  We state sufficient conditions for the existence, on a given open set, of the
extension, to nonlinear systems, of the Luenberger observer as it has been
proposed by Kazantzis and Kravaris. We prove it is sufficient to choose the
dimension of the system, giving the observer, less than or equal to 2 + twice
the dimension of the state to be observed. We show that it is sufficient to
know only an approximation of the solution of a PDE, needed for the
implementation. We establish a link with high gain observers. Finally we extend
our results to systems satisfying an unboundedness observability property.
