A new tool to study reducibility of a weak symplectic form to a constant on
e is introduced and used to prove a version of the Darboux theorem more gen
eral than previous ones. More precisely, at each point of the considered ma
nifold a Banach space is associated to the symplectic form (dual of the pha
se space with respect to the symplectic form), and it is shown that the Dar
boux theorem holds if such a space is locally constant. The following appli
cation is given. Consider a weak symplectic manifold M on which the Darboux
theorem is assumed to hold (e.g. a symplectic vector space). It is proved
that the Darboux theorem holds also for any finite codimension symplectic s
ubmanifolds of M, and for symplectic manifolds obtained from M by the Marsd
en-Weinstein reduction procedure.