  Given an ascending sequence of weak symplectic Banach manifolds on which the
Darboux theorem is true, we can ask about conditions under which the Darboux
Theorem is also true on the direct limit. We will show in general, without very
strong conditions, the answer is negative. In particular we give an example of
an ascending weak symplectic Banach manifolds on which the Darboux Theorem is
true but not on the direct limit. In a second part, we illustrate this
discussion in the context of an ascending sequences of Sobolev manifolds of
loops in symplectic finite dimensional manifolds. This context gives rise to an
example of direct limit of weak symplectic Banach manifolds on which the
Darboux theorem is true around any point.
