  The application of generalized inverses is usually neglected in pure
mathematical research. However, it is very effective for this paper. We expand
the famous matrix rank theorem due to R. Penrose to operators between Banach
paces. Therefore a modern perturbation analysis of generalized inverses is
built. Hereby, we find and prove a complete rank theorem in advanced calculus.
So a complete answer to the rank theorem problem presented by M. S. Berger is
given. Applying the co-final set and the perturbation analysis of generalized
inverses we prove the Frobenius theorem in Banach space, in the proof of which
the used vector field and flow theory are avoided. The co-final set is
essential to the Frobenius theorem. When the co-final set is trivial, the
theorem reduces to the differential equation with initial value in Banach
space. Also, we discuss a non-trivial family of subspaces and give its smooth
integral submanifold.
