{"paper":{"title":"The non-compact normed space of norms on a finite-dimensional Banach space","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CT","math.FA"],"primary_cat":"math.MG","authors_text":"Apoorva Khare","submitted_at":"2018-10-15T05:41:06Z","abstract_excerpt":"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 \\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.06188","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}