proposition A.611 The constraint on $\Gamma$

open in the book · appendices/A-long-proofs.tex:29266 · p. 3084

Rests on

depends_on lemma A.610 Block positivity ¶
- depends_on theorem 5.40 Rank–nullity ¶
  - depends_on definition 5.15 Basis ¶
    - depends_on definition 5.12 Subspace generated by a set of vectors ¶
      - depends_on definition 5.5 Linear combination ¶
      - depends_on definition 5.7 Vector subspace ¶
      - proves proof ch:03-linear-algebra-representations@proof-3 ¶
    - depends_on definition 5.14 Linear independence ¶
      - depends_on definition 5.5 Linear combination ¶ ↺
  - depends_on definition 5.39 Kernel, image, nullity, rank ¶
    - depends_on definition 5.37 Linear transformation ¶
      - depends_on definition 4.33 Vector space ¶
    - depends_on definition 5.7 Vector subspace ¶ ↺
    - proves proof ch:03-linear-algebra-representations@proof-12 ¶
  - depends_on lemma 5.38 Exchange and completion ¶
    - depends_on definition 5.15 Basis ¶ ↺
    - depends_on definition 5.12 Subspace generated by a set of vectors ¶ ↺
    - depends_on definition 5.14 Linear independence ¶ ↺
    - proves proof ch:03-linear-algebra-representations@proof-11 ¶
  - proves proof ch:03-linear-algebra-representations@proof-13 ¶
- depends_on theorem 5.84 Spectral theorem for a real symmetric operator ¶
  - depends_on proposition 5.28 Gram–Schmidt ¶
    - depends_on definition 5.14 Linear independence ¶ ↺
    - depends_on definition 5.18 Inner product ¶
      - depends_on definition 4.33 Vector space ¶ ↺
    - depends_on definition 5.27 Orthonormal basis ¶
      - depends_on definition 5.26 Orthogonal basis ¶
      - depends_on definition 5.25 Unit vector ¶
    - depends_on equation 5.44 eq:lin-norm-assoc ¶
    - proves proof ch:03-linear-algebra-representations@proof-5 ¶
  - depends_on theorem 5.71 The eigenvalues are the roots of the characteristic polynomial ¶
    - depends_on definition 5.70 Characteristic polynomial ¶
      - depends_on definition 5.56 Endomorphism ¶
      - depends_on equation 5.19 eq:lin-leibniz-det ¶
      - depends_on equation 5.74 eq:lin-matrix-rep ¶
      - proves proof ch:03-linear-algebra-representations@proof-25 ¶
    - depends_on definition 5.69 Eigenvector, eigenvalue, eigenspace ¶
      - depends_on definition 5.56 Endomorphism ¶ ↺
      - depends_on definition 5.39 Kernel, image, nullity, rank ¶ ↺
    - depends_on proposition 5.49 Injective, surjective, invertible ¶
      - depends_on definition 5.39 Kernel, image, nullity, rank ¶ ↺
      - depends_on proposition 5.48 Existence, uniqueness, and linearity of the inverse ¶
      - depends_on theorem 5.40 Rank–nullity ¶ ↺
      - proves proof ch:03-linear-algebra-representations@proof-18 ¶
    - proves proof ch:03-linear-algebra-representations@proof-26 ¶
  - depends_on theorem 5.79 Spectral theorem for a self-adjoint operator ¶
    - depends_on corollary 5.72 Existence of an eigenvalue over $\C$ ¶
      - depends_on theorem 8.19 Fundamental theorem of algebra ¶
      - depends_on theorem 5.71 The eigenvalues are the roots of the characteristic polynomial ¶ ↺
      - proves proof ch:03-linear-algebra-representations@proof-27 ¶
    - depends_on definition 5.41 Adjoint ¶
      - depends_on definition 5.18 Inner product ¶ ↺
      - depends_on definition 5.37 Linear transformation ¶ ↺
    - depends_on theorem 5.40 Rank–nullity ¶ ↺
    - proves proof ch:03-linear-algebra-representations@proof-31 ¶
  - proves proof ch:03-linear-algebra-representations@proof-34 ¶
- proves proof app:A-long-proofs@proof-368 ¶
depends_on lemma A.603 Overlaps and the resolution of unity ¶
- 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 6.27 Convergence; Cauchy sequence; completeness ¶
      - 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 5.18 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 ¶
depends_on proposition A.609 The exponent is a quadratic polynomial ¶
- depends_on lemma A.607 A zero-free entire function is an exponential ¶
  - depends_on lemma A.606 $F$ is entire, and is the overlap in disguise ¶
    - depends_on definition A.605 The analytic continuation of the overlaps ¶
      - depends_on definition 25.52 Wigner function ¶
      - depends_on equation A.918 eq:app-hudson-gaussian-coherent ¶ ↺
    - depends_on proposition 12.4 Cauchy–Schwarz and continuity of the inner product ¶ ↺
    - depends_on theorem 8.6 Cauchy–Riemann equations ¶
      - depends_on definition 7.98 Functions of class $C^{1}$ ¶
      - depends_on equation 8.4 eq:cpx-derivative ¶
      - proves proof ch:06-complex-analysis@proof-3 ¶
    - proves proof app:A-long-proofs@proof-364 ¶
  - depends_on theorem 7.109 Leibniz integral rule ¶
    - depends_on definition 7.125 Multiple integral ¶
      - depends_on axiom 7.1 Completeness of $\R$ ¶
      - depends_on definition 7.39 Darboux sums and the definite integral ¶
    - depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶
      - depends_on proposition 7.22 Sequential characterization ¶
      - depends_on theorem 7.7 Bolzano–Weierstrass ¶
      - proves proof ch:05-real-analysis@proof-10 ¶
    - depends_on theorem 7.35 Mean value theorem ¶
      - depends_on proposition 7.29 Linearity ¶
      - depends_on theorem 7.34 Rolle ¶
      - proves proof ch:05-real-analysis@proof-19 ¶
    - proves proof ch:05-real-analysis@proof-67 ¶
  - depends_on theorem 8.6 Cauchy–Riemann equations ¶ ↺
  - proves proof app:A-long-proofs@proof-365 ¶
- depends_on lemma A.608 An entire function with a quadratic bound on its real part ¶
  - depends_on equation 8.13 eq:cpx-derivative-formula ¶
  - depends_on theorem 8.18 Liouville ¶
    - depends_on proposition 8.10 Fundamental theorem for contours ¶
      - depends_on equation 8.6 eq:cpx-contour-integral ¶
      - depends_on theorem 7.43 Fundamental theorem of calculus, II ¶
      - proves proof ch:06-complex-analysis@proof-5 ¶
    - depends_on theorem 8.17 Derivatives of all orders; Cauchy estimates ¶
      - depends_on lemma 8.9 ML estimate ¶
      - depends_on theorem 8.16 Cauchy integral formula ¶
      - proves proof ch:06-complex-analysis@proof-11 ¶
    - proves proof ch:06-complex-analysis@proof-12 ¶
  - depends_on theorem 8.20 Taylor expansion ¶
    - depends_on equation 8.12 eq:cpx-cif ¶
    - depends_on equation 8.13 eq:cpx-derivative-formula ¶ ↺
    - depends_on lemma 8.9 ML estimate ¶ ↺
    - depends_on proposition 7.46 Geometric series ¶
      - depends_on corollary A.46 Monotone convergence ¶
      - depends_on definition 7.45 Series ¶
      - proves proof ch:05-real-analysis@proof-27 ¶
    - proves proof ch:06-complex-analysis@proof-14 ¶
  - proves proof app:A-long-proofs@proof-366 ¶
- depends_on proposition A.604 Non-negativity forbids a vanishing overlap ¶
  - depends_on equation 25.54 eq:pq-wigner-marginal-x ¶
  - depends_on lemma A.603 Overlaps and the resolution of unity ¶ ↺
  - depends_on proposition A.602 The Wigner function of a Gaussian is a positive Gaussian ¶
    - depends_on equation A.910 eq:app-hudson-gaussian-form ¶
    - depends_on equation 25.53 eq:pq-wigner ¶ ↺
    - depends_on lemma A.601 Several variables, complex symmetric matrix ¶
      - depends_on lemma A.599 One variable, complex coefficient ¶
      - depends_on lemma A.600 The real part of an inverse ¶
      - depends_on theorem 5.84 Spectral theorem for a real symmetric operator ¶ ↺
      - proves proof app:A-long-proofs@proof-360 ¶
    - proves proof app:A-long-proofs@proof-361 ¶
  - proves proof app:A-long-proofs@proof-363 ¶
- proves proof app:A-long-proofs@proof-367 ¶
proves proof app:A-long-proofs@proof-369 ¶

Supports

depends_on proposition A.612 The dictionary between $A$ and $\Gamma$ ¶

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`	→	Block positivity	declared	appendices/A-long-proofs.tex:29285
`depends_on`	→	Overlaps and the resolution of unity	declared	appendices/A-long-proofs.tex:29285
`depends_on`	→	The exponent is a quadratic polynomial	declared	appendices/A-long-proofs.tex:29285
`depends_on`	←	The dictionary between $A$ and $\Gamma$	declared	appendices/A-long-proofs.tex:29361
`proves`	←	app:A-long-proofs@proof-369	declared	appendices/A-long-proofs.tex:29289