theorem 12.46 Riesz representation

open in the book · parts/02-mathematical-methods/10-hilbert-spaces.tex:1211 · p. 426

Rests on

depends_on definition 12.45 Continuous linear functional; the dual ¶
- depends_on definition 12.35 Bounded operator; operator norm ¶
  - depends_on definition 5.37 Linear transformation ¶
    - depends_on definition 4.33 Vector space ¶
      - depends_on definition 4.8 Commutativity; abelian structure ¶
      - depends_on definition 4.32 Field ¶
      - depends_on definition 4.31 Module ¶
  - depends_on definition 5.19 Norm ¶
    - depends_on definition 4.33 Vector space ¶ ↺
- depends_on definition 5.46 Functional ¶
  - depends_on definition 5.37 Linear transformation ¶ ↺
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 ¶
proves proof ch:10-hilbert-spaces@proof-24 ¶

Supports

depends_on corollary 12.47 $\mathcal{H}$ is its own dual, antilinearly ¶
- 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 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.280 Direct-integral form of the spectral theorem ¶
    - depends_on proposition A.282 The fibre maps are continuous on $\Phi$ ¶ ↺
- 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 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 12.90 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 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 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 12.71 Adjoint of a densely defined operator ¶
  - 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 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 12.72 Symmetric; self-adjoint ¶
    - depends_on definition 12.78 Essential self-adjointness ¶
      - depends_on theorem A.266 von Neumann ¶ ↺
      - depends_on theorem 12.80 von Neumann's criterion ¶ ↺
    - depends_on lemma A.260 Cayley transform of a self-adjoint operator ¶
      - depends_on proposition A.262 Spectral theorem for an unbounded 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 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 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 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 definition A.248 Cyclic vector and cyclic subspace ¶
      - depends_on proposition A.246 Bounded Borel functional calculus ¶ ↺
      - depends_on proposition A.245 The measures $\mu_{x,y}$ ¶ ↺
  - 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

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Edges

type	direction	node	provenance	where
`depends_on`	→	Continuous linear functional; the dual	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1218
`depends_on`	→	Projection theorem	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1218
`depends_on`	←	$\mathcal{H}$ is its own dual, antilinearly	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1253
`depends_on`	←	Integration against a projection-valued measure	declared	appendices/A-long-proofs.tex:12373
`depends_on`	←	Bounded Borel functional calculus	declared	appendices/A-long-proofs.tex:12237
`depends_on`	←	The measures $\mu_{x,y}$	declared	appendices/A-long-proofs.tex:12191
`depends_on`	←	Existence and uniqueness of the adjoint	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:959
`proves`	←	ch:10-hilbert-spaces@proof-24	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1221