Projective limit of a sequence of compatible weak symplectic forms on a sequence of Banach bundles and Darboux Theorem

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Given a projective sequence of Banach bundles, each one provided with of a weak symplectic form, we look for conditions under which, the corresponding sequence of weak symplectic forms gives rise to weak symplectic form on the projective limit bundle. Then we apply this results to the tangent bundle of a projective limit of Banach manifolds. This naturally leads to ask about conditions under which the Darboux Theorem is also true on the projective limit of Banach manifolds. We will give some necessary and some sufficient conditions so that such a result is true. Then we discuss why, in general, the Moser's method can not work on projective limit of Banach weak symplectic Banach manifolds without very strong conditions like Kumar's results ([15]). In particular we give an analog result of Kumar' s one with weaker assumptions and we give an example for which such weaker conditions are satisfied. More generally, we produce examples of projective sequence of weak symplectic Banach manifolds on which the Darboux Theorem is true and an example for which the Darboux Theorem is true on each manifold, but is not true on the projective limit of these manifolds.