proposition 12.23 Gram–Schmidt in a Hilbert space

open in the book · parts/02-mathematical-methods/10-hilbert-spaces.tex:576 · p. 419

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 proposition 5.28 Gram–Schmidt ¶
- depends_on definition 5.14 Linear independence ¶ ↺
- depends_on definition 5.18 Inner product ¶ ↺
- depends_on definition 5.27 Orthonormal basis ¶ ↺
- depends_on equation 5.44 eq:lin-norm-assoc ¶
- proves proof ch:03-linear-algebra-representations@proof-5 ¶
proves proof ch:10-hilbert-spaces@proof-12 ¶

Supports

depends_on corollary 12.24 Separable spaces have countable orthonormal families ¶
- depends_on theorem 12.33 Every separable Hilbert space is $\ell^{2}$ ¶
depends_on example 12.25 Orthogonal polynomials ¶
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 ¶

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:587
`depends_on`	→	Gram–Schmidt	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:587
`depends_on`	←	Separable spaces have countable orthonormal families	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:609
`depends_on`	←	Orthogonal polynomials	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:637
`depends_on`	←	The tensor inner product is well defined and positive definite	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2608
`proves`	←	ch:10-hilbert-spaces@proof-12	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:590