Let

${C}_{1}$ and

${C}_{2}$ be two intersecting closed convex sets in a Hilbert space. Let

${P}_{1}$ and

${P}_{2}$ denote the corresponding projection operators. In 1933, von Neumann proved that the iterates produced by the sequence of alternating projections defined as

${y}_{n}={\left({P}_{1}{P}_{2}\right)}^{n}{y}_{0}$ converge in norm to

${P}_{{C}_{1}\cap {C}_{2}}\left({y}_{0}\right)$ when

${C}_{1}$ and

${C}_{2}$ are closed subspaces.

*L. M. Bregman* [Sov. Math., Dokl. 6, 688–692 (1965;

Zbl 0142.16804)] showed that the iterates converge weakly to a point in

${C}_{1}\cap {C}_{2}$ for any pair of closed convex sets. In the paper under review, the author shows that alternating projections not always converge in the norm by constructing an explicit counterexample.