On the Darboux theorem for weak symplectic manifolds

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.