theorem A.238 Spectral theorem, both forms

open in the book · appendices/A-long-proofs.tex:11944 · p. 2908

Rests on

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 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 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 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 proposition 12.4 Cauchy–Schwarz and continuity of the inner product ¶
      - 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 12.18 Projection theorem ¶ ↺
  - proves proof ch:10-hilbert-spaces@proof-10 ¶
- 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 definition 12.35 Bounded operator; operator norm ¶ ↺
      - depends_on definition 5.47 Inverse of a linear transformation ¶
    - 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 equation 8.6 eq:cpx-contour-integral ¶
      - depends_on theorem 7.43 Fundamental theorem of calculus, II ¶
      - proves proof ch:06-complex-analysis@proof-5 ¶
    - depends_on theorem 8.17 Derivatives of all orders; Cauchy estimates ¶
      - depends_on lemma 8.9 ML estimate ¶
      - depends_on theorem 8.16 Cauchy integral formula ¶
      - proves proof ch:06-complex-analysis@proof-11 ¶
    - 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 app:A-long-proofs@proof-151 ¶
proves proof app:A-long-proofs@proof-153 ¶
proves proof app:A-long-proofs@proof-156 ¶

Supports

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.280 Direct-integral form of the spectral theorem ¶ ↺
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 ¶

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`	→	Projection-valued measure	declared	appendices/A-long-proofs.tex:11966
`depends_on`	→	Norm of a self-adjoint operator	declared	appendices/A-long-proofs.tex:11966
`depends_on`	→	The spectrum of a self-adjoint operator is real	declared	appendices/A-long-proofs.tex:11966
`depends_on`	←	Direct-integral form of the spectral theorem	declared	appendices/A-long-proofs.tex:13962
`depends_on`	←	Spectral theorem for a unitary operator	declared	appendices/A-long-proofs.tex:12972
`depends_on`	←	Gelfand–Maurin	declared	appendices/A-long-proofs.tex:13939
`depends_on`	←	Stone	declared	appendices/A-long-proofs.tex:12645
`proves`	←	app:A-long-proofs@proof-151	declared	appendices/A-long-proofs.tex:12311
`proves`	←	app:A-long-proofs@proof-153	declared	appendices/A-long-proofs.tex:12400
`proves`	←	app:A-long-proofs@proof-156	declared	appendices/A-long-proofs.tex:12509