This paper studies the construction of observers for nonlinear time-varying discrete-time systems in a general context, where a certain function of the states must be estimated. Appropriate notions of robust complete observability are proposed, under which a constructive proof of existence of an observer is developed. Moreover, a "transitive observer property" is proven, according to which a state observer can be generated as the series connection of two observers. The analysis and the results are developed in general normed linear spaces, to cover both finite-dimensional and infinite-dimensional systems.