definition 12.105 Generalized eigenvector

open in the book · parts/02-mathematical-methods/10-hilbert-spaces.tex:2848 · p. 444

ground object -- no derivation owed

Rests on

depends_on definition 12.103 Gelfand triple ¶
- 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 5.19 Norm ¶
    - 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 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-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 ¶ ↺
depends_on definition 12.50 Point, continuous and residual spectrum ¶
- depends_on corollary 12.19 Double complement; the density criterion ¶ ↺
- depends_on definition 12.49 Resolvent set; spectrum ¶
  - depends_on definition 12.35 Bounded operator; operator norm ¶ ↺
  - depends_on definition 5.47 Inverse of a linear transformation ¶
    - depends_on definition 5.37 Linear transformation ¶ ↺

Supports

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 ¶

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`	→	Gelfand triple	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2857
`depends_on`	→	Point, continuous and residual spectrum	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2857
`depends_on`	←	The plane wave is a generalized momentum eigenvector	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2867
`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