definition 12.50 Point, continuous and residual spectrum

open in the book · parts/02-mathematical-methods/10-hilbert-spaces.tex:1312 · p. 427

ground object -- no derivation owed

Rests on

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 definition 5.18 Inner product ¶
    - depends_on proposition 12.4 Cauchy–Schwarz and continuity of the inner product ¶
      - depends_on definition 12.2 Hilbert space ¶
      - depends_on proposition 5.20 Cauchy–Schwarz inequality ¶
      - proves proof ch:10-hilbert-spaces@proof-1 ¶
    - 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 5.7 Vector subspace ¶
    - depends_on definition 12.2 Hilbert space ¶ ↺
    - depends_on proposition 12.6 Parallelogram law and polarization ¶
      - depends_on definition 5.18 Inner product ¶ ↺
      - depends_on equation 5.44 eq:lin-norm-assoc ¶
      - proves proof ch:10-hilbert-spaces@proof-3 ¶
    - 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 12.49 Resolvent set; spectrum ¶
- 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 4.8 Commutativity; abelian structure ¶
      - depends_on definition 4.32 Field ¶
      - depends_on definition 4.31 Module ¶
  - depends_on definition 5.19 Norm ¶
    - depends_on definition 4.33 Vector space ¶ ↺
- depends_on definition 5.47 Inverse of a linear transformation ¶
  - depends_on definition 5.37 Linear transformation ¶ ↺

Supports

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 12.56 Multiplication by the coordinate: spectrum without eigenvectors ¶
depends_on proposition 12.51 The three cases are exclusive and exhaustive ¶

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`	→	Double complement; the density criterion	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1325
`depends_on`	→	Resolvent set; spectrum	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1325
`depends_on`	←	Generalized eigenvector	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2857
`depends_on`	←	Multiplication by the coordinate: spectrum without eigenvectors	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1504
`depends_on`	←	The three cases are exclusive and exhaustive	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1332