lemma A.603 Overlaps and the resolution of unity

open in the book · appendices/A-long-proofs.tex:28899 · p. 3081

Rests on

depends_on equation A.918 eq:app-hudson-gaussian-coherent ¶
depends_on equation 25.56 eq:pq-wigner-overlap ¶
depends_on theorem 25.53 Properties of the Wigner function ¶
- depends_on equation 25.53 eq:pq-wigner ¶
- depends_on proposition 12.4 Cauchy–Schwarz and continuity of the inner product ¶
  - 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 5.20 Cauchy–Schwarz inequality ¶
    - depends_on definition 5.14 Linear independence ¶
      - depends_on definition 5.5 Linear combination ¶
    - depends_on definition 5.18 Inner product ¶ ↺
    - proves proof ch:03-linear-algebra-representations@proof-4 ¶
  - proves proof ch:10-hilbert-spaces@proof-1 ¶
- depends_on theorem 17.32 Fourier inversion ¶
  - depends_on definition 17.29 Fourier transform; the treatise convention ¶
  - depends_on equation 17.42 eq:ft-gaussian ¶
  - proves proof ch:15-fourier-integral-transforms@proof-20 ¶
- proves proof ch:08-poisson-quantum-bridge@proof-20 ¶
proves proof app:A-long-proofs@proof-362 ¶

Supports

depends_on proposition A.611 The constraint on $\Gamma$ ¶
- depends_on proposition A.612 The dictionary between $A$ and $\Gamma$ ¶
depends_on proposition A.604 Non-negativity forbids a vanishing overlap ¶
- depends_on proposition A.609 The exponent is a quadratic polynomial ¶
  - depends_on proposition A.611 The constraint on $\Gamma$ ¶ ↺

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-hudson-gaussian-coherent	declared	appendices/A-long-proofs.tex:28914
`depends_on`	→	eq:pq-wigner-overlap	declared	appendices/A-long-proofs.tex:28914
`depends_on`	→	Properties of the Wigner function	declared	appendices/A-long-proofs.tex:28914
`depends_on`	←	The constraint on $\Gamma$	declared	appendices/A-long-proofs.tex:29285
`depends_on`	←	Non-negativity forbids a vanishing overlap	declared	appendices/A-long-proofs.tex:28961
`proves`	←	app:A-long-proofs@proof-362	declared	appendices/A-long-proofs.tex:28918