proposition 12.95 The tensor inner product is well defined and positive definite

open in the book · parts/02-mathematical-methods/10-hilbert-spaces.tex:2600 · p. 441

Rests on

depends_on definition 12.94 Tensor product of Hilbert spaces ¶
- depends_on definition 12.2 Hilbert space ¶
  - depends_on definition 5.18 Inner product ¶
    - 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 6.27 Convergence; Cauchy sequence; completeness ¶
    - depends_on definition 6.24 Metric ¶
      - depends_on definition 3.43 Map ¶
      - depends_on definition 3.38 Ordered pair and Cartesian product ¶
  - depends_on equation 5.44 eq:lin-norm-assoc ¶
- depends_on definition 5.110 Bilinear map ¶
  - depends_on definition 5.37 Linear transformation ¶
    - depends_on definition 4.33 Vector space ¶ ↺
- depends_on definition 5.18 Inner product ¶ ↺
depends_on proposition 12.23 Gram–Schmidt in a Hilbert space ¶
- 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 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 ¶
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 ¶
      - 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 12.18 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 definition 5.27 Orthonormal basis ¶ ↺
    - depends_on equation 5.49 eq:lin-orthonormality ¶
  - 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-45 ¶

Supports

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.

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Edges

type	direction	node	provenance	where
`depends_on`	→	Tensor product of Hilbert spaces	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2608
`depends_on`	→	Gram–Schmidt in a Hilbert space	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2608
`depends_on`	→	Completeness, expansion, Parseval	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2608
`depends_on`	←	Entangled vectors exist	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2739
`depends_on`	←	Operators on a tensor product	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2653
`proves`	←	ch:10-hilbert-spaces@proof-45	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2612