The basic theory for the analysis of approximate methods of solution of nonlinear operator and functional equations is presented in the monographs [i-4], which also give extensive bibliography on these topics.The process of solving nonlinear equations with many isolated solutions consists of two stages: separation of solutions and iterative refinement of solutions. While the topic of refinement of solutions has been studied in considerable detail, the very difficult topic of separation of isolated solutions of nonlinear equations has been insufficiently studied.Separation of isolated solutions has been considered in [5] for the case when the nonlinear operators generating the exact equation and the sequence of approximate equations act in the same Hilbert space. In the present paper, we consider separation of isolated solutions when the nonlinear operators generating the exact and the approximate equations act in different spaces. The sequence of approximate equations is constructed by projection methods [2][3][4]. The solutions of the exact equation are separated using a local theorem of the iterative method [6].Thus, following [2, 3], we consider the operator equation r'u : = u -~u = 0 (1)

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