Skew-adjoint linear relatioins between Banach spaces

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