definition 12.29 Orthonormal basis

open in the book · parts/02-mathematical-methods/10-hilbert-spaces.tex:730 · p. 421

ground object -- no derivation owed

Rests on

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 definition 4.8 Commutativity; abelian structure ¶
      - depends_on definition 4.32 Field ¶
      - depends_on definition 4.31 Module ¶
- 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 5.12 Subspace generated by a set of vectors ¶
      - depends_on definition 5.14 Linear independence ¶
    - depends_on definition 5.24 Orthogonal vectors ¶ ↺
  - depends_on definition 5.25 Unit vector ¶
    - depends_on definition 5.19 Norm ¶
      - depends_on definition 4.33 Vector space ¶ ↺
- depends_on equation 5.49 eq:lin-orthonormality ¶

Supports

depends_on definition 12.86 Internal orthogonal decomposition ¶
- 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`	→	Orthogonal complement	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:734
`depends_on`	→	Orthonormal system; Fourier coefficients	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:734
`depends_on`	←	Internal orthogonal decomposition	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2392