proposition A.243 Continuous functional calculus

open in the book · appendices/A-long-proofs.tex:12103 · p. 2909

Rests on

depends_on proposition A.241 The polynomial calculus is isometric ¶
- depends_on lemma A.240 Spectral mapping for polynomials ¶
  - depends_on definition 12.49 Resolvent set; spectrum ¶
    - 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 5.47 Inverse of a linear transformation ¶
      - 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 ¶
      - 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 ¶
  - proves proof app:A-long-proofs@proof-146 ¶
- depends_on lemma A.239 The norm of a self-adjoint operator lies in its spectrum ¶
  - depends_on definition 12.49 Resolvent set; spectrum ¶ ↺
  - 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 12.43 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 ¶
      - 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 ¶
  - proves proof app:A-long-proofs@proof-145 ¶
- depends_on proposition 12.39 Algebra of the adjoint; the $C^{\ast}$ identity ¶ ↺
- proves proof app:A-long-proofs@proof-147 ¶
depends_on proposition 12.61 Uniqueness of the continuous functional calculus ¶
- depends_on definition 12.60 Functional calculus ¶
  - 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 definition 12.20 Orthogonal projection operator ¶
      - depends_on theorem 12.18 Projection theorem ¶
      - proves proof ch:10-hilbert-spaces@proof-11 ¶
  - depends_on theorem 12.59 Spectral theorem for a bounded self-adjoint operator ¶
    - depends_on definition 12.58 Projection-valued measure ¶ ↺
    - depends_on proposition 12.43 Norm of a self-adjoint operator ¶ ↺
    - 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 ¶
- depends_on proposition 12.37 $\mathcal{B}(\mathcal{H})$ is a Banach algebra ¶ ↺
- proves proof ch:10-hilbert-spaces@proof-32 ¶
depends_on theorem A.242 Stone–Weierstrass; quoted ¶
- 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 ¶
proves proof app:A-long-proofs@proof-148 ¶

Supports

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.280 Direct-integral form of the spectral theorem ¶
    - depends_on proposition A.282 The fibre maps are continuous on $\Phi$ ¶
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.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 ¶ ↺

Neighborhood

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Edges

type	direction	node	provenance	where
`depends_on`	→	The polynomial calculus is isometric	declared	appendices/A-long-proofs.tex:12117
`depends_on`	→	Uniqueness of the continuous functional calculus	declared	appendices/A-long-proofs.tex:12117
`depends_on`	→	Stone–Weierstrass; quoted	declared	appendices/A-long-proofs.tex:12117
`depends_on`	←	Cyclic vector and cyclic subspace	declared	appendices/A-long-proofs.tex:12430
`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
`proves`	←	app:A-long-proofs@proof-148	declared	appendices/A-long-proofs.tex:12121