  We extend the pseudoholomorphic curve methods from Floer theory to
infinite-dimensional phase spaces and use our results to prove the existence of
a forced time-periodic solution to a general Hamiltonian PDE with regularizing
nonlinearity. In particular, when the nonlinearity is sufficiently
regularizing, bounded and time-periodic, we prove an infinite-dimensional
version of Gromov-Floer compactness by using ideas from the theory of
Diophantine approximations to overcome the small divisor problem. Furthermore,
in the case when the infinite-dimensional phase space is a product of a
finite-dimensional closed symplectic manifold with linear symplectic Hilbert
space, we prove a cup-length estimate for the number of periodic solutions.
