# On the classification of positions and of complex structures in Banach spaces

## TL;DR

This paper investigates the complexity of classifying embeddings and complex structures in Banach spaces, revealing that these classification problems can reach high levels of complexity, including non-smooth and maximum complexity cases.

## Contribution

It introduces a topological framework to analyze the complexity of embedding and isomorphism relations in Banach spaces, demonstrating their potential for high complexity.

## Key findings

- Embedding relations can have non-smooth complexity levels.
- Certain spaces' embeddings reduce to complex equivalence relations like E_1.
- Some subspaces exhibit maximum possible complexity E_max.

## Abstract

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 $E_0$ and therefore is not smooth; b) the relation of position of $\ell_p$ inside $\ell_p$, or inside $L_p$, $p \neq 2$, reduces the relation $E_1$ 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 $E_{{\rm max}}$; d) there exists a subspace of $L_p, 1 \leq p <2$, on which isomorphism between complex structures reduces $E_1$ and therefore is not reducible to an orbit relation induced by the action of a Polish group.