equation 5.52 eq:lin-orthogonal-complement

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• depends_on definition 12.16 Orthogonal complement ¶
  • depends_on definition 12.29 Orthonormal basis ¶
    • depends_on definition 12.86 Internal orthogonal decomposition ¶
      • depends_on proposition 12.87 Expansion in an orthogonal decomposition ¶
        • depends_on lemma A.250 Decomposition into cyclic subspaces ¶
  • depends_on definition 12.86 Internal orthogonal decomposition ¶ ↺
  • depends_on proposition 12.17 The complement is always a closed subspace ¶
    • depends_on corollary 12.19 Double complement; the density criterion ¶
      • depends_on definition 12.79 Deficiency subspaces and indices ¶
        • depends_on definition A.269 Cayley transform ¶
        • depends_on example 12.82 Momentum on the half-line: no self-adjoint extension ¶
        • depends_on example 12.81 Momentum on a finite interval: a circle of self-adjoint momenta ¶
        • depends_on lemma A.267 Isometry of $A\pm\ii\mu$, and closed range ¶
        • depends_on lemma A.268 The indices do not depend on $\mu$ ¶
        • depends_on theorem A.266 von Neumann ¶
        • depends_on theorem 12.80 von Neumann's criterion ¶
      • depends_on definition 12.69 Operator with a domain ¶
        • depends_on definition 12.70 Graph; closed and closable operators ¶
        • depends_on definition 12.72 Symmetric; self-adjoint ¶
        • depends_on definition 12.71 Adjoint of a densely defined operator ¶
      • depends_on definition 12.103 Gelfand triple ¶
        • depends_on definition A.278 Countably Hilbert nuclear space ¶
        • depends_on definition 12.105 Generalized eigenvector ¶
        • depends_on example 12.104 The Schwartz triple ¶
        • depends_on theorem A.279 Gelfand–Maurin ¶
        • depends_on theorem 12.107 Nuclear spectral theorem ¶
      • depends_on definition 12.50 Point, continuous and residual spectrum ¶
        • depends_on definition 12.105 Generalized eigenvector ¶ ↺
        • depends_on example 12.56 Multiplication by the coordinate: spectrum without eigenvectors ¶
        • depends_on proposition 12.51 The three cases are exclusive and exhaustive ¶
      • depends_on definition 12.71 Adjoint of a densely defined operator ¶ ↺
      • depends_on theorem 12.30 Completeness, expansion, Parseval ¶
        • depends_on proposition 17.26 The lattice harmonics are an orthonormal basis ¶
        • depends_on proposition 12.87 Expansion in an orthogonal decomposition ¶ ↺
        • depends_on proposition 12.95 The tensor inner product is well defined and positive definite ¶
        • depends_on theorem A.229 Hilbert–Schmidt ¶
        • depends_on theorem 12.44 Hilbert–Schmidt: compact self-adjoint operators ¶
        • depends_on theorem 12.33 Every separable Hilbert space is $\ell^{2}$ ¶
      • depends_on theorem 12.55 The spectrum of a self-adjoint operator is real ¶
        • depends_on example 12.56 Multiplication by the coordinate: spectrum without eigenvectors ¶ ↺
        • depends_on theorem A.238 Spectral theorem, both forms ¶
        • depends_on theorem 12.59 Spectral theorem for a bounded self-adjoint operator ¶
    • depends_on theorem 12.18 Projection theorem ¶
      • depends_on corollary 12.19 Double complement; the density criterion ¶ ↺
      • depends_on definition A.248 Cyclic vector and cyclic subspace ¶
        • depends_on lemma A.249 The cyclic case ¶
        • depends_on lemma A.250 Decomposition into cyclic subspaces ¶ ↺
      • depends_on definition 12.20 Orthogonal projection operator ¶
        • depends_on definition 12.88 Reducing subspace ¶
        • depends_on proposition 12.21 Characterization of orthogonal projections ¶
      • depends_on lemma A.233 Construction of the system ¶
        • depends_on lemma A.234 The eigenvalues tend to zero, with finite multiplicity ¶
      • depends_on lemma A.231 Restriction to an invariant closed subspace ¶
        • depends_on lemma A.233 Construction of the system ¶ ↺
      • depends_on lemma A.267 Isometry of $A\pm\ii\mu$, and closed range ¶ ↺
      • depends_on proposition 12.27 Best approximation and Bessel's inequality ¶
        • depends_on theorem 12.30 Completeness, expansion, Parseval ¶ ↺
      • depends_on proposition 12.21 Characterization of orthogonal projections ¶ ↺
      • depends_on theorem 12.46 Riesz representation ¶
        • depends_on corollary 12.47 $\mathcal{H}$ is its own dual, antilinearly ¶
        • depends_on lemma A.247 Integration against a projection-valued measure ¶
        • depends_on proposition A.246 Bounded Borel functional calculus ¶
        • depends_on proposition A.245 The measures $\mu_{x,y}$ ¶
        • depends_on theorem 12.38 Existence and uniqueness of the adjoint ¶
• depends_on definition 5.149 Totally reducible representation ¶
  • depends_on proposition 5.151 prop:rep-not-totally-reducible ¶
  • depends_on proposition 5.153 prop:rep-unitary-completely-reducible ¶
    • depends_on theorem 101.62 The mixing matrix is unitary ¶
      • depends_on proposition 101.63 Parameter counting, and why three generations permit $CP$ violation ¶
      • depends_on theorem 103.11 GIM suppression at one loop ¶
        • depends_on phenomenon 103.42 $CP$ violation in charm decays ¶
• depends_on lemma 5.150 Invariance of the orthogonal complement ¶
  • depends_on proposition 5.151 prop:rep-not-totally-reducible ¶ ↺
  • depends_on proposition 5.153 prop:rep-unitary-completely-reducible ¶ ↺