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 $\mathcal{Q}_0={|\cdot|:E\to\mathbb{R}}$ 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.