On an arbitrary reflexive Banach space, we build asymptotic observers for anabstract class of nonlinear control systems with possible compact outputs. Animportant part of this paper is devoted to various examples, where we discussthe existence of persistent inputs which make the system observable. Theseresults make a wide generalization to a nonlinear framework of previous workson the observation problem in infinite dimension (see [11,18,22,26,27,38,40] and other references therein).