The Proof of Frobenius Theorem in a Banach Scale

In a previous paper published in “Comptes Rendus de 1‘Academie des Sciencest” of Paris ([4], MR 49#8065), we enunciated the Frobenius theorem and a simplification of a theorem of Treves ([9], MR 43#669), about differential equations in a Banach scale. Next we published examples and applications of the first theorem in “Proceedings of American Mathematical Society” ([5], MR 55#9181), in “Anais da Academia Brasileira de Ciências” [6], and in “Actas de la IV Elam” [8]. After appeals of several mathematicians now we give complete proofs and we want use it, in the sequel, in other applications.