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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.

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