A product of three projections

Tom 223 / 2014

Eva Kopecká, Vladimír Müller Studia Mathematica 223 (2014), 175-186 MSC: Primary 46C05; Secondary 54C20. DOI: 10.4064/sm223-2-4


Let $X$ and $Y$ be two closed subspaces of a Hilbert space. If we send a point back and forth between them by orthogonal projections, the iterates converge to the projection of the point onto the intersection of $X$ and $Y$ by a theorem of von Neumann.

Any sequence of orthoprojections of a point in a Hilbert space onto a finite family of closed subspaces converges weakly, according to Amemiya and Ando. The problem of norm convergence was open for a long time. Recently Adam Paszkiewicz constructed five subspaces of an infinite-dimensional Hilbert space and a sequence of projections on them which does not converge in norm. We construct three such subspaces, resolving the problem fully. As a corollary we observe that the Lipschitz constant of a certain Whitney-type extension does in general depend on the dimension of the underlying space.


  • Eva KopeckáDepartment of Mathematics
    University of Innsbruck
    A-6020 Innsbruck, Austria
  • Vladimír MüllerInstitute of Mathematics
    Academy of Sciences of the Czech Republic
    Žitná 25
    CZ-11567 Praha, Czech Republic

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