  We extend and generalize the result of Kalton and Swanson ($Z_2$ is a
symplectic Banach space with no Lagrangian subspace) by showing that all higher
order Rochgberg spaces $\mathfrak R^{(n)}$ are symplectic Banach spaces with no
Lagrangian subspaces. The nontrivial symplectic structure on even spaces is the
one induced by the natural duality; while the nontrivial symplectic structure
on odd spaces requires perturbation with a complex structure. We will also
study symplectic structures on general Banach spaces and, motivated by the
unexpected appearance of complex structures, we introduce and study almost
symplectic structures.
