2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/131280This is the first paper in a series introducing a generalized Fredholm theory in a new class of smooth spaces called polyfolds. The theory will be illustrated in upcoming papers by applications to Floer Theory, Gromov-Witten Theory and Symplectic Field Theory.Some typos corrected and some minor changes. To appear Journal of the European Mathematical SocietyFunctional AnalysisDifferential GeometrySymplectic Geometry46T99, 58B99, 53C99text