proposition A.259 The generator is self-adjoint

open in the book · appendices/A-long-proofs.tex:12842 · p. 2917

Rests on

depends_on proposition A.258 The generator is closed ¶
- depends_on equation A.470 eq:app-stone-domain ¶
- depends_on lemma A.257 Integrated form of the equation of motion ¶
  - depends_on lemma A.254 Riemann integral of a continuous curve ¶
    - depends_on definition 12.2 Hilbert space ¶
      - depends_on definition 5.18 Inner product ¶
      - depends_on definition 6.27 Convergence; Cauchy sequence; completeness ¶
      - depends_on equation 5.44 eq:lin-norm-assoc ¶
    - depends_on proposition 12.4 Cauchy–Schwarz and continuity of the inner product ¶
      - depends_on definition 12.2 Hilbert space ¶ ↺
      - depends_on proposition 5.20 Cauchy–Schwarz inequality ¶
      - proves proof ch:10-hilbert-spaces@proof-1 ¶
    - proves proof app:A-long-proofs@proof-157 ¶
  - 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 definition 12.41 The operator classes ¶
      - depends_on definition 6.6 Continuous map ¶
    - 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 theorem 12.59 Spectral theorem for a bounded self-adjoint operator ¶
      - proves proof ch:10-hilbert-spaces@prooflink-3 ¶
    - proves proof ch:10-hilbert-spaces@proof-34 ¶
  - proves proof app:A-long-proofs@proof-160 ¶
- proves proof app:A-long-proofs@proof-161 ¶
depends_on proposition A.256 Density ¶
- depends_on definition 12.64 Strongly continuous one-parameter unitary group ¶ ↺
- depends_on lemma A.255 Smoothed vectors lie in the domain ¶
  - depends_on definition 12.64 Strongly continuous one-parameter unitary group ¶ ↺
  - depends_on equation A.470 eq:app-stone-domain ¶ ↺
  - depends_on lemma A.254 Riemann integral of a continuous curve ¶ ↺
  - proves proof app:A-long-proofs@proof-158 ¶
- proves proof app:A-long-proofs@proof-159 ¶
depends_on proposition 12.67 The generator is symmetric, and generates the motion ¶ ↺
proves proof app:A-long-proofs@proof-162 ¶

Supports

depends_on lemma A.260 Cayley transform of a self-adjoint operator ¶
- 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.282 The fibre maps are continuous on $\Phi$ ¶
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`	→	The generator is closed	declared	appendices/A-long-proofs.tex:12845
`depends_on`	→	Density	declared	appendices/A-long-proofs.tex:12845
`depends_on`	→	The generator is symmetric, and generates the motion	declared	appendices/A-long-proofs.tex:12845
`depends_on`	←	Cayley transform of a self-adjoint operator	declared	appendices/A-long-proofs.tex:12927
`depends_on`	←	The two constructions are inverse	declared	appendices/A-long-proofs.tex:13194
`proves`	←	app:A-long-proofs@proof-162	declared	appendices/A-long-proofs.tex:12849