  We consider generalized inverses of linear operators on arbitrary vector
spaces and study the question when their product in reverse order is again a
generalized inverse. This problem is equivalent to the question when the
product of two projectors is again a projector, and we discuss necessary and
sufficient conditions in terms of their kernels and images alone. We give a new
representation of the product of generalized inverses that does not require
explicit knowledge of the factors. Our approach is based on implicit
representations of subspaces via their orthogonals in the dual space. For
Fredholm operators, the corresponding computations reduce to finite-dimensional
problems. We illustrate our results with examples for matrices and linear
ordinary boundary problems.
