proposition 12.73 The adjoint is always closed

open in the book · parts/02-mathematical-methods/10-hilbert-spaces.tex:2001 · p. 435

Rests on

depends_on definition 12.70 Graph; closed and closable operators ¶
- 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 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 ¶
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 4.33 Vector space ¶ ↺
    - 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 definition 5.37 Linear transformation ¶ ↺
      - depends_on definition 5.19 Norm ¶
    - depends_on proposition 12.8 Absolutely convergent series test ¶
      - depends_on definition 5.19 Norm ¶ ↺
      - depends_on definition 6.27 Convergence; Cauchy sequence; completeness ¶
      - proves proof ch:10-hilbert-spaces@proof-4 ¶
    - 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 definition 12.35 Bounded operator; operator norm ¶ ↺
      - depends_on definition 5.46 Functional ¶
    - depends_on theorem 12.18 Projection theorem ¶ ↺
    - proves proof ch:10-hilbert-spaces@proof-24 ¶
  - proves proof ch:10-hilbert-spaces@proof-20 ¶
proves proof ch:10-hilbert-spaces@proof-35 ¶

Supports

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`	→	Graph; closed and closable operators	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2006
`depends_on`	→	Adjoint of a densely defined operator	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2006
`depends_on`	←	Essential self-adjointness	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2135
`depends_on`	←	Cayley transform of a self-adjoint operator	declared	appendices/A-long-proofs.tex:12927
`proves`	←	ch:10-hilbert-spaces@proof-35	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2009