definition 12.86 Internal orthogonal decomposition

open in the book · parts/02-mathematical-methods/10-hilbert-spaces.tex:2386 · p. 439

ground object -- no derivation owed

Rests on

depends_on definition 12.29 Orthonormal basis ¶
- depends_on definition 12.16 Orthogonal complement ¶
  - depends_on definition 5.24 Orthogonal vectors ¶
    - depends_on definition 5.18 Inner product ¶
      - depends_on definition 4.33 Vector space ¶
  - depends_on equation 5.52 eq:lin-orthogonal-complement ¶
- depends_on definition 12.26 Orthonormal system; Fourier coefficients ¶
  - depends_on definition 5.27 Orthonormal basis ¶
    - depends_on definition 5.26 Orthogonal basis ¶
      - depends_on definition 5.15 Basis ¶
      - depends_on definition 5.24 Orthogonal vectors ¶ ↺
    - depends_on definition 5.25 Unit vector ¶
      - depends_on definition 5.19 Norm ¶
  - depends_on equation 5.49 eq:lin-orthonormality ¶
depends_on definition 12.16 Orthogonal complement ¶ ↺

Supports

depends_on proposition 12.87 Expansion in an orthogonal decomposition ¶
- depends_on lemma A.250 Decomposition into cyclic subspaces ¶
  - depends_on proposition A.280 Direct-integral form of the spectral theorem ¶
    - depends_on proposition A.282 The fibre maps are continuous on $\Phi$ ¶

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`	→	Orthonormal basis	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2392
`depends_on`	→	Orthogonal complement	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2392
`depends_on`	←	Expansion in an orthogonal decomposition	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2408