proposition 12.27 Best approximation and Bessel's inequality

open in the book · parts/02-mathematical-methods/10-hilbert-spaces.tex:660 · 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 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 5.24 Orthogonal vectors ¶ ↺
    - depends_on proposition 12.4 Cauchy–Schwarz and continuity of the inner product ¶
      - depends_on definition 12.2 Hilbert space ¶
      - depends_on proposition 5.20 Cauchy–Schwarz inequality ¶
      - proves proof ch:10-hilbert-spaces@proof-1 ¶
    - proves proof ch:10-hilbert-spaces@proof-2 ¶
  - depends_on definition 12.16 Orthogonal complement ¶
    - depends_on definition 5.24 Orthogonal vectors ¶ ↺
    - depends_on equation 5.52 eq:lin-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 5.7 Vector subspace ¶
      - depends_on definition 5.5 Linear combination ¶
  - depends_on definition 12.2 Hilbert space ¶ ↺
  - depends_on proposition 12.6 Parallelogram law and polarization ¶
    - depends_on definition 5.18 Inner product ¶ ↺
    - depends_on equation 5.44 eq:lin-norm-assoc ¶
    - proves proof ch:10-hilbert-spaces@proof-3 ¶
  - proves proof ch:10-hilbert-spaces@proof-7 ¶
- proves proof ch:10-hilbert-spaces@proof-9 ¶
proves proof ch:10-hilbert-spaces@proof-14 ¶

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.

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partly declared
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Edges

type	direction	node	provenance	where
`depends_on`	→	Orthonormal system; Fourier coefficients	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:669
`depends_on`	→	Projection theorem	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:669
`depends_on`	←	Completeness, expansion, Parseval	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:761
`proves`	←	ch:10-hilbert-spaces@proof-14	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:672