  We formulate and prove a de Finetti representation theorem for finitely
exchangeable states of a quantum system consisting of k infinite-dimensional
subsystems. The theorem is valid for states that can be written as the partial
trace of a pure state chosen from a family of subsets C_n of the full symmetric
subspace for $n$ subsystems. We show that such states become arbitrarily close
to mixtures of pure power states as n increases. We give a second equivalent
characterization of the family C_n.
