definition 12.20 Orthogonal projection operator

open in the book · parts/02-mathematical-methods/10-hilbert-spaces.tex:504 · p. 418

ground object -- no derivation owed

Rests on

depends_on theorem 12.18 Projection theorem ¶
- depends_on proposition 12.17 The complement is always a closed subspace ¶
  - depends_on corollary 12.5 Continuity of the norm and of orthogonality ¶
    - depends_on definition 5.24 Orthogonal vectors ¶
      - depends_on definition 5.18 Inner product ¶
    - depends_on proposition 12.4 Cauchy–Schwarz and continuity of the inner product ¶
      - depends_on definition 12.2 Hilbert space ¶
      - depends_on proposition 5.20 Cauchy–Schwarz inequality ¶
      - proves proof ch:10-hilbert-spaces@proof-1 ¶
    - proves proof ch:10-hilbert-spaces@proof-2 ¶
  - depends_on definition 12.16 Orthogonal complement ¶
    - depends_on definition 5.24 Orthogonal vectors ¶ ↺
    - depends_on equation 5.52 eq:lin-orthogonal-complement ¶
  - proves proof ch:10-hilbert-spaces@proof-8 ¶
- depends_on theorem 12.14 Closest point in a closed convex set ¶
  - depends_on definition 12.13 Convex set ¶
    - depends_on definition 5.7 Vector subspace ¶
      - depends_on definition 5.5 Linear combination ¶
  - depends_on definition 12.2 Hilbert space ¶ ↺
  - depends_on proposition 12.6 Parallelogram law and polarization ¶
    - depends_on definition 5.18 Inner product ¶ ↺
    - depends_on equation 5.44 eq:lin-norm-assoc ¶
    - proves proof ch:10-hilbert-spaces@proof-3 ¶
  - proves proof ch:10-hilbert-spaces@proof-7 ¶
- proves proof ch:10-hilbert-spaces@proof-9 ¶

Supports

depends_on definition 12.88 Reducing subspace ¶
- depends_on definition 12.90 Self-adjoint family; commutant; irreducibility ¶
  - depends_on proposition A.589 Cyclic subspaces and the rank of the average ¶
    - depends_on lemma A.590 The Gram matrix is universal ¶
      - depends_on proposition A.591 Any two irreducible Weyl systems are equivalent ¶
    - depends_on proposition A.591 Any two irreducible Weyl systems are equivalent ¶ ↺
  - depends_on theorem A.579 Stone–von Neumann ¶
  - depends_on theorem 12.91 Schur's lemma, commutant form ¶
    - depends_on proposition A.591 Any two irreducible Weyl systems are equivalent ¶ ↺
    - depends_on theorem 12.114 Stone–von Neumann ¶
  - depends_on theorem 12.114 Stone–von Neumann ¶ ↺
- depends_on proposition 12.89 Reduction is commutation ¶
  - depends_on proposition A.589 Cyclic subspaces and the rank of the average ¶ ↺
  - depends_on theorem 12.91 Schur's lemma, commutant form ¶ ↺
depends_on proposition 12.21 Characterization of orthogonal projections ¶
- depends_on definition 12.41 The operator classes ¶
  - depends_on definition 12.90 Self-adjoint family; commutant; irreducibility ¶ ↺
  - depends_on definition 12.58 Projection-valued measure ¶
    - depends_on definition 12.60 Functional calculus ¶
      - depends_on proposition 12.61 Uniqueness of the continuous functional calculus ¶
    - depends_on lemma A.247 Integration against a projection-valued measure ¶
    - depends_on theorem A.238 Spectral theorem, both forms ¶
      - depends_on proposition A.280 Direct-integral form of the spectral theorem ¶
      - depends_on proposition A.261 Spectral theorem for a unitary operator ¶
      - depends_on theorem A.279 Gelfand–Maurin ¶
      - depends_on theorem A.253 Stone ¶
    - depends_on theorem 12.59 Spectral theorem for a bounded self-adjoint operator ¶
      - depends_on definition 12.60 Functional calculus ¶ ↺
      - depends_on theorem 12.107 Nuclear spectral theorem ¶
      - depends_on theorem 12.91 Schur's lemma, commutant form ¶ ↺
      - depends_on theorem 12.66 Stone ¶
  - depends_on definition 12.64 Strongly continuous one-parameter unitary group ¶
    - depends_on definition A.580 Weyl operator ¶
      - depends_on definition A.585 The Gaussian average of the Weyl operators ¶
      - depends_on lemma A.581 Composition law ¶
    - depends_on definition 12.109 Weyl system ¶
      - depends_on corollary 12.112 Commutator of the momentum with a function of the position ¶
      - depends_on definition A.580 Weyl operator ¶ ↺
      - depends_on example 12.110 The Schrödinger system ¶
      - depends_on proposition 12.111 The Weyl relation is a covariance statement ¶
      - depends_on theorem A.579 Stone–von Neumann ¶ ↺
      - depends_on theorem 12.114 Stone–von Neumann ¶ ↺
    - depends_on lemma A.255 Smoothed vectors lie in the domain ¶
      - depends_on proposition A.256 Density ¶
    - depends_on lemma A.581 Composition law ¶ ↺
    - depends_on lemma A.582 Joint strong continuity ¶
      - depends_on lemma A.583 Absolutely convergent operator-valued integrals ¶
    - depends_on proposition A.256 Density ¶ ↺
    - depends_on proposition 12.65 Exponential of a bounded self-adjoint operator ¶
      - depends_on theorem 12.66 Stone ¶ ↺
    - depends_on proposition 12.67 The generator is symmetric, and generates the motion ¶
      - depends_on lemma A.257 Integrated form of the equation of motion ¶
      - depends_on proposition A.263 The two constructions are inverse ¶
      - depends_on proposition A.259 The generator is self-adjoint ¶
      - depends_on proposition 12.111 The Weyl relation is a covariance statement ¶ ↺
      - depends_on theorem A.253 Stone ¶ ↺
    - depends_on theorem A.253 Stone ¶ ↺
    - depends_on theorem 12.66 Stone ¶ ↺
    - depends_on theorem 25.34 Stone–von Neumann ¶
      - depends_on lemma 25.39 The low-degree images are forced ¶
      - depends_on theorem 25.38 Groenewold–van Hove ¶
  - depends_on lemma A.231 Restriction to an invariant closed subspace ¶
    - 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.230 Sequential characterisation ¶
    - depends_on lemma A.232 Attainment ¶
      - depends_on lemma A.233 Construction of the system ¶ ↺
    - depends_on lemma A.231 Restriction to an invariant closed subspace ¶ ↺
    - depends_on lemma A.234 The eigenvalues tend to zero, with finite multiplicity ¶ ↺
  - depends_on proposition 12.42 Elementary consequences ¶
    - depends_on lemma A.232 Attainment ¶ ↺
    - depends_on proposition 12.43 Norm of a self-adjoint operator ¶
      - depends_on lemma A.232 Attainment ¶ ↺
      - depends_on lemma A.239 The norm of a self-adjoint operator lies in its spectrum ¶
      - depends_on theorem A.229 Hilbert–Schmidt ¶
      - depends_on theorem A.238 Spectral theorem, both forms ¶ ↺
      - depends_on theorem 12.44 Hilbert–Schmidt: compact self-adjoint operators ¶
      - depends_on theorem 12.59 Spectral theorem for a bounded self-adjoint operator ¶ ↺
  - depends_on theorem A.229 Hilbert–Schmidt ¶ ↺
  - depends_on theorem 12.44 Hilbert–Schmidt: compact self-adjoint operators ¶ ↺
  - 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 definition 12.58 Projection-valued measure ¶ ↺
- depends_on proposition 12.89 Reduction is commutation ¶ ↺

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
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Edges

type	direction	node	provenance	where
`depends_on`	→	Projection theorem	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:510
`depends_on`	←	Reducing subspace	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2441
`depends_on`	←	Characterization of orthogonal projections	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:524