proposition 12.51 The three cases are exclusive and exhaustive

open in the book · parts/02-mathematical-methods/10-hilbert-spaces.tex:1328 · p. 427

Rests on

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 4.4 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 6.27 Convergence; Cauchy sequence; completeness ¶
  - depends_on definition 6.24 Metric ¶
    - depends_on definition 3.43 Map ¶
      - depends_on definition 3.24 Quantifiers ¶
      - depends_on definition 3.28 Set ¶
    - depends_on definition 3.38 Ordered pair and Cartesian product ¶
- depends_on equation 5.44 eq:lin-norm-assoc ¶
depends_on definition 12.50 Point, continuous and residual spectrum ¶
- 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 definition 12.49 Resolvent set; spectrum ¶
  - depends_on definition 12.35 Bounded operator; operator norm ¶
    - depends_on definition 5.37 Linear transformation ¶
      - depends_on definition 4.33 Vector space ¶ ↺
    - depends_on definition 5.19 Norm ¶
      - depends_on definition 4.33 Vector space ¶ ↺
  - depends_on definition 5.47 Inverse of a linear transformation ¶
    - depends_on definition 5.37 Linear transformation ¶ ↺
proves proof ch:10-hilbert-spaces@proof-26 ¶

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type	direction	node	provenance	where
`depends_on`	→	Hilbert space	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1332
`depends_on`	→	Point, continuous and residual spectrum	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1332
`proves`	←	ch:10-hilbert-spaces@proof-26	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1335