ABSTRACT. When A and B are closed subspaces of a Hilbert space X, and P_{A}, P_{B} are the corresponding orthogonal projections, von Neumann showed that

(α) [(I − P_{A})(I − P_{B})]^{n}(x) → (I − P\overline{A+B})(x) (x ∈ X).

This result is extended by replacing X with any smooth and uniformly convex Banach space, and A and B any closed subspaces whose corresponding metric projections P_{A}, P_{B} are linear. Further, it is shown that (α) holds whenever X is a uniformly smooth and uniformly convex Banach space and A, B closed subspace such that A+B is closed.