contain a set of unit vectors whose cardinality equals the density of $\mathcal {C}(K)$, and such that the distances between any two distinct vectors are always greater than $1$. We prove that this is the case if the density is at most $\mathfrak {c}$, and that for several classes of $\mathcal {C}(K)$ spaces (of arbitrary density) it is even possible to find such a set which is $2$-equilateral, that is, the distance between two distinct vectors is exactly 2.

Authors

  • Marek CúthCharles University
    Faculty of Mathematics and Physics
    Department of Mathematical Analysis
    Sokolovská 83
    186 75 Praha 8, Czech Republic
    e-mail
  • Ondřej KurkaCharles University
    Faculty of Mathematics and Physics
    Department of Mathematical Analysis
    Sokolovská 83
    186 75 Praha 8, Czech Republic
    and
    Institute of Mathematics
    Czech Academy of Sciences
    Žitná 25
    115 67 Praha 1, Czech Republic
    e-mail
  • Benjamin VejnarCharles University
    Faculty of Mathematics and Physics
    Department of Mathematical Analysis
    Sokolovská 83
    186 75 Praha 8, Czech Republic
    e-mail

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