Large Banach spaces with no infinite equilateral sets

2021/04/12 by Koszmider, Piotr, Wark, Hugh
#FOS: Mathematics #Functional Analysis (math.FA) #Logic (math.LO)

paper · doi:10.48550/arxiv.2104.05330

Abstract

A subset of a Banach space is called equilateral if the distances between any two of its distinct elements are the same. It is proved that there exist non-separable Banach spaces (in fact of density continuum) with no infinite equilateral subset. These examples are strictly convex renormings of ℓ₁([0,1]). A wider class of renormings of ℓ₁([0,1]) which admit no uncountable equilateral sets is also considered.

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