proposition A.589 Cyclic subspaces and the rank of the average

open in the book · appendices/A-long-proofs.tex:28223 · p. 3074

Rests on

depends_on definition 12.90 Self-adjoint family; commutant; irreducibility ¶
- depends_on definition 12.41 The operator classes ¶
  - depends_on definition 6.9 Compact set ¶
    - depends_on definition 6.5 Open cover ¶
      - depends_on definition 6.2 Open set ¶
      - depends_on equation 3.51 eq:set-indexed ¶
  - depends_on proposition 12.21 Characterization of orthogonal projections ¶
    - depends_on definition 12.20 Orthogonal projection operator ¶
      - depends_on theorem 12.18 Projection theorem ¶
    - depends_on theorem 12.18 Projection theorem ¶ ↺
    - proves proof ch:10-hilbert-spaces@proof-11 ¶
  - depends_on theorem 12.38 Existence and uniqueness of the adjoint ¶
    - depends_on definition 5.41 Adjoint ¶
      - depends_on definition 5.18 Inner product ¶
      - depends_on definition 5.37 Linear transformation ¶
    - depends_on proposition 12.37 $\mathcal{B}(\mathcal{H})$ is a Banach algebra ¶
      - depends_on definition 12.35 Bounded operator; operator norm ¶
      - depends_on proposition 12.8 Absolutely convergent series test ¶
      - proves proof ch:10-hilbert-spaces@proof-19 ¶
    - depends_on theorem 12.46 Riesz representation ¶
      - depends_on definition 12.45 Continuous linear functional; the dual ¶
      - depends_on theorem 12.18 Projection theorem ¶ ↺
      - proves proof ch:10-hilbert-spaces@proof-24 ¶
    - proves proof ch:10-hilbert-spaces@proof-20 ¶
- depends_on definition 12.88 Reducing subspace ¶
  - depends_on definition 12.35 Bounded operator; operator norm ¶ ↺
  - depends_on definition 12.20 Orthogonal projection operator ¶ ↺
depends_on proposition 12.89 Reduction is commutation ¶
- depends_on definition 12.88 Reducing subspace ¶ ↺
- depends_on proposition 12.21 Characterization of orthogonal projections ¶ ↺
- proves proof ch:10-hilbert-spaces@proof-43 ¶
depends_on theorem A.587 The Gaussian average is a non-zero projector that absorbs the Weyl operators ¶
- depends_on definition A.585 The Gaussian average of the Weyl operators ¶
  - depends_on definition A.580 Weyl operator ¶
    - 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 definition 12.109 Weyl system ¶
      - depends_on definition 12.64 Strongly continuous one-parameter unitary group ¶ ↺
      - depends_on theorem 12.75 The canonical commutation relation admits no bounded solution ¶
      - depends_on theorem 12.66 Stone ¶
    - depends_on equation 12.71 eq:hilbert-weyl-relation ¶
  - depends_on lemma A.583 Absolutely convergent operator-valued integrals ¶
    - depends_on lemma A.254 Riemann integral of a continuous curve ¶
      - depends_on definition 12.2 Hilbert space ¶
      - depends_on proposition 12.4 Cauchy–Schwarz and continuity of the inner product ¶
      - proves proof app:A-long-proofs@proof-157 ¶
    - depends_on lemma A.582 Joint strong continuity ¶
      - depends_on definition 12.64 Strongly continuous one-parameter unitary group ¶ ↺
      - depends_on lemma A.581 Composition law ¶
      - proves proof app:A-long-proofs@proof-348 ¶
    - depends_on proposition 12.4 Cauchy–Schwarz and continuity of the inner product ¶ ↺
    - proves proof app:A-long-proofs@proof-349 ¶
- depends_on lemma A.584 Fourier injectivity in $n$ variables ¶
  - depends_on definition 17.29 Fourier transform; the treatise convention ¶
  - depends_on equation 17.42 eq:ft-gaussian ¶
  - 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 app:A-long-proofs@proof-350 ¶
- depends_on lemma A.586 Gaussian integral with a complex linear term ¶
  - depends_on equation 7.151 eq:ana-iterated ¶
  - depends_on equation 17.42 eq:ft-gaussian ¶ ↺
  - depends_on equation 17.35 eq:ft-translation ¶
  - proves proof app:A-long-proofs@proof-351 ¶
- proves proof app:A-long-proofs@proof-352 ¶
proves proof app:A-long-proofs@proof-353 ¶

Supports

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.591 Any two irreducible Weyl systems are equivalent ¶ ↺

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`	→	Self-adjoint family; commutant; irreducibility	declared	appendices/A-long-proofs.tex:28237
`depends_on`	→	Reduction is commutation	declared	appendices/A-long-proofs.tex:28237
`depends_on`	→	The Gaussian average is a non-zero projector that absorbs the Weyl operators	declared	appendices/A-long-proofs.tex:28237
`depends_on`	←	The Gram matrix is universal	declared	appendices/A-long-proofs.tex:28296
`depends_on`	←	Any two irreducible Weyl systems are equivalent	declared	appendices/A-long-proofs.tex:28332
`proves`	←	app:A-long-proofs@proof-353	declared	appendices/A-long-proofs.tex:28241