theorem A.253 Stone

open in the book · appendices/A-long-proofs.tex:12632 · p. 2915

Rests on

depends_on definition 12.64 Strongly continuous one-parameter unitary group ¶
- depends_on definition 12.41 The operator classes ¶
  - depends_on definition 6.9 Compact set ¶
    - depends_on definition 6.5 Open cover ¶
      - depends_on definition 6.2 Open set ¶
      - depends_on equation 3.51 eq:set-indexed ¶
  - depends_on proposition 12.21 Characterization of orthogonal projections ¶
    - depends_on definition 12.20 Orthogonal projection operator ¶
      - depends_on theorem 12.18 Projection theorem ¶
    - 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 definition 5.18 Inner product ¶
      - 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 ¶
      - proves proof ch:10-hilbert-spaces@proof-19 ¶
    - depends_on theorem 12.46 Riesz representation ¶
      - depends_on definition 12.45 Continuous linear functional; the dual ¶
      - depends_on theorem 12.18 Projection theorem ¶ ↺
      - proves proof ch:10-hilbert-spaces@proof-24 ¶
    - proves proof ch:10-hilbert-spaces@proof-20 ¶
- depends_on definition 6.6 Continuous map ¶
  - depends_on definition 3.53 Preimage ¶
  - depends_on definition 6.2 Open set ¶ ↺
  - depends_on definition 6.1 Topological space ¶
    - depends_on definition 3.31 Empty set ¶
    - depends_on definition 3.35 Union, intersection, difference ¶
      - depends_on definition 3.8 Conjunction ¶
      - depends_on definition 3.9 Disjunction ¶
      - depends_on definition 3.7 Negation ¶
    - depends_on definition 3.30 Subset ¶
    - depends_on equation 3.51 eq:set-indexed ¶ ↺
depends_on proposition 12.67 The generator is symmetric, and generates the motion ¶
- depends_on definition 12.64 Strongly continuous one-parameter unitary group ¶ ↺
- depends_on theorem 12.66 Stone ¶
  - depends_on definition 12.64 Strongly continuous one-parameter unitary group ¶ ↺
  - depends_on proposition 12.65 Exponential of a bounded self-adjoint operator ¶
    - depends_on definition 12.64 Strongly continuous one-parameter unitary group ¶ ↺
    - depends_on proposition 12.37 $\mathcal{B}(\mathcal{H})$ is a Banach algebra ¶ ↺
    - 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-33 ¶
  - depends_on theorem 12.59 Spectral theorem for a bounded self-adjoint operator ¶
    - 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 proposition 12.43 Norm of a self-adjoint operator ¶
      - depends_on proposition 12.42 Elementary consequences ¶
      - 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 ¶
      - proves proof ch:10-hilbert-spaces@proof-30 ¶
    - proves proof ch:10-hilbert-spaces@prooflink-2 ¶
  - proves proof ch:10-hilbert-spaces@prooflink-3 ¶
- proves proof ch:10-hilbert-spaces@proof-34 ¶
depends_on theorem A.238 Spectral theorem, both forms ¶
- 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 ¶ ↺
- 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-166 ¶

Supports

depends_on proposition A.263 The two constructions are inverse ¶

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`	→	Strongly continuous one-parameter unitary group	declared	appendices/A-long-proofs.tex:12645
`depends_on`	→	The generator is symmetric, and generates the motion	declared	appendices/A-long-proofs.tex:12645
`depends_on`	→	Spectral theorem, both forms	declared	appendices/A-long-proofs.tex:12645
`depends_on`	←	The two constructions are inverse	declared	appendices/A-long-proofs.tex:13194
`proves`	←	app:A-long-proofs@proof-166	declared	appendices/A-long-proofs.tex:13134