  In this paper we prove the existence of an almost invariant symplectic slow
manifold for analytic Hamiltonian slow-fast systems with finitely many slow
degrees of freedom for which the error field is exponentially small. We allow
for infinitely many fast degrees of freedom. The method we use is motivated by
a paper of MacKay from 2004. The method does not notice resonances, and
therefore we do not pose any restrictions on the motion normal to the slow
manifold other than it being fast and analytic. We also present a stability
result and obtain a generalization of a result of Gelfreich and Lerman on an
invariant slow manifold to (finitely) many fast degrees of freedom.
