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

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 $k⩾2$ the metric subspace $\{\Vert \cdot {\Vert }_p:p\in [1,\infty ]\}$ 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.

arXiv comment17 pages, no figures. Added Section 4 and updated Section 5. The Banach-Mazur compactum is indeed a quotient of our non-compact metric space of norms