# Analytic P-ideals and Banach spaces

## TL;DR

This paper explores the deep connections between Banach space theory and analytic P-ideals, revealing symmetries, new examples, and characterizations that advance understanding in both areas.

## Contribution

It uncovers symmetries between Banach spaces with unconditional bases and analytic P-ideals, introduces new examples, and provides novel characterizations impacting the theory.

## Key findings

- Identified symmetries between Banach spaces and analytic P-ideals
- Constructed new examples of Banach spaces and P-ideals
- Developed a new characterization of precompact families

## Abstract

We study the interplay between Banach space theory and theory of analytic P-ideals. Applying the observation that, up to isomorphism, all Banach spaces with unconditional bases can be constructed in a way very similar to the construction of analytic P-ideals from submeasures, we point out numerous symmetries between the two theories. Also, we investigate a special case, the interactions between combinatorics of families of finite sets, topological properties of the "Schreier type" Banach spaces associated to these families, and the complexity of ideals generated by the canonical bases in these spaces. Among other results, we present some new examples of Banach spaces and analytic P-ideals, a new characterization of precompact families and its applications to enhance Pt\'ak's Lemma and Mazur's Lemma.