This paper considers the state observation problem for nonlinear dynamical systems. The proposed framework is a direct generalization of a method introduced in a recent paper for autonomous system. Its characteristic feature is that the dynamic part of the observer is linear and, as a consequence, that convergence takes place globally in the observer coordinates. The observer is completed by a static nonlinearity which maps the observer state in the original state space. An associated observation mapping is introduced and is interpreted in terms of an orthonormal expansion of the input and the output with respect to a certain basis in a suitable Hilbert space. It is shown that, by choosing the observer dimension properly, an observer with arbitrary small asymptotic observation error is obtained, provided that some compactness properties for the subset to be observed and the set of input signals hold. Under a stronger condition, the finite complexity property, an exact observer is achieved. Finally, an integral formula representation for the observer nonlinearity is given.