This paper discusses the design problem of distributed H-infinity Luenberger-type partial differential equation (PDE) observer for state estimation of a linear unstable parabolic distributed parameter system (DPS) with external disturbance and measurement disturbance. Both pointwise measurement in space and local piecewise uniform measurement in space are considered; that is, sensors are only active at some specified points or applied at part thereof of the spatial domain. The spatial domain is decomposed into multiple subdomains according to the location of the sensors such that only one sensor is located at each subdomain. By using Lyapunov technique, Wirtinger's inequality at each subdomain, and integration by parts, a Lyapunov-based design of Luenberger-type PDE observer is developed such that the resulting estimation error system is exponentially stable with an H-infinity performance constraint, and presented in terms of standard linear matrix inequalities (LMIs). For the case of local piecewise uniform measurement in space, the first mean value theorem for integrals is utilised in the observer design development. Moreover, the problem of optimal H-infinity observer design is also addressed in the sense of minimising the attenuation level. Numerical simulation results are presented to show the satisfactory performance of the proposed design method.