A note on point-finite coverings by balls

Abstract: We provide an elementary proof of a result by V.P.~Fonf and C.~Zanco on point-finite coverings of separable Hilbert spaces. Indeed, by using a variation of the famous argument introduced by J.~Lindenstrauss and R.R.~Phelps \cite{LP} to prove that the unit ball of a reflexive infinite-dimensional Banach space has uncountably many extreme points, we prove the following result: Let $X$ be an infinite-dimensional Hilbert space satisfying $\mathrm{dens}(X)<2^{\aleph_0}$, then $X$ does not admit point-finite coverings by open or closed balls, each of positive radius. In the second part of the paper, we follow the argument introduced by V.P. Fonf, M. Levin, and C. Zanco in \cite{FonfLevZan14} to prove that the previous result holds also in infinite-dimensional Banach spaces that are both uniformly rotund and uniformly smooth.