The Banach space of quasinorms on a finite-dimensional space

TL;DR

This paper proves that the set of continuous quasinorms on a finite-dimensional space forms a complete Banach space under a specific pseudometric, and explores its relation to the Banach-Mazur compactum.

Contribution

It introduces a new Banach space structure on the set of quasinorms on finite-dimensional spaces and clarifies its connection to the Banach-Mazur compactum.

Findings

The pseudometric induces a complete norm on the quotient space of quasinorms.

The space of quasinorms is shown to be a Banach space.

Relation between this space and the Banach-Mazur compactum is explained.

Abstract

Our main result states that, given a finite-dimensional vector space $E$, the pseudometric defined in the set of continuous quasinorms $\mathcal{Q}_0=\{\|\cdot\|:E\to\mathbb{R}\}$ as $$d(\|\cdot\|_X,\|\cdot\|_Y)=\min\{\mu:\|\cdot\|_X \leq\lambda\|\cdot\|_Y\leq\mu\|\cdot\|_X\text{ for some }\lambda \}$$ induces, in fact, a complete norm when we take the obvious quotient $\mathcal{Q}=\mathcal{Q}_0/\!\sim$ and define the appropriate operations on $\mathcal{Q}$. We finish the paper with a little explanation of how this space and the Banach-Mazur compactum are related.