theorem 12.107 Nuclear spectral theorem

Rests on

• depends_on definition 12.103 Gelfand triple ¶
  • depends_on corollary 12.47 $\mathcal{H}$ is its own dual, antilinearly ¶
    • 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 5.19 Norm ¶
      • depends_on definition 5.46 Functional ¶
        • depends_on definition 5.37 Linear transformation ¶ ↺
    • depends_on theorem 12.46 Riesz representation ¶
      • depends_on definition 12.45 Continuous linear functional; the dual ¶ ↺
      • 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 ¶
      • proves proof ch:10-hilbert-spaces@proof-24 ¶
    • proves proof ch:10-hilbert-spaces@proof-25 ¶
  • depends_on corollary 12.19 Double complement; the density criterion ¶
    • depends_on proposition 12.17 The complement is always a closed subspace ¶ ↺
    • depends_on theorem 12.18 Projection theorem ¶ ↺
    • proves proof ch:10-hilbert-spaces@proof-10 ¶
  • depends_on definition 12.45 Continuous linear functional; the dual ¶ ↺
• depends_on definition 12.105 Generalized eigenvector ¶
  • depends_on definition 12.103 Gelfand triple ¶ ↺
  • depends_on definition 12.50 Point, continuous and residual spectrum ¶
    • depends_on corollary 12.19 Double complement; the density criterion ¶ ↺
    • depends_on definition 12.49 Resolvent set; spectrum ¶
      • depends_on definition 12.35 Bounded operator; operator norm ¶ ↺
      • depends_on definition 5.47 Inverse of a linear transformation ¶
        • depends_on definition 5.37 Linear transformation ¶ ↺
• 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 definition 6.9 Compact set ¶
        • depends_on definition 6.5 Open cover ¶
      • depends_on proposition 12.21 Characterization of orthogonal projections ¶
        • depends_on definition 12.20 Orthogonal projection operator ¶
        • 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 proposition 12.37 $\mathcal{B}(\mathcal{H})$ is a Banach algebra ¶
        • depends_on theorem 12.46 Riesz representation ¶ ↺
        • proves proof ch:10-hilbert-spaces@proof-20 ¶
    • 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 definition 12.41 The operator classes ¶ ↺
      • 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-22 ¶
    • 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 ¶
      • depends_on proposition 12.52 Neumann series; the spectrum is bounded ¶
        • depends_on definition 12.49 Resolvent set; spectrum ¶ ↺
        • depends_on proposition 12.37 $\mathcal{B}(\mathcal{H})$ is a Banach algebra ¶ ↺
        • proves proof ch:10-hilbert-spaces@proof-27 ¶
      • depends_on proposition 12.53 The resolvent set is open, the resolvent analytic ¶
        • depends_on definition 8.5 Complex derivative ¶
        • depends_on proposition 12.52 Neumann series; the spectrum is bounded ¶ ↺
        • proves proof ch:10-hilbert-spaces@proof-28 ¶
      • depends_on theorem 8.18 Liouville ¶
        • depends_on proposition 8.10 Fundamental theorem for contours ¶
        • depends_on theorem 8.17 Derivatives of all orders; Cauchy estimates ¶
        • proves proof ch:06-complex-analysis@proof-12 ¶
      • proves proof ch:10-hilbert-spaces@proof-29 ¶
    • proves proof ch:10-hilbert-spaces@proof-30 ¶
  • proves proof ch:10-hilbert-spaces@prooflink-2 ¶
• proves proof ch:10-hilbert-spaces@prooflink-5 ¶

Supports

Nothing declares a dependency on this node yet.