Strictly convex abelian metric groups are normed spaces

Authors

  • T.O. Banakh Ivan Franko National Univerisy of Lviv, 1 Universytetska str., 79000, Lviv, Ukraine; Jan Kochanowski University of Kielce, 5 Żeromskiego str., 25369, Kielce, Poland https://orcid.org/0000-0001-6710-4611
  • O.V. Mazurenko Ivan Franko National Univerisy of Lviv, 1 Universytetska str., 79000, Lviv, Ukraine https://orcid.org/0009-0009-3639-0171
https://doi.org/10.15330/cmp.18.1.198-210

Keywords:

strictly convex metric space, normed space, abelian metric group, geodesic metric space, finite-dimensional normed space, locally compact metric group, $\mathbb R$-normable metric group
Published online: 2026-06-08

Abstract

We prove that every strictly convex abelian metric group has a canonical structure of a normed space over the field of real numbers. We deduce this fact from the $\mathbb R$-normability of strictly convex metric groups. Moreover, we prove that a strictly convex (more generaly, $\mathbb R$-normable) metric group is a finite-dimensional normed space if and only if it is locally compact if and only if it is (compactly) finite-dimensional. Also we prove that every strictly convex metric space is geodesic.

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How to Cite
(1)
Banakh, T.; Mazurenko, O. Strictly Convex Abelian Metric Groups Are Normed Spaces. Carpathian Math. Publ. 2026, 18, 198-210.