theorem A.279 Gelfand–Maurin

open in the book · appendices/A-long-proofs.tex:13906 · p. 2928

Rests on

depends_on definition 12.103 Gelfand triple ¶
- depends_on corollary 12.47 $\mathcal{H}$ is its own dual, antilinearly ¶
  - depends_on definition 12.45 Continuous linear functional; the dual ¶
    - 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 5.46 Functional ¶
      - depends_on definition 5.37 Linear transformation ¶ ↺
  - depends_on theorem 12.46 Riesz representation ¶
    - depends_on definition 12.45 Continuous linear functional; the dual ¶ ↺
    - depends_on theorem 12.18 Projection theorem ¶
      - depends_on proposition 12.17 The complement is always a closed subspace ¶
      - depends_on theorem 12.14 Closest point in a closed convex set ¶
      - proves proof ch:10-hilbert-spaces@proof-9 ¶
    - proves proof ch:10-hilbert-spaces@proof-24 ¶
  - proves proof ch:10-hilbert-spaces@proof-25 ¶
- depends_on corollary 12.19 Double complement; the density criterion ¶
  - depends_on proposition 12.17 The complement is always a closed subspace ¶ ↺
  - depends_on theorem 12.18 Projection theorem ¶ ↺
  - proves proof ch:10-hilbert-spaces@proof-10 ¶
- depends_on definition 12.45 Continuous linear functional; the dual ¶ ↺
depends_on definition 12.105 Generalized eigenvector ¶
- depends_on definition 12.103 Gelfand triple ¶ ↺
- depends_on definition 12.50 Point, continuous and residual spectrum ¶
  - depends_on corollary 12.19 Double complement; the density criterion ¶ ↺
  - 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 definition 5.37 Linear transformation ¶ ↺
depends_on theorem A.238 Spectral theorem, both forms ¶
- 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 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 12.21 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 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 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 theorem 8.17 Derivatives of all orders; Cauchy estimates ¶
      - 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 ¶
proves proof app:A-long-proofs@proof-179 ¶

Supports

depends_on example A.283 Momentum on the line ¶

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`	→	Gelfand triple	declared	appendices/A-long-proofs.tex:13939
`depends_on`	→	Generalized eigenvector	declared	appendices/A-long-proofs.tex:13939
`depends_on`	→	Spectral theorem, both forms	declared	appendices/A-long-proofs.tex:13939
`depends_on`	←	Momentum on the line	declared	appendices/A-long-proofs.tex:14226
`proves`	←	app:A-long-proofs@proof-179	declared	appendices/A-long-proofs.tex:14130