The Banach space of quasinorms on a finite-dimensional space

Abstract

Our main result states that, given a finite-dimensional vector space $E$, the pseudometric defined in the set of continuous quasinorms ${\mathcal{Q}}_0=\{\Vert \cdot \Vert :E\to \mathbb{R}\}$ as $$d(\Vert \cdot {\Vert }_X,\Vert \cdot {\Vert }_Y)=\min \{\mu :\Vert \cdot {\Vert }_X\le \lambda \Vert \cdot {\Vert }_Y\le \mu \Vert \cdot {\Vert }_X\text{ for some }\lambda \}$$ 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.