We establish rather general and simple theorems implying, among other things, the pseudo-(un)stable manifold theorem, Sternberg's theorem on smooth conjugacy between hyperbolic germs of maps or vector fields, and results of Fenichel, Hirsch, Pugh and Shub on existence, uniqueness and structural stability of stable or unstable manifolds at compact invariant manifolds. The Perron–Irwin approach via sequence spaces plays a crucial role.