We present a rigorous discussion for abelian BF theories in which the
base manifold of the U(1) bundle is homeomorphic to a Hilbert space. T
he theory has an infinite number of stages of reducibility. We specify
conditions on the base manifold under which the covariant quantizatio
n of the system can be performed unambiguously. Applications of the fo
rmulation to the superparticle and the supertstring are also discussed
.