proposition 12.52 Neumann series; the spectrum is bounded

open in the book · parts/02-mathematical-methods/10-hilbert-spaces.tex:1354 · p. 428

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 12.37 $\mathcal{B}(\mathcal{H})$ 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 ¶
      - depends_on definition 3.43 Map ¶
      - depends_on definition 3.38 Ordered pair and Cartesian product ¶
  - proves proof ch:10-hilbert-spaces@proof-4 ¶
- proves proof ch:10-hilbert-spaces@proof-19 ¶
proves proof ch:10-hilbert-spaces@proof-27 ¶

Supports

depends_on lemma A.239 The norm of a self-adjoint operator lies in its spectrum ¶
- 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.246 Bounded Borel functional calculus ¶
      - depends_on proposition A.262 Spectral theorem for an unbounded self-adjoint operator ¶
      - depends_on proposition A.261 Spectral theorem for a unitary 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 ¶ ↺
depends_on proposition A.261 Spectral theorem for a unitary operator ¶ ↺
depends_on proposition 12.53 The resolvent set is open, the resolvent analytic ¶
- depends_on theorem 12.54 The spectrum is compact and non-empty ¶
  - depends_on theorem 12.55 The spectrum of a self-adjoint operator is real ¶
    - depends_on example 12.56 Multiplication by the coordinate: spectrum without eigenvectors ¶
    - depends_on theorem A.238 Spectral theorem, both forms ¶
      - 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 theorem A.279 Gelfand–Maurin ¶
      - depends_on theorem A.253 Stone ¶
    - depends_on theorem 12.59 Spectral theorem for a bounded self-adjoint operator ¶
      - depends_on definition 12.60 Functional calculus ¶
      - depends_on theorem 12.107 Nuclear spectral theorem ¶
      - depends_on theorem 12.91 Schur's lemma, commutant form ¶
      - depends_on theorem 12.66 Stone ¶
depends_on theorem 12.54 The spectrum is compact and non-empty ¶ ↺

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	parts/02-mathematical-methods/10-hilbert-spaces.tex:1364
`depends_on`	→	$\mathcal{B}(\mathcal{H})$ is a Banach algebra	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1364
`depends_on`	←	The norm of a self-adjoint operator lies in its spectrum	declared	appendices/A-long-proofs.tex:11985
`depends_on`	←	Spectral theorem for a unitary operator	declared	appendices/A-long-proofs.tex:12972
`depends_on`	←	The resolvent set is open, the resolvent analytic	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1391
`depends_on`	←	The spectrum is compact and non-empty	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1418
`proves`	←	ch:10-hilbert-spaces@proof-27	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1367