definition 12.58 Projection-valued measure

open in the book · parts/02-mathematical-methods/10-hilbert-spaces.tex:1568 · p. 430

ground object -- no derivation owed

Rests on

depends_on definition 12.41 The operator classes ¶
- depends_on definition 6.9 Compact set ¶
  - depends_on definition 6.5 Open cover ¶
    - depends_on definition 6.2 Open set ¶
      - depends_on definition 6.1 Topological space ¶
    - depends_on equation 3.51 eq:set-indexed ¶
- depends_on proposition 12.21 Characterization of orthogonal projections ¶
  - depends_on definition 12.20 Orthogonal projection operator ¶
    - depends_on theorem 12.18 Projection theorem ¶
      - depends_on proposition 12.17 The complement is always a closed subspace ¶
      - depends_on theorem 12.14 Closest point in a closed convex set ¶
      - proves proof ch:10-hilbert-spaces@proof-9 ¶
  - depends_on theorem 12.18 Projection theorem ¶ ↺
  - proves proof ch:10-hilbert-spaces@proof-11 ¶
- depends_on theorem 12.38 Existence and uniqueness of the adjoint ¶
  - depends_on definition 5.41 Adjoint ¶
    - depends_on definition 5.18 Inner product ¶
      - depends_on definition 4.33 Vector space ¶
    - depends_on definition 5.37 Linear transformation ¶
      - depends_on definition 4.33 Vector space ¶ ↺
  - depends_on proposition 12.37 $\mathcal{B}(\mathcal{H})$ is a Banach algebra ¶
    - depends_on definition 12.35 Bounded operator; operator norm ¶
      - depends_on definition 5.37 Linear transformation ¶ ↺
      - depends_on definition 5.19 Norm ¶
    - depends_on proposition 12.8 Absolutely convergent series test ¶
      - depends_on definition 5.19 Norm ¶ ↺
      - depends_on definition 6.27 Convergence; Cauchy sequence; completeness ¶
      - proves proof ch:10-hilbert-spaces@proof-4 ¶
    - proves proof ch:10-hilbert-spaces@proof-19 ¶
  - depends_on theorem 12.46 Riesz representation ¶
    - depends_on definition 12.45 Continuous linear functional; the dual ¶
      - depends_on definition 12.35 Bounded operator; operator norm ¶ ↺
      - depends_on definition 5.46 Functional ¶
    - depends_on theorem 12.18 Projection theorem ¶ ↺
    - proves proof ch:10-hilbert-spaces@proof-24 ¶
  - proves proof ch:10-hilbert-spaces@proof-20 ¶
depends_on proposition 12.21 Characterization of orthogonal projections ¶ ↺

Supports

depends_on definition 12.60 Functional calculus ¶
- depends_on proposition 12.61 Uniqueness of the continuous functional calculus ¶
  - depends_on proposition A.243 Continuous functional calculus ¶
    - 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 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 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.282 The fibre maps are continuous on $\Phi$ ¶
- depends_on proposition A.261 Spectral theorem for a unitary operator ¶ ↺
- depends_on theorem A.279 Gelfand–Maurin ¶
  - depends_on example A.283 Momentum on the line ¶
- 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 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 corollary 12.112 Commutator of the momentum with a function of the position ¶ ↺
    - 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 example 12.110 The Schrödinger system ¶
      - depends_on proposition A.592 The vacuum of the Schrödinger system ¶
      - depends_on theorem A.579 Stone–von Neumann ¶
    - depends_on proposition 12.111 The Weyl relation is a covariance statement ¶
      - depends_on corollary 12.112 Commutator of the momentum with a function of the position ¶ ↺
    - depends_on theorem A.579 Stone–von Neumann ¶ ↺
    - depends_on theorem 12.114 Stone–von Neumann ¶ ↺
  - 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.258 The generator is closed ¶
    - depends_on proposition A.263 The two constructions are inverse ¶ ↺
    - depends_on proposition A.259 The generator is self-adjoint ¶
      - depends_on lemma A.260 Cayley transform of a self-adjoint operator ¶
      - depends_on proposition A.263 The two constructions are inverse ¶ ↺
    - depends_on proposition 12.111 The Weyl relation is a covariance statement ¶ ↺
    - depends_on theorem A.253 Stone ¶ ↺
  - depends_on proposition 12.111 The Weyl relation is a covariance statement ¶ ↺
  - depends_on theorem 25.34 Stone–von Neumann ¶
    - depends_on lemma 25.39 The low-degree images are forced ¶
      - depends_on example 25.45 Weyl ordering collapses ¶
    - depends_on theorem 25.38 Groenewold–van Hove ¶
      - depends_on corollary 25.41 Quantization is not a functor ¶
      - depends_on remark 24.57 The half-integer, and the honest status of the construction ¶
      - depends_on remark 24.55 What polarization costs ¶

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 operator classes	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1588
`depends_on`	→	Characterization of orthogonal projections	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1588
`depends_on`	←	Functional calculus	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1658
`depends_on`	←	Integration against a projection-valued measure	declared	appendices/A-long-proofs.tex:12373
`depends_on`	←	Spectral theorem, both forms	declared	appendices/A-long-proofs.tex:11966
`depends_on`	←	Spectral theorem for a bounded self-adjoint operator	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1620