Locating Ax, where A is a subspace of B(H)

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].