Large separated sets of unit vectors in Banach spaces of continuous functions
Volume 157 / 2019
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.
