In this paper, we develop a new approach for the observer design of switched linear systems by ergodic theory and matrix analysis. Differently from most current results, we formulate the admissible switching sequences into a compact metric space and establish linear cocycles corresponding to the switched dynamics on this metric space under the framework of dynamical system theory. The necessary and sufficient condition for the almost surely convergent observer is given. Observer gains that minimize the unitarily invariant norm of the error dynamics during each switching process are obtained via pseudoinverse setting. An example is given to illustrate the proposed approach. (C) 2018 Elsevier Ltd. All rights reserved.