proposition 12.28 Convergence criterion for orthogonal series

open in the book · parts/02-mathematical-methods/10-hilbert-spaces.tex:701 · p. 420

Rests on

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 5.18 Inner product ¶
  - 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 ¶
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 definition 6.24 Metric ¶
    - depends_on definition 3.43 Map ¶
      - depends_on definition 3.24 Quantifiers ¶
      - depends_on definition 3.28 Set ¶
    - depends_on definition 3.38 Ordered pair and Cartesian product ¶
- depends_on equation 5.44 eq:lin-norm-assoc ¶
proves proof ch:10-hilbert-spaces@proof-15 ¶

Supports

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 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 $\Phi$ ¶
- 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 $\ell^{2}$ ¶

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 system; Fourier coefficients	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:708
`depends_on`	→	Hilbert space	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:708
`depends_on`	←	Completeness, expansion, Parseval	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:761
`proves`	←	ch:10-hilbert-spaces@proof-15	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:711