  This article is concerned with analytic Hamiltonian dynamical systems in
infinite dimension in a neighborhood of an elliptic fixed point. Given a
quadratic Hamiltonian, we consider the set of its analytic higher order
perturbations. We first define the subset of elements which are formally
symplectically conjugacted to a (formal) Birkhoff normal form. We prove that if
the quadratic Hamiltonian satisfies a Diophantine-like condition and if such a
perturbation is formally symplectically conjugated to the quadratic
Hamiltonian, then it is also analytically symplectically conjugated to it. Of
course what is an analytic symplectic change of variables depends strongly on
the choice of the phase space. Here we work on periodic functions with Gevrey
regularity.
