proposition 12.6 Parallelogram law and polarization

open in the book · parts/02-mathematical-methods/10-hilbert-spaces.tex:157 · p. 414

Rests on

depends_on definition 5.18 Inner product ¶
- 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 ¶ ↺
depends_on equation 5.44 eq:lin-norm-assoc ¶
proves proof ch:10-hilbert-spaces@proof-3 ¶

Supports

depends_on theorem 12.14 Closest point in a closed convex set ¶
- 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 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 12.69 Operator with a domain ¶
      - depends_on definition 12.70 Graph; closed and closable operators ¶
      - depends_on definition 12.72 Symmetric; self-adjoint ¶
      - depends_on definition 12.71 Adjoint of a densely defined operator ¶
    - depends_on definition 12.103 Gelfand triple ¶
      - depends_on definition A.278 Countably Hilbert nuclear space ¶
      - depends_on definition 12.105 Generalized eigenvector ¶
      - depends_on example 12.104 The Schwartz triple ¶
      - 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 proposition 12.95 The tensor inner product is well defined and positive definite ¶
      - depends_on theorem A.229 Hilbert–Schmidt ¶
      - depends_on theorem 12.44 Hilbert–Schmidt: compact self-adjoint operators ¶
      - 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 theorem 12.59 Spectral theorem for a bounded self-adjoint operator ¶
  - 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 proposition A.280 Direct-integral form of the spectral theorem ¶
  - 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 12.89 Reduction is commutation ¶
    - depends_on proposition 12.21 Characterization of orthogonal projections ¶
      - depends_on definition 12.41 The operator classes ¶
      - 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.233 Construction of the system ¶ ↺
  - 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.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 ¶

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
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declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	Inner product	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:168
`depends_on`	→	eq:lin-norm-assoc	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:168
`depends_on`	←	Closest point in a closed convex set	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:348
`proves`	←	ch:10-hilbert-spaces@proof-3	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:171