Adaptive observers using projection-operator-based parameter update laws are considered for a class of linear infinite-dimensional systems with bounded input operator and full state measurement and subject to time-varying matched uncertainties and disturbances. The adaptive control architecture, introduced recently for finite-dimensional plants to provide guaranteed transient performance via fast adaptation, is then extended to this class using the proposed observers. Existence and uniqueness of solutions for the resulting closed loop system and uniform boundedness of the observation error are established first. Then, provided certain assumptions on the plant transfer function and the solution of a Lyapunov inequality hold, uniform guaranteed transient performance bounds on the plant state and control signal under the architecture are derived. Two examples satisfying the assumptions-control of a heat equation and a wave equation-are presented. Reference input tracking simulation results for the heat equation under the adaptive control subject to time-varying matched uncertainties and disturbances are presented in support of the theory.