definition 12.26 Orthonormal system; Fourier coefficients

open in the book · parts/02-mathematical-methods/10-hilbert-spaces.tex:648 · p. 420

ground object -- no derivation owed

Rests on

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 5.5 Linear combination ¶
      - depends_on definition 5.7 Vector subspace ¶
      - proves proof ch:03-linear-algebra-representations@proof-3 ¶
    - depends_on definition 5.14 Linear independence ¶
      - depends_on definition 5.5 Linear combination ¶ ↺
  - depends_on definition 5.24 Orthogonal vectors ¶
    - depends_on definition 5.18 Inner product ¶
      - depends_on definition 4.33 Vector space ¶
- 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 12.29 Orthonormal basis ¶
- 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 12.27 Best approximation and Bessel's inequality ¶
- depends_on theorem 12.30 Completeness, expansion, Parseval ¶
  - depends_on proposition 17.26 The lattice harmonics are an orthonormal basis ¶
  - depends_on proposition 12.87 Expansion in an orthogonal decomposition ¶ ↺
  - depends_on proposition 12.95 The tensor inner product is well defined and positive definite ¶
    - depends_on example 12.100 Entangled vectors exist ¶
    - depends_on proposition 12.96 Operators on a tensor product ¶
  - depends_on theorem A.229 Hilbert–Schmidt ¶
  - depends_on theorem 12.44 Hilbert–Schmidt: compact self-adjoint operators ¶
    - depends_on theorem A.461 Completeness in the weighted and in the energy norm ¶
    - depends_on theorem A.471 Spectral decomposition and completeness in $L^{2}_{r}$ ¶
      - depends_on lemma A.472 The pairing identity ¶
  - depends_on theorem 12.33 Every separable Hilbert space is $\ell^{2}$ ¶
depends_on proposition 12.28 Convergence criterion for orthogonal series ¶
- depends_on theorem 12.30 Completeness, expansion, Parseval ¶ ↺

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:657
`depends_on`	→	eq:lin-orthonormality	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:657
`depends_on`	←	Orthonormal basis	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:734
`depends_on`	←	Best approximation and Bessel's inequality	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:669
`depends_on`	←	Convergence criterion for orthogonal series	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:708