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 quasinormsinduces, in fact, a complete norm when we take the obvious quotient Q = Q0/ ∼ and define the appropriate operations on Q.We finish the paper with a little explanation of how this space and the Banach-Mazur compactum are related.