lemma A.239 The norm of a self-adjoint operator lies in its spectrum

open in the book · appendices/A-long-proofs.tex:11978 · p. 2908

Rests on

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 4.8 Commutativity; abelian structure ¶
      - depends_on definition 4.32 Field ¶
      - depends_on definition 4.31 Module ¶
  - 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 12.52 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 ¶
  - 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 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 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 12.39 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 ¶

Supports

depends_on proposition A.241 The polynomial calculus is isometric ¶
- 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.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

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`	→	Resolvent set; spectrum	declared	appendices/A-long-proofs.tex:11985
`depends_on`	→	Neumann series; the spectrum is bounded	declared	appendices/A-long-proofs.tex:11985
`depends_on`	→	Norm of a self-adjoint operator	declared	appendices/A-long-proofs.tex:11985
`depends_on`	←	The polynomial calculus is isometric	declared	appendices/A-long-proofs.tex:12068
`proves`	←	app:A-long-proofs@proof-145	declared	appendices/A-long-proofs.tex:11989