The Banach space of quasinorms on a finite-dimensional space

2020/07/08 by Sánchez, Javier Cabello, González, Daniel Morales
#46B20 #47A30 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2007.04382

Abstract

Our main result states that, given a finite-dimensional vector space E, the pseudometric defined in the set of continuous quasinorms Q₀=\‖⋅‖:E→ℝ\ as d(‖⋅‖_X,‖⋅‖_Y)=min\μ:‖⋅‖_X ≤λ‖⋅‖_Y≤μ‖⋅‖_X for some λ\ induces, 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.

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