Suppose M be the projective limit of weak symplectic Banach manifolds {(Mi, øij)}i, j en, where Mi are modeled over reflexive Banach space and Hx is compatible with the projective system (defined in the article). We associate to each point x M, a Fréchet space Hx. We prove that if Hx are locally identical, then with certain smoothness and boundedness condition, there exists a Darboux chart for the weak symplectic structure. © 2015 World Scientific Publishing Company.