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  • 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.
  • 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. (en)
Title
  • A product of three projections
  • A product of three projections (en)
skos:prefLabel
  • A product of three projections
  • A product of three projections (en)
skos:notation
  • RIV/67985840:_____/14:00434096!RIV15-GA0-67985840
http://linked.open...avai/riv/aktivita
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  • I, P(GA14-07880S)
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  • 2
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  • 1065
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  • RIV/67985840:_____/14:00434096
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  • Hilbert space; projection; extension (en)
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  • PL - Polská republika
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  • [2F31B96CFCAA]
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  • Studia mathematica
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  • 223
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  • Müller, Vladimír
  • Kopecká, E.
http://linked.open...ain/vavai/riv/wos
  • 000348884900004
issn
  • 0039-3223
number of pages
http://bibframe.org/vocab/doi
  • 10.4064/sm223-2-4
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