proposition A.241 The polynomial calculus is isometric

open in the book · appendices/A-long-proofs.tex:12060 · p. 2909

Rests on

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 4.33 Vector space ¶
    - depends_on definition 5.19 Norm ¶
      - depends_on definition 4.33 Vector space ¶ ↺
  - 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 ¶
      - depends_on definition 6.24 Metric ¶
    - 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 definition 6.9 Compact set ¶
      - depends_on proposition 12.21 Characterization of orthogonal projections ¶
      - depends_on theorem 12.38 Existence and uniqueness of the adjoint ¶
    - 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 ¶
- 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 ¶

Supports

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.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`	→	Spectral mapping for polynomials	declared	appendices/A-long-proofs.tex:12068
`depends_on`	→	The norm of a self-adjoint operator lies in its spectrum	declared	appendices/A-long-proofs.tex:12068
`depends_on`	→	Algebra of the adjoint; the $C^{\ast}$ identity	declared	appendices/A-long-proofs.tex:12068
`depends_on`	←	Continuous functional calculus	declared	appendices/A-long-proofs.tex:12117
`proves`	←	app:A-long-proofs@proof-147	declared	appendices/A-long-proofs.tex:12072