Reducing Subspaces

Carlos S. Kubrusly
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Carlos S. Kubrusly: Catholic University of Rio de Janeiro

Chapter 4 in Hilbert Space Operators, 2003, pp 33-40 from Springer

Abstract: Abstract If М is a linear manifold of an inner product space, then М ⊥ М⊥, and therefore. М ∩ М⊥ = {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 (search for similar items in EconPapers)
Date: 2003
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DOI: 10.1007/978-1-4612-2064-0_4

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