definition 12.103 Gelfand triple

open in the book · parts/02-mathematical-methods/10-hilbert-spaces.tex:2805 · p. 443

ground object -- no derivation owed

Rests on

depends_on corollary 12.47 $\mathcal{H}$ is its own dual, antilinearly ¶
- depends_on definition 12.45 Continuous linear functional; the dual ¶
  - 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 5.46 Functional ¶
    - depends_on definition 5.37 Linear transformation ¶ ↺
- depends_on theorem 12.46 Riesz representation ¶
  - depends_on definition 12.45 Continuous linear functional; the dual ¶ ↺
  - depends_on theorem 12.18 Projection theorem ¶
    - 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 12.14 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-24 ¶
- proves proof ch:10-hilbert-spaces@proof-25 ¶
depends_on corollary 12.19 Double complement; the density criterion ¶
- depends_on proposition 12.17 The complement is always a closed subspace ¶ ↺
- depends_on theorem 12.18 Projection theorem ¶ ↺
- proves proof ch:10-hilbert-spaces@proof-10 ¶
depends_on definition 12.45 Continuous linear functional; the dual ¶ ↺

Supports

depends_on definition A.278 Countably Hilbert nuclear space ¶
- depends_on theorem A.281 Nuclear spaces embed by Hilbert–Schmidt maps; quoted ¶
  - depends_on proposition A.282 The fibre maps are continuous on $\Phi$ ¶
depends_on definition 12.105 Generalized eigenvector ¶
- depends_on proposition 12.106 The plane wave is a generalized momentum eigenvector ¶
  - depends_on example A.283 Momentum on the line ¶
- depends_on theorem A.279 Gelfand–Maurin ¶
  - depends_on example A.283 Momentum on the line ¶ ↺
- depends_on theorem 12.107 Nuclear spectral theorem ¶
depends_on example 12.104 The Schwartz triple ¶
- depends_on example A.283 Momentum on the line ¶ ↺
- depends_on proposition 12.106 The plane wave is a generalized momentum eigenvector ¶ ↺
depends_on theorem A.279 Gelfand–Maurin ¶ ↺
depends_on theorem 12.107 Nuclear spectral theorem ¶ ↺

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`	→	$\mathcal{H}$ is its own dual, antilinearly	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2821
`depends_on`	→	Double complement; the density criterion	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2821
`depends_on`	→	Continuous linear functional; the dual	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2821
`depends_on`	←	Countably Hilbert nuclear space	declared	appendices/A-long-proofs.tex:13903
`depends_on`	←	Generalized eigenvector	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2857
`depends_on`	←	The Schwartz triple	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2844
`depends_on`	←	Gelfand–Maurin	declared	appendices/A-long-proofs.tex:13939
`depends_on`	←	Nuclear spectral theorem	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2901