theorem 12.114 Stone–von Neumann

Rests on

• depends_on definition 12.90 Self-adjoint family; commutant; irreducibility ¶
  • 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 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 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 5.37 Linear transformation ¶
      • depends_on proposition 12.37 $\mathcal{B}(\mathcal{H})$ is a Banach algebra ¶
        • depends_on definition 12.35 Bounded operator; operator norm ¶
        • depends_on proposition 12.8 Absolutely convergent series test ¶
        • 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 theorem 12.18 Projection theorem ¶ ↺
        • proves proof ch:10-hilbert-spaces@proof-24 ¶
      • proves proof ch:10-hilbert-spaces@proof-20 ¶
  • depends_on definition 12.88 Reducing subspace ¶
    • depends_on definition 12.35 Bounded operator; operator norm ¶ ↺
    • depends_on definition 12.20 Orthogonal projection operator ¶ ↺
• depends_on definition 12.109 Weyl system ¶
  • depends_on definition 12.64 Strongly continuous one-parameter unitary group ¶
    • depends_on definition 12.41 The operator classes ¶ ↺
    • depends_on definition 6.6 Continuous map ¶
      • depends_on definition 3.53 Preimage ¶
      • depends_on definition 6.2 Open set ¶ ↺
      • depends_on definition 6.1 Topological space ¶
        • depends_on definition 3.31 Empty set ¶
        • depends_on definition 3.35 Union, intersection, difference ¶
        • depends_on definition 3.30 Subset ¶
        • depends_on equation 3.51 eq:set-indexed ¶ ↺
  • depends_on theorem 12.75 The canonical commutation relation admits no bounded solution ¶
    • depends_on definition 12.35 Bounded operator; operator norm ¶ ↺
    • depends_on proposition 12.37 $\mathcal{B}(\mathcal{H})$ is a Banach algebra ¶ ↺
    • proves proof ch:10-hilbert-spaces@proof-37 ¶
  • depends_on theorem 12.66 Stone ¶
    • depends_on definition 12.64 Strongly continuous one-parameter unitary group ¶ ↺
    • depends_on proposition 12.65 Exponential of a bounded self-adjoint operator ¶
      • depends_on definition 12.64 Strongly continuous one-parameter unitary group ¶ ↺
      • depends_on proposition 12.37 $\mathcal{B}(\mathcal{H})$ is a Banach algebra ¶ ↺
      • depends_on proposition 12.39 Algebra of the adjoint; the $C^{\ast}$ identity ¶
        • depends_on proposition 12.37 $\mathcal{B}(\mathcal{H})$ is a Banach algebra ¶ ↺
        • depends_on theorem 12.38 Existence and uniqueness of the adjoint ¶ ↺
        • proves proof ch:10-hilbert-spaces@proof-21 ¶
      • proves proof ch:10-hilbert-spaces@proof-33 ¶
    • depends_on theorem 12.59 Spectral theorem for a bounded self-adjoint operator ¶
      • depends_on definition 12.58 Projection-valued measure ¶
        • depends_on definition 12.41 The operator classes ¶ ↺
        • depends_on proposition 12.21 Characterization of orthogonal projections ¶ ↺
      • depends_on proposition 12.43 Norm of a self-adjoint operator ¶
        • depends_on proposition 12.42 Elementary consequences ¶
        • depends_on proposition 12.39 Algebra of the adjoint; the $C^{\ast}$ identity ¶ ↺
        • proves proof ch:10-hilbert-spaces@proof-23 ¶
      • depends_on theorem 12.55 The spectrum of a self-adjoint operator is real ¶
        • depends_on corollary 12.19 Double complement; the density criterion ¶
        • depends_on definition 12.41 The operator classes ¶ ↺
        • depends_on theorem 12.54 The spectrum is compact and non-empty ¶
        • proves proof ch:10-hilbert-spaces@proof-30 ¶
      • proves proof ch:10-hilbert-spaces@prooflink-2 ¶
    • proves proof ch:10-hilbert-spaces@prooflink-3 ¶
• depends_on theorem 12.91 Schur's lemma, commutant form ¶
  • depends_on definition 12.90 Self-adjoint family; commutant; irreducibility ¶ ↺
  • depends_on proposition 12.89 Reduction is commutation ¶
    • depends_on definition 12.88 Reducing subspace ¶ ↺
    • depends_on proposition 12.21 Characterization of orthogonal projections ¶ ↺
    • proves proof ch:10-hilbert-spaces@proof-43 ¶
  • depends_on theorem 12.59 Spectral theorem for a bounded self-adjoint operator ¶ ↺
  • proves proof ch:10-hilbert-spaces@proof-44 ¶

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