proposition A.258 The generator is closed

open in the book · appendices/A-long-proofs.tex:12814 · p. 2917

Rests on

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 4.33 Vector space ¶
    - depends_on definition 6.27 Convergence; Cauchy sequence; completeness ¶
      - depends_on definition 6.24 Metric ¶
    - 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 ¶
      - depends_on definition 5.14 Linear independence ¶
      - depends_on definition 5.18 Inner product ¶ ↺
      - proves proof ch:03-linear-algebra-representations@proof-4 ¶
    - 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 6.9 Compact set ¶
      - depends_on proposition 12.21 Characterization of orthogonal projections ¶
      - depends_on theorem 12.38 Existence and uniqueness of the adjoint ¶
    - 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 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 ¶
      - 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 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 ch:10-hilbert-spaces@prooflink-2 ¶
    - 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 ¶

Supports

depends_on proposition A.259 The generator is self-adjoint ¶
- 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`	→	eq:app-stone-domain	declared	appendices/A-long-proofs.tex:12816
`depends_on`	→	Integrated form of the equation of motion	declared	appendices/A-long-proofs.tex:12816
`depends_on`	←	The generator is self-adjoint	declared	appendices/A-long-proofs.tex:12845
`proves`	←	app:A-long-proofs@proof-161	declared	appendices/A-long-proofs.tex:12819