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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm223-2-4,
author = {Eva Kopeck\'a and Vladim\'\i r M\"uller},
title = {A product of three projections},
journal = {Studia Mathematica},
volume = {223},
year = {2014},
pages = {175-186},
zbl = {1314.46031},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm223-2-4}
}
Eva Kopecká; Vladimír Müller. A product of three projections. Studia Mathematica, Tome 223 (2014) pp. 175-186. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm223-2-4/