In this paper, a controllable, observable, asymptotically stable, finite-dimensional linear system is considered. The input-output problem considered is whether the input in a fairly general class of Hilbert spaces will produce the output in the same class. The problem in this generality appears to be very difficult and in this paper a large class of Hilbert spaces is determined for which the result is true and a series of counter examples are given to the more obvious conjectures.