  In this paper, we prove the stability theorems for the isotropic
perturbations of Lagrangian subspaces and maximal isotropic subspaces in strong
symplectic Banach spaces. Then we prove a stability theorem for the mod $2$
dimensions of kernel of skew-adjoint Fredholm linear relations between real
Banach spaces with index $0$. Finally we gives the path components of the set
of skew-adjoint Fredholm linear relations between real Banach spaces with
indices $0$.
