  We present a rigurous disscusion for abelian $BF$ theories in which the base
manifold of the $U(1)$ bundle is homeomorphic to a Hilbert space. The theory
has an infinte number of stages of reducibility. We specify conditions on the
base manifold under which the covarinat quantization of the system can be
performed unambiguously. Applications of the formulation to the superparticle
and the supertstring are also discussed.
