Locating the Range of an Operator on a Hilbert Space

exists (is computable) for each x in H; if P is that projection, / is the identity operator on H, and the adjoint T*ofT exists, then / P is the projection of H on ker(3*), the kernel of T*. (For an example to show that the existence of the adjoint of a bounded operator on a Hilbert space is not automatic in constructive mathematics, see Brouwerian Example 3 in [7]; see also Example 2 below.) This much is neither surprising nor particularly interesting. Of more interest is the observation that, constructively, the existence of the projection on ker(T*) is no guarantee of the existence of the projection on the closure of ran(7), unless we are prepared to accept within our constructive mathematics Markov's principle: