Darboux chart on Projective limit of weak symplectic Banach manifold

Abstract

Suppose M be the projective limit of weak symplectic Banach manifolds \{(M_i,\phi_{ij})\}_{i,j\in\mathbb N}, where M_i are modeled over reflexive Banach space and \sigma is compatible with the inverse system(defined in the article). We associate to each point x\in M, a Fr\'{e}chet space H_x(defined in section 3). We prove that if H_x are locally constant, then with certain smoothness and boundedness condition, there exists Darboux chart for the weak symplectic structure.