theorem 12.14 Closest point in a closed convex set

open in the book · parts/02-mathematical-methods/10-hilbert-spaces.tex:341 · p. 417

Rests on

depends_on definition 12.13 Convex set ¶
- depends_on definition 5.7 Vector subspace ¶
  - depends_on definition 5.5 Linear combination ¶
    - 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 12.2 Hilbert space ¶
- depends_on definition 5.18 Inner product ¶
  - depends_on definition 4.33 Vector space ¶ ↺
- depends_on definition 6.27 Convergence; Cauchy sequence; completeness ¶
  - depends_on definition 6.24 Metric ¶
    - depends_on definition 3.43 Map ¶
      - depends_on definition 3.24 Quantifiers ¶
      - depends_on definition 3.28 Set ¶
    - depends_on definition 3.38 Ordered pair and Cartesian product ¶
- depends_on equation 5.44 eq:lin-norm-assoc ¶
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 ¶

Supports

depends_on theorem 12.18 Projection theorem ¶
- depends_on corollary 12.19 Double complement; the density criterion ¶
  - depends_on definition 12.79 Deficiency subspaces and indices ¶
    - depends_on definition A.269 Cayley transform ¶
      - depends_on proposition A.270 Properties of the transform ¶
    - depends_on example 12.82 Momentum on the half-line: no self-adjoint extension ¶
      - depends_on corollary A.275 Momentum on $[0,\infty)$: no extension ¶
    - depends_on example 12.81 Momentum on a finite interval: a circle of self-adjoint momenta ¶
      - depends_on corollary A.274 Momentum on ${[}0,L{]}$: the circle of extensions ¶
      - depends_on example 12.82 Momentum on the half-line: no self-adjoint extension ¶ ↺
    - depends_on lemma A.267 Isometry of $A\pm\ii\mu$, and closed range ¶
      - depends_on definition A.269 Cayley transform ¶ ↺
      - depends_on lemma A.268 The indices do not depend on $\mu$ ¶
      - depends_on proposition A.270 Properties of the transform ¶ ↺
      - depends_on proposition A.273 Self-adjoint means unitary ¶
    - depends_on lemma A.268 The indices do not depend on $\mu$ ¶ ↺
    - depends_on theorem A.266 von Neumann ¶
      - depends_on corollary A.275 Momentum on $[0,\infty)$: no extension ¶ ↺
      - depends_on corollary A.274 Momentum on ${[}0,L{]}$: the circle of extensions ¶ ↺
    - depends_on theorem 12.80 von Neumann's criterion ¶
      - 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 definition 12.69 Operator with a domain ¶
    - depends_on definition 12.70 Graph; closed and closable operators ¶
      - depends_on proposition 12.73 The adjoint is always closed ¶
    - 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 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 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 12.50 Point, continuous and residual spectrum ¶
    - depends_on definition 12.105 Generalized eigenvector ¶ ↺
    - depends_on example 12.56 Multiplication by the coordinate: spectrum without eigenvectors ¶
    - depends_on proposition 12.51 The three cases are exclusive and exhaustive ¶
  - depends_on definition 12.71 Adjoint of a densely defined operator ¶ ↺
  - depends_on theorem 12.30 Completeness, expansion, Parseval ¶
    - depends_on proposition 17.26 The lattice harmonics are an orthonormal basis ¶
    - depends_on proposition 12.87 Expansion in an orthogonal decomposition ¶
      - depends_on lemma A.250 Decomposition into cyclic subspaces ¶
    - depends_on proposition 12.95 The tensor inner product is well defined and positive definite ¶
      - depends_on example 12.100 Entangled vectors exist ¶
      - depends_on proposition 12.96 Operators on a tensor product ¶
    - 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 12.33 Every separable Hilbert space is $\ell^{2}$ ¶
  - 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 proposition A.280 Direct-integral form of the spectral theorem ¶
      - depends_on proposition A.261 Spectral theorem for a unitary operator ¶
      - depends_on theorem A.279 Gelfand–Maurin ¶ ↺
      - depends_on theorem A.253 Stone ¶
    - depends_on theorem 12.59 Spectral theorem for a bounded self-adjoint operator ¶
      - depends_on definition 12.60 Functional calculus ¶
      - depends_on theorem 12.107 Nuclear spectral theorem ¶ ↺
      - depends_on theorem 12.91 Schur's lemma, commutant form ¶
      - depends_on theorem 12.66 Stone ¶
- depends_on definition A.248 Cyclic vector and cyclic subspace ¶
  - depends_on lemma A.249 The cyclic case ¶
  - depends_on lemma A.250 Decomposition into cyclic subspaces ¶ ↺
- depends_on definition 12.20 Orthogonal projection operator ¶
  - depends_on definition 12.88 Reducing subspace ¶
    - 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 proposition 12.89 Reduction is commutation ¶
      - depends_on proposition A.589 Cyclic subspaces and the rank of the average ¶ ↺
      - depends_on theorem 12.91 Schur's lemma, commutant form ¶ ↺
  - depends_on proposition 12.21 Characterization of orthogonal projections ¶
    - 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 12.58 Projection-valued measure ¶ ↺
    - depends_on proposition 12.89 Reduction is commutation ¶ ↺
- depends_on lemma A.233 Construction of the system ¶
  - depends_on lemma A.234 The eigenvalues tend to zero, with finite multiplicity ¶
- depends_on lemma A.231 Restriction to an invariant closed subspace ¶ ↺
- depends_on lemma A.267 Isometry of $A\pm\ii\mu$, and closed range ¶ ↺
- depends_on proposition 12.27 Best approximation and Bessel's inequality ¶
  - depends_on theorem 12.30 Completeness, expansion, Parseval ¶ ↺
- depends_on proposition 12.21 Characterization of orthogonal projections ¶ ↺
- depends_on theorem 12.46 Riesz representation ¶
  - depends_on corollary 12.47 $\mathcal{H}$ is its own dual, antilinearly ¶
    - depends_on definition 12.103 Gelfand triple ¶ ↺
  - 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.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 12.71 Adjoint of a densely defined 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 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`	→	Convex set	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:348
`depends_on`	→	Hilbert space	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:348
`depends_on`	→	Parallelogram law and polarization	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:348
`depends_on`	←	Projection theorem	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:436
`proves`	←	ch:10-hilbert-spaces@proof-7	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:351