definition 12.45 Continuous linear functional; the dual

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Rests on

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 4.4 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 5.19 Norm ¶
  - depends_on definition 4.33 Vector space ¶ ↺
depends_on definition 5.46 Functional ¶
- depends_on definition 5.37 Linear transformation ¶ ↺

Supports

depends_on corollary 12.47 $\mathcal{H}$ is its own dual, antilinearly ¶
- depends_on definition 12.103 Gelfand triple ¶
  - 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 ¶ ↺
depends_on definition 12.103 Gelfand triple ¶ ↺
depends_on theorem 12.46 Riesz representation ¶
- depends_on corollary 12.47 $\mathcal{H}$ is its own dual, antilinearly ¶ ↺
- depends_on lemma A.247 Integration against a projection-valued measure ¶
- depends_on proposition A.246 Bounded Borel functional calculus ¶
  - 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 $\Phi$ ¶ ↺
  - depends_on proposition A.261 Spectral theorem for a unitary operator ¶
    - depends_on proposition A.262 Spectral theorem for an unbounded self-adjoint operator ¶ ↺
- depends_on proposition A.245 The measures $\mu_{x,y}$ ¶
  - depends_on lemma A.249 The cyclic case ¶
  - depends_on proposition A.246 Bounded Borel functional calculus ¶ ↺
- depends_on theorem 12.38 Existence and uniqueness of the adjoint ¶
  - depends_on definition 12.41 The operator classes ¶
    - depends_on definition 12.90 Self-adjoint family; commutant; irreducibility ¶
      - depends_on proposition A.589 Cyclic subspaces and the rank of the average ¶
      - depends_on theorem A.579 Stone–von Neumann ¶
      - depends_on theorem 12.91 Schur's lemma, commutant form ¶
      - depends_on theorem 12.114 Stone–von Neumann ¶
    - depends_on definition 12.58 Projection-valued measure ¶
      - depends_on definition 12.60 Functional calculus ¶
      - depends_on lemma A.247 Integration against a projection-valued measure ¶ ↺
      - depends_on theorem A.238 Spectral theorem, both forms ¶
      - depends_on theorem 12.59 Spectral theorem for a bounded self-adjoint operator ¶
    - depends_on definition 12.64 Strongly continuous one-parameter unitary group ¶
      - depends_on definition A.580 Weyl operator ¶
      - depends_on definition 12.109 Weyl system ¶
      - depends_on lemma A.255 Smoothed vectors lie in the domain ¶
      - depends_on lemma A.581 Composition law ¶
      - depends_on lemma A.582 Joint strong continuity ¶
      - depends_on proposition A.256 Density ¶
      - depends_on proposition 12.65 Exponential of a bounded self-adjoint operator ¶
      - depends_on proposition 12.67 The generator is symmetric, and generates the motion ¶
      - depends_on theorem A.253 Stone ¶
      - depends_on theorem 12.66 Stone ¶
      - depends_on theorem 25.34 Stone–von Neumann ¶
    - depends_on lemma A.231 Restriction to an invariant closed subspace ¶
      - depends_on lemma A.233 Construction of the system ¶
    - depends_on lemma A.230 Sequential characterisation ¶
      - depends_on lemma A.232 Attainment ¶
      - depends_on lemma A.231 Restriction to an invariant closed subspace ¶ ↺
      - depends_on lemma A.234 The eigenvalues tend to zero, with finite multiplicity ¶
    - depends_on proposition 12.42 Elementary consequences ¶
      - depends_on lemma A.232 Attainment ¶ ↺
      - depends_on proposition 12.43 Norm of a self-adjoint operator ¶
    - depends_on theorem A.229 Hilbert–Schmidt ¶
    - depends_on theorem 12.44 Hilbert–Schmidt: compact self-adjoint operators ¶
      - depends_on theorem A.461 Completeness in the weighted and in the energy norm ¶
      - depends_on theorem A.471 Spectral decomposition and completeness in $L^{2}_{r}$ ¶
    - depends_on theorem 12.55 The spectrum of a self-adjoint operator is real ¶
      - depends_on example 12.56 Multiplication by the coordinate: spectrum without eigenvectors ¶
      - depends_on theorem A.238 Spectral theorem, both forms ¶ ↺
      - depends_on theorem 12.59 Spectral theorem for a bounded self-adjoint operator ¶ ↺
  - depends_on definition 12.71 Adjoint of a densely defined operator ¶
    - depends_on definition 12.79 Deficiency subspaces and indices ¶
      - depends_on definition A.269 Cayley transform ¶
      - depends_on example 12.82 Momentum on the half-line: no self-adjoint extension ¶
      - depends_on example 12.81 Momentum on a finite interval: a circle of self-adjoint momenta ¶
      - depends_on lemma A.267 Isometry of $A\pm\ii\mu$, and closed range ¶
      - depends_on lemma A.268 The indices do not depend on $\mu$ ¶
      - depends_on theorem A.266 von Neumann ¶
      - depends_on theorem 12.80 von Neumann's criterion ¶
    - depends_on definition 12.72 Symmetric; self-adjoint ¶
      - depends_on definition 12.78 Essential self-adjointness ¶
      - depends_on lemma A.260 Cayley transform of a self-adjoint operator ¶
      - depends_on lemma A.267 Isometry of $A\pm\ii\mu$, and closed range ¶ ↺
      - depends_on lemma A.271 Injectivity of $\identity-V$ for any isometric extension ¶
      - depends_on lemma A.272 The operator attached to an isometry ¶
      - depends_on proposition A.273 Self-adjoint means unitary ¶
      - depends_on theorem A.266 von Neumann ¶ ↺
      - depends_on theorem 12.74 Hellinger–Toeplitz ¶
      - depends_on theorem 12.80 von Neumann's criterion ¶ ↺
    - depends_on proposition 12.73 The adjoint is always closed ¶
      - depends_on definition 12.78 Essential self-adjointness ¶ ↺
      - depends_on lemma A.260 Cayley transform of a self-adjoint operator ¶ ↺
  - depends_on proposition 12.39 Algebra of the adjoint; the $C^{\ast}$ identity ¶
    - depends_on definition 12.79 Deficiency subspaces and indices ¶ ↺
    - depends_on proposition A.241 The polynomial calculus is isometric ¶
      - depends_on proposition A.243 Continuous functional calculus ¶
    - depends_on proposition 12.65 Exponential of a bounded self-adjoint operator ¶ ↺
    - depends_on proposition 12.42 Elementary consequences ¶ ↺
    - depends_on proposition 12.43 Norm of a self-adjoint operator ¶ ↺
    - depends_on proposition 12.96 Operators on a tensor product ¶

Neighborhood

Every logical edge within two steps of this node.

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partly declared
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Edges

type	direction	node	provenance	where
`depends_on`	→	Bounded operator; operator norm	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1208
`depends_on`	→	Functional	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1208
`depends_on`	←	$\mathcal{H}$ is its own dual, antilinearly	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1253
`depends_on`	←	Gelfand triple	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2821
`depends_on`	←	Riesz representation	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1218