definition 12.90 Self-adjoint family; commutant; irreducibility

open in the book · parts/02-mathematical-methods/10-hilbert-spaces.tex:2468 · p. 440

ground object -- no derivation owed

Rests on

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 definition 6.1 Topological space ¶
    - 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 proposition 12.17 The complement is always a closed subspace ¶
      - depends_on theorem 12.14 Closest point in a closed convex set ¶
      - proves proof ch:10-hilbert-spaces@proof-9 ¶
  - 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 4.33 Vector space ¶
    - depends_on definition 5.37 Linear transformation ¶
      - depends_on definition 4.33 Vector space ¶ ↺
  - depends_on proposition 12.37 $\mathcal{B}(\mathcal{H})$ is a Banach algebra ¶
    - depends_on definition 12.35 Bounded operator; operator norm ¶
      - depends_on definition 5.37 Linear transformation ¶ ↺
      - depends_on definition 5.19 Norm ¶
    - depends_on proposition 12.8 Absolutely convergent series test ¶
      - depends_on definition 5.19 Norm ¶ ↺
      - depends_on definition 6.27 Convergence; Cauchy sequence; completeness ¶
      - proves proof ch:10-hilbert-spaces@proof-4 ¶
    - 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 definition 12.35 Bounded operator; operator norm ¶ ↺
      - depends_on definition 5.46 Functional ¶
    - 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 ¶ ↺

Supports

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.591 Any two irreducible Weyl systems are equivalent ¶ ↺
depends_on theorem A.579 Stone–von Neumann ¶
depends_on theorem 12.91 Schur's lemma, commutant form ¶
- depends_on proposition A.591 Any two irreducible Weyl systems are equivalent ¶ ↺
- depends_on theorem 12.114 Stone–von Neumann ¶
depends_on theorem 12.114 Stone–von Neumann ¶ ↺

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`	→	The operator classes	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2480
`depends_on`	→	Reducing subspace	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2480
`depends_on`	←	Cyclic subspaces and the rank of the average	declared	appendices/A-long-proofs.tex:28237
`depends_on`	←	Stone–von Neumann	declared	appendices/A-long-proofs.tex:27706
`depends_on`	←	Schur's lemma, commutant form	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2490
`depends_on`	←	Stone–von Neumann	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:3196