  Consider a physical system for which a mathematically rigorous geometric
quantization procedure exists. Now subject the system to a finite set of
irreducible first class (bosonic) constraints. It is shown that there is a
mathematically rigorous BRST quantization of the constrained system whose
cohomology at ghost number zero recovers the constrained quantum states.
Moreover this space of constrained states has a well-defined Hilbert space
structure inherited from that of the original system. Treatments of these ideas
in the Physics literature are more general but suffer from having states with
infinite or zero "norms" and thus are not admissible as states. Also the BRST
operator for many systems require regularization to be well-defined. In our
more restricted context we show that our treatment does not suffer from any of
these difficulties. This work was submitted for publication March 21,2006.
