  This is part II of our book on KAM theory. We start by defining functorial
analysis and then switch to the particular case of Kolmogorov spaces. We
develop functional calculus based on the notion of local operators. This allows
to define the exponential and therefore relation between Lie algebra and Lie
group actions in the infinite dimensional context. Then we introduce a notion
of finite dimensional reduction and use it to prove a fixed point theorem for
Kolmogorov spaces. We conclude by proving general normal theorems.
