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An alternating projection that does not converge in norm. (English) Zbl 1070.46013
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 =(P 1 P 2 ) n y 0 converge in norm to P C 1 C 2 (y 0 ) 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 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.

MSC:
46C05Hilbert and pre-Hilbert spaces: geometry and topology
41A65Abstract approximation theory