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Reducing Subspaces

  • Carlos S. Kubrusly

Abstract

If М is a linear manifold of an inner product space, then ММ, and therefore. ММ = {0}. A central result of Hilbert space geometry (the Projection Theorem) says that, if \(\mathcal{H}\) is a Hilbert space and М is a subspace of \(\mathcal{H}\), then М + М = \(\mathcal{H}\). In other words, the orthogonal complement of a subspace М of a Hilbert space is a complementary subspace of М (see e.g., [32, pp. 339,368]). Thus, in a Hilbert space, every subspace has a complementary subspace, and this only happens in a Hilbert space [39].

Keywords

Hilbert Space Orthogonal Projection Direct Summand Linear Manifold Orthogonal Subspace 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Birkhäuser Boston 2003

Authors and Affiliations

  • Carlos S. Kubrusly
    • 1
  1. 1.Catholic University of Rio de JaneiroRio de JaneiroBrazil

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