The optimal control of an infinite dimensional linear control system with unbounded time interval is considered. The results obtained are an extension to a Hilbert space setting of the results reported in Brock and Haurie [7]. In particular, it is shown that (under appropriate conditions) overtaking optimal trajectories asymptotically approach a unique optimal steady state. This is the so-called turnpike property found in the economics liteerature. When this condition is combined with an associated optimal control problem, results concerning the existence of overtaking optimal solutions are obtained. Finally we demonstrate the applicability of our results by considering two examples arising in the context of economics.