theorem 12.18 Projection theorem

open in the book · parts/02-mathematical-methods/10-hilbert-spaces.tex:426 · p. 418

Rests on

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 definition 4.33 Vector space ¶
  - depends_on proposition 12.4 Cauchy–Schwarz and continuity of the inner product ¶
    - depends_on definition 12.2 Hilbert space ¶
      - depends_on definition 5.18 Inner product ¶ ↺
      - depends_on definition 6.27 Convergence; Cauchy sequence; completeness ¶
      - depends_on equation 5.44 eq:lin-norm-assoc ¶
    - depends_on proposition 5.20 Cauchy–Schwarz inequality ¶
      - depends_on definition 5.14 Linear independence ¶
      - depends_on definition 5.18 Inner product ¶ ↺
      - proves proof ch:03-linear-algebra-representations@proof-4 ¶
    - 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 4.33 Vector space ¶ ↺
- 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 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 proposition A.270 Properties of the transform ¶
      - depends_on lemma A.271 Injectivity of $\identity-V$ for any isometric extension ¶
      - depends_on lemma A.272 The operator attached to an isometry ¶
      - depends_on proposition A.273 Self-adjoint means unitary ¶
  - depends_on example 12.82 Momentum on the half-line: no self-adjoint extension ¶
    - depends_on corollary A.275 Momentum on $[0,\infty)$: no extension ¶
  - depends_on example 12.81 Momentum on a finite interval: a circle of self-adjoint momenta ¶
    - depends_on corollary A.274 Momentum on ${[}0,L{]}$: the circle of extensions ¶
    - depends_on example 12.82 Momentum on the half-line: no self-adjoint extension ¶ ↺
  - depends_on lemma A.267 Isometry of $A\pm\ii\mu$, and closed range ¶
    - depends_on definition A.269 Cayley transform ¶ ↺
    - depends_on lemma A.268 The indices do not depend on $\mu$ ¶
    - depends_on proposition A.270 Properties of the transform ¶ ↺
    - depends_on proposition A.273 Self-adjoint means unitary ¶ ↺
  - depends_on lemma A.268 The indices do not depend on $\mu$ ¶ ↺
  - depends_on theorem A.266 von Neumann ¶
    - depends_on corollary A.275 Momentum on $[0,\infty)$: no extension ¶ ↺
    - depends_on corollary A.274 Momentum on ${[}0,L{]}$: the circle of extensions ¶ ↺
  - depends_on theorem 12.80 von Neumann's criterion ¶
    - 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 definition 12.69 Operator with a domain ¶
  - depends_on definition 12.70 Graph; closed and closable operators ¶
    - depends_on proposition 12.73 The adjoint is always closed ¶
      - depends_on definition 12.78 Essential self-adjointness ¶
      - depends_on lemma A.260 Cayley transform of a self-adjoint operator ¶
  - depends_on definition 12.72 Symmetric; self-adjoint ¶
    - depends_on definition 12.78 Essential self-adjointness ¶ ↺
    - depends_on lemma A.260 Cayley transform of a self-adjoint operator ¶ ↺
    - depends_on lemma A.267 Isometry of $A\pm\ii\mu$, and closed range ¶ ↺
    - depends_on lemma A.271 Injectivity of $\identity-V$ for any isometric extension ¶ ↺
    - depends_on lemma A.272 The operator attached to an isometry ¶ ↺
    - depends_on proposition A.273 Self-adjoint means unitary ¶ ↺
    - depends_on theorem A.266 von Neumann ¶ ↺
    - depends_on theorem 12.74 Hellinger–Toeplitz ¶
      - depends_on corollary 12.76 Position and momentum are unbounded, and cannot be everywhere defined ¶
    - depends_on theorem 12.80 von Neumann's criterion ¶ ↺
  - depends_on definition 12.71 Adjoint of a densely defined operator ¶
    - depends_on definition 12.79 Deficiency subspaces and indices ¶ ↺
    - depends_on definition 12.72 Symmetric; self-adjoint ¶ ↺
    - depends_on proposition 12.73 The adjoint is always closed ¶ ↺
- depends_on definition 12.103 Gelfand triple ¶
  - depends_on definition A.278 Countably Hilbert nuclear space ¶
    - depends_on theorem A.281 Nuclear spaces embed by Hilbert–Schmidt maps; quoted ¶
      - depends_on proposition A.282 The fibre maps are continuous on $\Phi$ ¶
  - depends_on definition 12.105 Generalized eigenvector ¶
    - depends_on proposition 12.106 The plane wave is a generalized momentum eigenvector ¶
      - depends_on example A.283 Momentum on the line ¶
    - depends_on theorem A.279 Gelfand–Maurin ¶
      - depends_on example A.283 Momentum on the line ¶ ↺
    - depends_on theorem 12.107 Nuclear spectral theorem ¶
  - depends_on example 12.104 The Schwartz triple ¶
    - depends_on example A.283 Momentum on the line ¶ ↺
    - depends_on proposition 12.106 The plane wave is a generalized momentum eigenvector ¶ ↺
  - 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 lemma A.250 Decomposition into cyclic subspaces ¶
      - depends_on proposition A.280 Direct-integral form of the spectral theorem ¶
  - depends_on proposition 12.95 The tensor inner product is well defined and positive definite ¶
    - depends_on example 12.100 Entangled vectors exist ¶
    - depends_on proposition 12.96 Operators on a tensor product ¶
  - depends_on theorem A.229 Hilbert–Schmidt ¶
  - depends_on theorem 12.44 Hilbert–Schmidt: compact self-adjoint operators ¶
    - depends_on theorem A.461 Completeness in the weighted and in the energy norm ¶
    - depends_on theorem A.471 Spectral decomposition and completeness in $L^{2}_{r}$ ¶
      - depends_on lemma A.472 The pairing identity ¶
  - 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 proposition A.280 Direct-integral form of the spectral theorem ¶ ↺
    - depends_on proposition A.261 Spectral theorem for a unitary operator ¶
      - depends_on proposition A.262 Spectral theorem for an unbounded self-adjoint operator ¶
    - depends_on theorem A.279 Gelfand–Maurin ¶ ↺
    - depends_on theorem A.253 Stone ¶
      - depends_on proposition A.263 The two constructions are inverse ¶
  - depends_on theorem 12.59 Spectral theorem for a bounded self-adjoint operator ¶
    - depends_on definition 12.60 Functional calculus ¶
      - depends_on proposition 12.61 Uniqueness of the continuous functional calculus ¶
    - depends_on theorem 12.107 Nuclear spectral theorem ¶ ↺
    - 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 12.66 Stone ¶
      - depends_on corollary 12.112 Commutator of the momentum with a function of the position ¶
      - depends_on definition 12.109 Weyl system ¶
      - depends_on proposition 12.67 The generator is symmetric, and generates the motion ¶
      - depends_on proposition 12.111 The Weyl relation is a covariance statement ¶
      - depends_on theorem 25.34 Stone–von Neumann ¶
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 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 theorem A.579 Stone–von Neumann ¶
    - depends_on theorem 12.91 Schur's lemma, commutant form ¶ ↺
    - 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 lemma A.247 Integration against a projection-valued measure ¶
      - 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 12.64 Strongly continuous one-parameter unitary group ¶
      - depends_on definition A.580 Weyl operator ¶
      - depends_on definition 12.109 Weyl system ¶ ↺
      - depends_on lemma A.255 Smoothed vectors lie in the domain ¶
      - depends_on lemma A.581 Composition law ¶
      - depends_on lemma A.582 Joint strong continuity ¶
      - depends_on proposition A.256 Density ¶
      - depends_on proposition 12.65 Exponential of a bounded self-adjoint operator ¶
      - depends_on proposition 12.67 The generator is symmetric, and generates the motion ¶ ↺
      - depends_on theorem A.253 Stone ¶ ↺
      - depends_on theorem 12.66 Stone ¶ ↺
      - depends_on theorem 25.34 Stone–von Neumann ¶ ↺
    - depends_on lemma A.231 Restriction to an invariant closed subspace ¶
      - depends_on lemma A.233 Construction of the system ¶
    - depends_on lemma A.230 Sequential characterisation ¶
      - depends_on lemma A.232 Attainment ¶
      - 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 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 definition 12.58 Projection-valued measure ¶ ↺
  - depends_on proposition 12.89 Reduction is commutation ¶ ↺
depends_on lemma A.233 Construction of the system ¶ ↺
depends_on lemma A.231 Restriction to an invariant closed subspace ¶ ↺
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 definition 12.103 Gelfand triple ¶ ↺
- depends_on lemma A.247 Integration against a projection-valued measure ¶ ↺
- depends_on proposition A.246 Bounded Borel functional calculus ¶
  - depends_on proposition A.262 Spectral theorem for an unbounded self-adjoint operator ¶ ↺
  - depends_on proposition A.261 Spectral theorem for a unitary operator ¶ ↺
- depends_on proposition A.245 The measures $\mu_{x,y}$ ¶
  - depends_on lemma A.249 The cyclic case ¶ ↺
  - depends_on proposition A.246 Bounded Borel functional calculus ¶ ↺
- depends_on theorem 12.38 Existence and uniqueness of the adjoint ¶
  - depends_on definition 12.41 The operator classes ¶ ↺
  - depends_on definition 12.71 Adjoint of a densely defined operator ¶ ↺
  - depends_on proposition 12.39 Algebra of the adjoint; the $C^{\ast}$ identity ¶
    - depends_on definition 12.79 Deficiency subspaces and indices ¶ ↺
    - depends_on proposition A.241 The polynomial calculus is isometric ¶
      - depends_on proposition A.243 Continuous functional calculus ¶
    - depends_on proposition 12.65 Exponential of a bounded self-adjoint operator ¶ ↺
    - depends_on proposition 12.42 Elementary consequences ¶ ↺
    - depends_on proposition 12.43 Norm of a self-adjoint operator ¶ ↺
    - depends_on proposition 12.96 Operators on a tensor product ¶ ↺

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	The complement is always a closed subspace	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:436
`depends_on`	→	Closest point in a closed convex set	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:436
`depends_on`	←	Double complement; the density criterion	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:478
`depends_on`	←	Cyclic vector and cyclic subspace	declared	appendices/A-long-proofs.tex:12430
`depends_on`	←	Orthogonal projection operator	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:510
`depends_on`	←	Construction of the system	declared	appendices/A-long-proofs.tex:11706
`depends_on`	←	Restriction to an invariant closed subspace	declared	appendices/A-long-proofs.tex:11601
`depends_on`	←	Isometry of $A\pm\ii\mu$, and closed range	declared	appendices/A-long-proofs.tex:13342
`depends_on`	←	Best approximation and Bessel's inequality	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:669
`depends_on`	←	Characterization of orthogonal projections	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:524
`depends_on`	←	Riesz representation	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1218
`proves`	←	ch:10-hilbert-spaces@proof-9	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:439