definition 5.46 Functional

open in the book · parts/02-mathematical-methods/03-linear-algebra-representations.tex:2128 · p. 121

ground object -- no derivation owed

Rests on

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 4.3 Algebraic structure ¶
      - depends_on definition 4.2 Binary operation ¶
  - depends_on definition 4.32 Field ¶
    - depends_on definition 4.11 Inverse element ¶
      - depends_on definition 4.9 Neutral element ¶
    - depends_on definition 4.20 Monoid ¶
      - depends_on definition 4.9 Neutral element ¶ ↺
      - depends_on definition 4.18 Semigroup ¶
    - depends_on definition 4.30 Ring ¶
      - depends_on definition 4.8 Commutativity; abelian structure ¶ ↺
      - depends_on definition 4.28 Distributivity ¶
      - depends_on definition 4.21 Group ¶
      - depends_on definition 4.18 Semigroup ¶ ↺
  - depends_on definition 4.31 Module ¶
    - depends_on definition 4.29 Action ¶
      - depends_on definition 4.9 Neutral element ¶ ↺
    - depends_on definition 4.28 Distributivity ¶ ↺
    - depends_on definition 4.21 Group ¶ ↺
    - depends_on definition 4.30 Ring ¶ ↺

Supports

depends_on definition 12.45 Continuous linear functional; the dual ¶
- 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 definition 12.105 Generalized eigenvector ¶
      - 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 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.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 definition 12.58 Projection-valued measure ¶
      - depends_on definition 12.64 Strongly continuous one-parameter unitary group ¶
      - depends_on lemma A.231 Restriction to an invariant closed subspace ¶
      - depends_on lemma A.230 Sequential characterisation ¶
      - depends_on proposition 12.42 Elementary consequences ¶
      - depends_on theorem A.229 Hilbert–Schmidt ¶
      - depends_on theorem 12.44 Hilbert–Schmidt: compact self-adjoint operators ¶
      - depends_on theorem 12.55 The spectrum of a self-adjoint operator is real ¶
    - depends_on definition 12.71 Adjoint of a densely defined operator ¶
      - depends_on definition 12.79 Deficiency subspaces and indices ¶
      - depends_on definition 12.72 Symmetric; self-adjoint ¶
      - depends_on proposition 12.73 The adjoint is always closed ¶
    - 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 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 ¶
depends_on definition 5.59 Dual space ¶
- depends_on definition 5.66 Pushforward ¶
  - depends_on proposition 5.67 Adjointness of pullback and pushforward ¶
- depends_on proposition 5.61 Annihilator of a subspace ¶
  - depends_on corollary 5.62 A functional that annihilates a set of constraints ¶
- depends_on proposition 5.114 prop:lin-bilinear-dual ¶
- depends_on proposition 5.63 prop:lin-dual-inner-product ¶
- depends_on proposition 5.67 Adjointness of pullback and pushforward ¶ ↺
depends_on definition 5.60 Functional specified on a basis ¶

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`	→	Linear transformation	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:2131
`depends_on`	←	Continuous linear functional; the dual	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1208
`depends_on`	←	Dual space	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:2589
`depends_on`	←	Functional specified on a basis	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:2602