2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/99496We give an alternative proof of the stable manifold theorem as an application of the (right and left) inverse mapping theorem on a space of sequences. We investigate the diffeomorphism class of the global stable manifold, a problem which in the general Banach setting gives rise to subtle questions about the possibility of extending germs of diffeomorphisms.18 pages, LaTeX2e file. Revised version (corrected typos, introduction slightly extended). To appear in Studia MathematicaDynamical SystemsFunctional Analysis37D10; 46B20; 58B10text