Enriched purity and presentability in Banach spaces

TL;DR

This work develops a Calderón–type model-theoretic program for Banach spaces by enriching the category of Banach spaces over complete metric spaces $\operatorname{CMet}$. It proves that finite-dimensional Banach spaces are finitely presentable in the enriched sense, implying that $Ban$ is locally finitely presentable as a $\operatorname{CMet}$-enriched category, with pure morphisms coinciding with ideals. It then characterizes approximately injective Banach spaces in terms of finite-dimensional domains and separable codomains, and frames purity via a logical/positively bounded language, connecting ideals, ultrapowers, and pure-injectivity. The paper further advances the theory by relating Lindenstrauss spaces to approximate injectivity with respect to isometries between finite-dimensional spaces and by introducing saturation notions that tie to Gurarii spaces and almost universal disposition, yielding a cohesive enrichment-based perspective on functional-analytic structure and model-theoretic behavior in Banach spaces.

Abstract

The category $Ban$ of Banach spaces and linear maps of norm $\leq 1$ is locally $\aleph_1$-presentable but not locally finitely presentable. We prove, however, that $Ban$ is locally finitely presentable in the enriched sense over complete metric spaces. Moreover, in this sense, pure morphisms are just ideals of Banach spaces. We characterize classes of Banach spaces approximately injective to sets of morphisms having finite-dimensional domains and separable codomains.

Theorems & Definitions (81)

• Definition 2.1
• Remark 2.2
• Remark 2.3
• Lemma 2.4
• proof
• Lemma 2.5
• proof
• Lemma 2.6
• proof
• Remark 2.7