Reducing Subspaces

  • Carlos S. Kubrusly


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].


Hilbert Space Orthogonal Projection Direct Summand Linear Manifold Orthogonal Subspace 
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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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