  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.
