proposition 12.89 Reduction is commutation

open in the book · parts/02-mathematical-methods/10-hilbert-spaces.tex:2444 · p. 440

Rests on

depends_on definition 12.88 Reducing subspace ¶
- 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 4.32 Field ¶
      - depends_on definition 4.31 Module ¶
  - 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 12.16 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 12.2 Hilbert space ¶
      - depends_on proposition 12.6 Parallelogram law and polarization ¶
      - proves proof ch:10-hilbert-spaces@proof-7 ¶
    - proves proof ch:10-hilbert-spaces@proof-9 ¶
depends_on proposition 12.21 Characterization of orthogonal projections ¶
- depends_on definition 12.20 Orthogonal projection operator ¶ ↺
- depends_on theorem 12.18 Projection theorem ¶ ↺
- proves proof ch:10-hilbert-spaces@proof-11 ¶
proves proof ch:10-hilbert-spaces@proof-43 ¶

Every logical edge within two steps of this node.

type	direction	node	provenance	where
`depends_on`	→	Reducing subspace	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2448
`depends_on`	→	Characterization of orthogonal projections	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2448
`depends_on`	←	Cyclic subspaces and the rank of the average	declared	appendices/A-long-proofs.tex:28237
`depends_on`	←	Schur's lemma, commutant form	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2490
`proves`	←	ch:10-hilbert-spaces@proof-43	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2451