On the classification of positions and complex structures in Banach spaces

A topological setting is defined to study the complexities of the relation of equivalence of embeddings (or “position”) of a Banach space into another and of the relation of isomorphism of complex structures on a real Banach space. The following results are obtained: a) if X is not uniformly finitely extensible, then there exists a space Y for which the relation of position of Y inside X reduces the relation E0 and therefore is not smooth; b) the relation of position of ℓp inside ℓp, or inside Lp, p≠2, reduces the relation E1 and therefore is not reducible to an orbit relation induced by the action of a Polish group; c) the relation of position of a space inside another can attain the maximum complexity Emax; d) there exists a subspace of Lp, 1≤p<2, on which isomorphism between complex structures reduces E1 and therefore is not reducible to an orbit relation induced by the action of a Polish group.