lemma A.254 Riemann integral of a continuous curve

open in the book · appendices/A-long-proofs.tex:12655 · p. 2915

Rests on

depends_on definition 12.2 Hilbert space ¶
- depends_on definition 5.18 Inner product ¶
  - depends_on definition 4.33 Vector space ¶
    - depends_on definition 4.8 Commutativity; abelian structure ¶
      - depends_on definition 4.4 Internal binary operation; magma ¶
    - depends_on definition 4.32 Field ¶
      - depends_on definition 4.11 Inverse element ¶
      - depends_on definition 4.20 Monoid ¶
      - depends_on definition 4.30 Ring ¶
    - depends_on definition 4.31 Module ¶
      - depends_on definition 4.29 Action ¶
      - depends_on definition 4.28 Distributivity ¶
      - depends_on definition 4.21 Group ¶
      - depends_on definition 4.30 Ring ¶ ↺
- depends_on definition 6.27 Convergence; Cauchy sequence; completeness ¶
  - depends_on definition 6.24 Metric ¶
    - depends_on definition 3.43 Map ¶
      - depends_on definition 3.24 Quantifiers ¶
      - depends_on definition 3.28 Set ¶
    - depends_on definition 3.38 Ordered pair and Cartesian product ¶
- 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 5.5 Linear combination ¶
      - depends_on definition 4.33 Vector space ¶ ↺
  - 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 ¶

Supports

depends_on lemma A.255 Smoothed vectors lie in the domain ¶
- depends_on proposition A.256 Density ¶
  - 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.263 The two constructions are inverse ¶
depends_on lemma A.257 Integrated form of the equation of motion ¶
- depends_on proposition A.258 The generator is closed ¶
  - depends_on proposition A.259 The generator is self-adjoint ¶ ↺
depends_on lemma A.583 Absolutely convergent operator-valued integrals ¶
- depends_on definition A.585 The Gaussian average of the Weyl operators ¶
  - depends_on theorem A.587 The Gaussian average is a non-zero projector that absorbs the Weyl operators ¶
    - depends_on lemma A.590 The Gram matrix is universal ¶
      - depends_on proposition A.591 Any two irreducible Weyl systems are equivalent ¶
    - depends_on proposition A.589 Cyclic subspaces and the rank of the average ¶
      - depends_on lemma A.590 The Gram matrix is universal ¶ ↺
      - depends_on proposition A.591 Any two irreducible Weyl systems are equivalent ¶ ↺
    - depends_on proposition A.592 The vacuum of the Schrödinger system ¶

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`	→	Hilbert space	declared	appendices/A-long-proofs.tex:12667
`depends_on`	→	Cauchy–Schwarz and continuity of the inner product	declared	appendices/A-long-proofs.tex:12667
`depends_on`	←	Smoothed vectors lie in the domain	declared	appendices/A-long-proofs.tex:12704
`depends_on`	←	Integrated form of the equation of motion	declared	appendices/A-long-proofs.tex:12794
`depends_on`	←	Absolutely convergent operator-valued integrals	declared	appendices/A-long-proofs.tex:27912
`proves`	←	app:A-long-proofs@proof-157	declared	appendices/A-long-proofs.tex:12670