definition 12.70 Graph; closed and closable operators

open in the book · parts/02-mathematical-methods/10-hilbert-spaces.tex:1954 · p. 434

ground object -- no derivation owed

Rests on

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 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 5.37 Linear transformation ¶
  - 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 6.3 Closed set ¶
- depends_on definition 3.35 Union, intersection, difference ¶
  - depends_on definition 3.8 Conjunction ¶
  - depends_on definition 3.9 Disjunction ¶
  - depends_on definition 3.7 Negation ¶
- depends_on definition 6.2 Open set ¶
  - depends_on definition 6.1 Topological space ¶
    - depends_on definition 3.31 Empty set ¶
    - depends_on definition 3.35 Union, intersection, difference ¶ ↺
    - depends_on definition 3.30 Subset ¶
    - depends_on equation 3.51 eq:set-indexed ¶

Supports

depends_on proposition 12.73 The adjoint is always closed ¶
- depends_on definition 12.78 Essential self-adjointness ¶
  - depends_on theorem A.266 von Neumann ¶
    - depends_on corollary A.275 Momentum on $[0,\infty)$: no extension ¶
    - depends_on corollary A.274 Momentum on ${[}0,L{]}$: the circle of extensions ¶
  - depends_on theorem 12.80 von Neumann's criterion ¶
    - depends_on example 12.82 Momentum on the half-line: no self-adjoint extension ¶
      - depends_on corollary A.275 Momentum on $[0,\infty)$: no extension ¶ ↺
    - depends_on example 12.81 Momentum on a finite interval: a circle of self-adjoint momenta ¶
      - depends_on corollary A.274 Momentum on ${[}0,L{]}$: the circle of extensions ¶ ↺
      - depends_on example 12.82 Momentum on the half-line: no self-adjoint extension ¶ ↺
- depends_on lemma A.260 Cayley transform of a self-adjoint operator ¶
  - depends_on proposition A.262 Spectral theorem for an unbounded self-adjoint operator ¶
    - 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`	→	Operator with a domain	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1967
`depends_on`	→	Closed set	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1967
`depends_on`	←	The adjoint is always closed	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2006