The Banach Space of Quasinorms on a Finite-Dimensional Space

Abstract

We will show that the set of quasinorms, after taking quotient by the dilations, on a finite-dimensional linear space has a natural structure of Banach space. Our main result states that, given a finite-dimensional vector space $E$, the pseudometric defined in the set of 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 $\mathcal{Q}=\mathcal{Q}_0/\negthinspace\sim$ 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.