Università Roma Tre

The realization of a Hilbert space as a real symplectic space. Finite size example. The spaces L ^ 2 (C) and L ^ 2 (T) as symplectic spaces. Bounded polynomials on a Hilbert space. Formulation through multilinear operators, the polarization identity. Space of regular polynomials. The rule of the majority. Poisson parenthesis between two regular polynomials. Regular analytic functions. Poisson algebra structure. Symplectic coordinate changes generated by a regular Hamiltonian. Regular Hamiltonians on a space of sequences. The spaces h ^ p. The NLS equation. Good local position. Regular Hamiltonians on spaces h_ {s, p}. Immersions, solution of the homological equation in the Gevrey case. The formal series in infinite dimension. Poisson's parenthesis. Filtered Lie algebra structure. Baker Campbell Hausdorf's formula. To the normal form of formal Birkhoff. Uniqueness of Birkhoff's normal form. Formal vs analytic linearizability.

notes, Biasco Massetti Procesi. Abstract Birkhoff Normal Form. C.M.P. 375 (2020)