proposition 12.85 The direct sum is a Hilbert space

open in the book · parts/02-mathematical-methods/10-hilbert-spaces.tex:2363 · p. 439

Rests on

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 4.4 Internal binary operation; magma ¶
    - depends_on definition 4.32 Field ¶
      - depends_on definition 4.11 Inverse element ¶
      - depends_on definition 4.20 Monoid ¶
      - depends_on definition 4.30 Ring ¶
    - depends_on definition 4.31 Module ¶
      - depends_on definition 4.29 Action ¶
      - depends_on definition 4.28 Distributivity ¶
      - depends_on definition 4.21 Group ¶
      - depends_on definition 4.30 Ring ¶ ↺
  - 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 4.33 Vector space ¶ ↺
    - depends_on definition 6.27 Convergence; Cauchy sequence; completeness ¶
      - depends_on definition 6.24 Metric ¶
    - 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 ¶

Supports

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`	→	External direct sum	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2369
`depends_on`	→	$\ell^{2}$ is complete	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2369
`depends_on`	←	Expansion in an orthogonal decomposition	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2408
`proves`	←	ch:10-hilbert-spaces@proof-41	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2372