theorem 12.74 Hellinger–Toeplitz

open in the book · parts/02-mathematical-methods/10-hilbert-spaces.tex:2023 · p. 435

Rests on

depends_on definition 12.72 Symmetric; self-adjoint ¶
- depends_on definition 12.69 Operator with a domain ¶
  - 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 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 ¶
      - proves proof ch:10-hilbert-spaces@proof-9 ¶
    - proves proof ch:10-hilbert-spaces@proof-10 ¶
  - depends_on definition 5.37 Linear transformation ¶
    - 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 12.71 Adjoint of a densely defined operator ¶
  - depends_on corollary 12.19 Double complement; the density criterion ¶ ↺
  - depends_on definition 12.69 Operator with a domain ¶ ↺
  - depends_on theorem 12.38 Existence and uniqueness of the adjoint ¶
    - depends_on definition 5.41 Adjoint ¶
      - depends_on definition 5.18 Inner product ¶
      - depends_on definition 5.37 Linear transformation ¶ ↺
    - depends_on proposition 12.37 $\mathcal{B}(\mathcal{H})$ is a Banach algebra ¶
      - depends_on definition 12.35 Bounded operator; operator norm ¶
      - depends_on proposition 12.8 Absolutely convergent series test ¶
      - proves proof ch:10-hilbert-spaces@proof-19 ¶
    - depends_on theorem 12.46 Riesz representation ¶
      - depends_on definition 12.45 Continuous linear functional; the dual ¶
      - depends_on theorem 12.18 Projection theorem ¶ ↺
      - proves proof ch:10-hilbert-spaces@proof-24 ¶
    - proves proof ch:10-hilbert-spaces@proof-20 ¶
depends_on proposition 12.36 Boundedness is continuity ¶
- depends_on definition 12.35 Bounded operator; operator norm ¶ ↺
- depends_on proposition 6.28 $\varepsilon$–$\delta$ characterization ¶
  - depends_on definition 6.6 Continuous map ¶
    - depends_on definition 3.53 Preimage ¶
    - depends_on definition 6.2 Open set ¶
      - depends_on definition 6.1 Topological space ¶
    - depends_on definition 6.1 Topological space ¶ ↺
  - depends_on definition 6.26 Open ball; metric topology ¶
    - depends_on definition 6.24 Metric ¶
      - depends_on definition 3.43 Map ¶
      - depends_on definition 3.38 Ordered pair and Cartesian product ¶
    - depends_on definition 6.2 Open set ¶ ↺
    - depends_on definition 6.1 Topological space ¶ ↺
  - proves proof ch:04-topology@proof-9 ¶
- proves proof ch:10-hilbert-spaces@proof-18 ¶
depends_on remark 12.1 Analytical inputs quoted, not proved ¶
proves proof ch:10-hilbert-spaces@proof-36 ¶

Supports

depends_on corollary 12.76 Position and momentum are unbounded, and cannot be everywhere defined ¶

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`	→	Symmetric; self-adjoint	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2028
`depends_on`	→	Boundedness is continuity	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2028
`depends_on`	→	Analytical inputs quoted, not proved	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2028
`depends_on`	←	Position and momentum are unbounded, and cannot be everywhere defined	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2091
`proves`	←	ch:10-hilbert-spaces@proof-36	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2032