On symplectic Banach spaces

Abstract We extend and generalize the result of Kalton Swanson ( $$Z_2$$ Z 2 is a symplectic Banach space with no Lagrangian subspace) by showing that all higher order Rochgberg spaces $${\mathfrak {R}}^{(n)}$$ R ( n ) are subspaces. The nontrivial structure on Rochberg even one induced natural duality; while odd requires perturbation complex structure. will also study structures general and, motivated unexpected appearance structures, we introduce almost structures.