proposition 12.87 Expansion in an orthogonal decomposition

open in the book · parts/02-mathematical-methods/10-hilbert-spaces.tex:2395 · p. 439

Rests on

depends_on definition 12.86 Internal orthogonal decomposition ¶
- 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 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 5.25 Unit vector ¶
    - depends_on equation 5.49 eq:lin-orthonormality ¶
- depends_on definition 12.16 Orthogonal complement ¶ ↺
depends_on proposition 12.85 The direct sum is a Hilbert space ¶
- depends_on definition 12.84 External direct sum ¶
  - depends_on definition 5.101 External direct sum ¶
    - 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 definition 5.5 Linear combination ¶
      - depends_on definition 4.33 Vector space ¶ ↺
  - depends_on example 12.9 The sequence space $\ell^{2}$ ¶
    - 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 equation 5.44 eq:lin-norm-assoc ¶
    - depends_on definition 5.18 Inner product ¶ ↺
- depends_on proposition 12.10 $\ell^{2}$ is complete ¶
  - depends_on axiom 7.1 Completeness of $\R$ ¶
  - depends_on example 12.9 The sequence space $\ell^{2}$ ¶ ↺
  - proves proof ch:10-hilbert-spaces@proof-5 ¶
- proves proof ch:10-hilbert-spaces@proof-41 ¶
depends_on theorem 12.30 Completeness, expansion, Parseval ¶
- depends_on corollary 12.19 Double complement; the density criterion ¶
  - 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 ¶
      - proves proof ch:10-hilbert-spaces@proof-2 ¶
    - depends_on definition 12.16 Orthogonal complement ¶ ↺
    - proves proof ch:10-hilbert-spaces@proof-8 ¶
  - depends_on theorem 12.18 Projection theorem ¶
    - depends_on proposition 12.17 The complement is always a closed subspace ¶ ↺
    - depends_on theorem 12.14 Closest point in a closed convex set ¶
      - depends_on definition 12.13 Convex set ¶
      - depends_on definition 12.2 Hilbert space ¶ ↺
      - depends_on proposition 12.6 Parallelogram law and polarization ¶
      - proves proof ch:10-hilbert-spaces@proof-7 ¶
    - proves proof ch:10-hilbert-spaces@proof-9 ¶
  - proves proof ch:10-hilbert-spaces@proof-10 ¶
- depends_on proposition 12.27 Best approximation and Bessel's inequality ¶
  - depends_on definition 12.26 Orthonormal system; Fourier coefficients ¶ ↺
  - depends_on theorem 12.18 Projection theorem ¶ ↺
  - proves proof ch:10-hilbert-spaces@proof-14 ¶
- depends_on proposition 12.28 Convergence criterion for orthogonal series ¶
  - depends_on definition 12.26 Orthonormal system; Fourier coefficients ¶ ↺
  - depends_on definition 12.2 Hilbert space ¶ ↺
  - proves proof ch:10-hilbert-spaces@proof-15 ¶
- proves proof ch:10-hilbert-spaces@proof-16 ¶
proves proof ch:10-hilbert-spaces@proof-42 ¶

Supports

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

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Edges

type	direction	node	provenance	where
`depends_on`	→	Internal orthogonal decomposition	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2408
`depends_on`	→	The direct sum is a Hilbert space	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2408
`depends_on`	→	Completeness, expansion, Parseval	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2408
`depends_on`	←	Decomposition into cyclic subspaces	declared	appendices/A-long-proofs.tex:12478
`proves`	←	ch:10-hilbert-spaces@proof-42	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2412