On the Darboux theorem for weak symplectic manifolds

A new tool to study reducibility of a weak symplectic form to a constant one is introduced and used to prove a version of the Darboux theorem more general than previous ones. More precisely, at each point of the considered manifold a Banach space is associated to the symplectic form (dual of the phase space with respect to the symplectic form), and it is shown that the Darboux theorem holds if such a space is locally constant. The following application 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 submanifolds of $M$, and for symplectic manifolds obtained from $M$ by the Marsden-Weinstein reduction procedure.