  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.
