  Let A be a linear space of operators on a Hilbert space H, x a vector in H,
and Ax the subspace of H comprising all vectors of the form Tx with T in A. We
discuss, within a Bishop-style constructive framework, conditions under which
the projection [Ax] of H on the closure of Ax exists. We derive a general
result that leads directly to both the open mapping theorem and our main
theorem on the existence of [Ax].
