Tracking an ensemble of basic signals is often required of control systems in general. Here we are given a linear continuous-time infinite-dimensional plant on a Hilbert space and a space of tracking signals generated by a finite basis, and we show that there exists a stabilizing direct adaptive control law that will stabilize the plant and cause it to asymptotically track any member of this collection of signals. The plant is described by a closed, densely defined linear operator that generates a continuous semigroup of bounded operators on the Hilbert space of states. There is no state or parameter estimation used in this adaptive approach. Our results are illustrated by adaptive control of general linear diffusion systems.