KAM theory with applications to Hamiltonian partial differential equations

In these notes I present a KAM theorem on the existence of lower dimensional invariant tori for a class of nearly integrable Hamiltonian systems of infinite dimensions, where the second Melnikov's conditions are completely eliminated and the algebraic structure of the normal frequencies is not required. This theorem call be used to construct invariant tori and quasi-periodic Solutions for nonlinear wave equations, Schrodinger equations and other equations of any spatial dimensions.