The normed space of norms on Euclidean space

Apoorva Khare

Published 2018-10-15Version 1

We study the space of all norms on a finite-dimensional vector space (or free module) over the real, complex, or quaternion numbers. As is well-known, all such norms are Lipschitz-equivalent; as is perhaps less well-known, these norms form a pseudo-metric space in an alternate, incomparable way to the Banach-Mazur paradigm (and in a sense "coarser"). We study the associated quotient metric space, and show in particular that it is complete, connected, and non-compact unlike the Banach-Mazur compactum. Along the way, our analysis reveals an implicit functorial construction of function spaces with diameter norms. We end with some questions, including in studying the parallel setting of the metric space of all metrics on a finite set.