On the partial derivative-equation in a Banach space

We define a separable Banach space X and prove the existence of a partial derivative-closed C-infinity-smooth (0, I)-form f on the unit ball B of X, which is not partial derivative-exact on any open subset. Further, we show that the sheaf cohomology groups H-q (Omega, 0) = 0, q greater than or equal to 1, where 0 is the sheaf of germs of holomorphic functions on X, and Omega is any pseudoconvex domain in X, e.g., Omega = B. As the Dolbeault group H-partial derivative(0.1)(B) not equal 0, the Dolbeault isomorphism theorem does not generalize to arbitrary Banach spaces. Lastly, we construct a C-infinity-smooth integrable almost complex structure on M = B x C such that no open subset of M is biholomorphic to an open subset of a Banach space. Hence the Newlander-Nirenberg theorem does not generalize to arbitrary Banach manifolds.