The non-compact normed space of norms on a finite-dimensional Banach space

TL;DR

This paper introduces a new pseudometric on the space of all norms on finite-dimensional vector spaces, revealing it as a complete, connected, non-compact normed space with novel embedding properties and connections to metric space theory.

Contribution

It defines and studies a new coarser topology on norms, embedding the space into a normed space, and explores analogous structures for metrics, establishing new connections and embeddings.

Findings

The quotient space of norms is complete, connected, and non-compact.

The space of all metrics on a finite set forms a normed space.

Embeddings of finite metric spaces into the space of all metrics are constructed.

Abstract

We discuss a new pseudometric on the space of all norms on a finite-dimensional vector space (or free module) $\mathbb{F}^k$, with $\mathbb{F}$ the real, complex, or quaternion numbers. This metric arises from the Lipschitz-equivalence of all norms on $\mathbb{F}^k$, and seems to be unexplored in the literature. We initiate the study of the associated quotient metric space, and show that it is complete, connected, and non-compact. In particular, the new topology is strictly coarser than that of the Banach-Mazur compactum. For example, for each $k \geqslant 2$ the metric subspace $\{ \| \cdot \|_p : p \in [1,\infty] \}$ maps isometrically and monotonically to $[0, \log k]$ (or $[0,1]$ by scaling the norm), again unlike in the Banach-Mazur compactum. Our analysis goes through embedding the above quotient space into a normed space, and reveals an implicit functorial construction of function spaces with diameter norms (as well as a variant of the distortion). In particular, we realize the above quotient space of norms as a normed space. We next study the parallel setting of the - also hitherto unexplored - metric space $\mathcal{S}([n])$ of all metrics on a finite set of $n$ elements, revealing the connection between log-distortion and diameter norms. In particular, we show that $\mathcal{S}([n])$ is also a normed space. We demonstrate embeddings of equivalence classes of finite metric spaces (parallel to the Gromov-Hausdorff setting), as well as of $\mathcal{S}([n-1])$, into $\mathcal{S}([n])$. We conclude by discussing extensions to norms on an arbitrary Banach space and to discrete metrics on any set, as well as some questions in both settings above.