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 Ptak's Lemma and Mazur's Lemma.

中文翻译：

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解析 P 理想和巴拿赫空间

我们研究 Banach 空间理论和解析 P 理想理论之间的相互作用。应用观察，直到同构，所有具有无条件基的 Banach 空间都可以以非常类似于从子测度构建解析 P 理想的方式构建，我们指出了两种理论之间的许多对称性。此外，我们研究了一个特殊情况，有限集族的组合、与这些族相关的“Schreier 型”Banach 空间的拓扑性质以及这些空间中规范基生成的理想的复杂性之间的相互作用。在其他结果中，我们展示了 Banach 空间和解析 P 理想的一些新示例、预紧族的新特征及其在增强 Ptak 引理和 Mazur 引理方面的应用。