Inverse Function Theorem in Fr\'echet Spaces

We consider the classical Inverse Function Theorem of Nash and Moser from the angle of some recent development by Ekeland and the authors. Geometrisation of tame estimates coupled with certain ideas coming from Variational Analysis when applied to a directionally differentiable function, produce very general surjectivity result and, if injectivity can be ensured, Inverse Function Theorem with the expected Lipschitz-like continuity of the inverse. We also present a brief application to differential equations.
Comment: This is revised version. Implicit Mapping Theorem and an application to ODE were added