The theorem on which this paper is based is an easy generalization of the fact that the nullspace of an operator is the orthocomplement of (the closure of) the range of its adjoint. Its significance, here, is the observation that this may be applied to give a computationall y feasible algorithm for the problem of the title. Consider a pair of hilbert spaces U, V and a linear transformation A: U-+V. For any b in the range of A there is, by definition, a solution of the equation: (1) Ax = b . It is specifically not assumed here that the range of A is closed in V so, in general, the solution cannot be taken to depend continuously on b. Since, in many contexts, the inhomogeneous term b is known only through 'measurement' hence, only to 'arbitrary but finite' accuracy this lack of continuous dependence has heretofore been taken to preclude useful computation; see, e.g., Hadamard's discussion of a 'well-posed' problem [4]. There has been, however, considerable interest (some references are noted in § 5) in computational approaches to various problems which are ill-posed in Hadamard's sense. Even for well-posed problems it has proved desirable to distinguish in principle (cf. [9], p. 224) between the notions of solvability (existence) and approximation-solvability (obtaining a solution — granting solvability — as a limit of solutions of finite dimensional problems). Numerous examples of such singular situations as we consider might easily be adduced at this point: the backward heat equation, integral equations of the first kind, analytic continuation, inversion of the Laplace transform, etc. The particular application through which the author came to the algorithm described here concerned synthesis of a boundary null-control for the heat equation. This has already been discussed in greater detail elsewhere [2] but will be treated briefly here §6 as an example. The author would like to dedicate this paper to the memory of W. C. Chewning, who initiated that work, with thanks and in regret for his untimely death.