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Our main result states that, given a finite-dimensional vector space $E$, thepseudometric 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 theBanach-Mazur compactum are related.

The Banach space of quasinorms on a finite-dimensional space