definition 12.88 Reducing subspace

open in the book · parts/02-mathematical-methods/10-hilbert-spaces.tex:2438 · p. 440

ground object -- no derivation owed

Rests on

depends_on definition 12.35 Bounded operator; operator norm ¶
- depends_on definition 5.37 Linear transformation ¶
  - 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 5.19 Norm ¶
  - depends_on definition 4.33 Vector space ¶ ↺
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 corollary 12.5 Continuity of the norm and of orthogonality ¶
      - depends_on definition 5.24 Orthogonal vectors ¶
      - depends_on proposition 12.4 Cauchy–Schwarz and continuity of the inner product ¶
      - proves proof ch:10-hilbert-spaces@proof-2 ¶
    - depends_on definition 12.16 Orthogonal complement ¶
      - depends_on definition 5.24 Orthogonal vectors ¶ ↺
      - depends_on equation 5.52 eq:lin-orthogonal-complement ¶
    - proves proof ch:10-hilbert-spaces@proof-8 ¶
  - depends_on theorem 12.14 Closest point in a closed convex set ¶
    - depends_on definition 12.13 Convex set ¶
      - depends_on definition 5.7 Vector subspace ¶
    - 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 12.6 Parallelogram law and polarization ¶
      - depends_on definition 5.18 Inner product ¶ ↺
      - depends_on equation 5.44 eq:lin-norm-assoc ¶ ↺
      - proves proof ch:10-hilbert-spaces@proof-3 ¶
    - proves proof ch:10-hilbert-spaces@proof-7 ¶
  - proves proof ch:10-hilbert-spaces@proof-9 ¶

Supports

depends_on definition 12.90 Self-adjoint family; commutant; irreducibility ¶
- 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 ¶ ↺
depends_on proposition 12.89 Reduction is commutation ¶
- depends_on proposition A.589 Cyclic subspaces and the rank of the average ¶ ↺
- depends_on theorem 12.91 Schur's lemma, commutant form ¶ ↺

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`	→	Bounded operator; operator norm	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2441
`depends_on`	→	Orthogonal projection operator	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2441
`depends_on`	←	Self-adjoint family; commutant; irreducibility	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2480
`depends_on`	←	Reduction is commutation	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2448