proposition 12.4 Cauchy–Schwarz and continuity of the inner product

open in the book · parts/02-mathematical-methods/10-hilbert-spaces.tex:105 · p. 414

Rests on

depends_on definition 12.2 Hilbert space ¶
- 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 4.32 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 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 5.20 Cauchy–Schwarz inequality ¶
- depends_on definition 5.14 Linear independence ¶
  - depends_on definition 5.5 Linear combination ¶
    - depends_on definition 4.33 Vector space ¶ ↺
- depends_on definition 5.18 Inner product ¶ ↺
- proves proof ch:03-linear-algebra-representations@proof-4 ¶
proves proof ch:10-hilbert-spaces@proof-1 ¶

Supports

depends_on corollary 12.5 Continuity of the norm and of orthogonality ¶
- depends_on proposition 12.17 The complement is always a closed subspace ¶
  - 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 theorem 12.18 Projection theorem ¶
    - depends_on corollary 12.19 Double complement; the density criterion ¶ ↺
    - 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 proposition 12.21 Characterization of orthogonal projections ¶
    - 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 lemma A.247 Integration against a projection-valued measure ¶
      - depends_on proposition A.246 Bounded Borel functional calculus ¶
      - depends_on proposition A.245 The measures $\mu_{x,y}$ ¶
      - depends_on theorem 12.38 Existence and uniqueness of the adjoint ¶
depends_on lemma A.606 $F$ is entire, and is the overlap in disguise ¶
- depends_on lemma A.607 A zero-free entire function is an exponential ¶
  - depends_on proposition A.609 The exponent is a quadratic polynomial ¶
    - depends_on proposition A.611 The constraint on $\Gamma$ ¶
      - depends_on proposition A.612 The dictionary between $A$ and $\Gamma$ ¶
depends_on lemma A.254 Riemann integral of a continuous curve ¶
- depends_on lemma A.255 Smoothed vectors lie in the domain ¶
  - depends_on proposition A.256 Density ¶
    - depends_on proposition A.259 The generator is self-adjoint ¶
      - depends_on lemma A.260 Cayley transform of a self-adjoint operator ¶
      - depends_on proposition A.263 The two constructions are inverse ¶
- depends_on lemma A.257 Integrated form of the equation of motion ¶
  - depends_on proposition A.258 The generator is closed ¶
    - depends_on proposition A.259 The generator is self-adjoint ¶ ↺
- depends_on lemma A.583 Absolutely convergent operator-valued integrals ¶
  - depends_on definition A.585 The Gaussian average of the Weyl operators ¶
    - depends_on theorem A.587 The Gaussian average is a non-zero projector that absorbs the Weyl operators ¶
      - depends_on lemma A.590 The Gram matrix is universal ¶
      - depends_on proposition A.589 Cyclic subspaces and the rank of the average ¶
      - depends_on proposition A.592 The vacuum of the Schrödinger system ¶
depends_on lemma A.583 Absolutely convergent operator-valued integrals ¶ ↺
depends_on proposition 12.102 Neither plane waves nor deltas are in $L^{2}$ ¶
- depends_on example 12.104 The Schwartz triple ¶ ↺
depends_on theorem 25.53 Properties of the Wigner function ¶
- depends_on lemma A.603 Overlaps and the resolution of unity ¶
  - depends_on proposition A.611 The constraint on $\Gamma$ ¶ ↺
  - depends_on proposition A.604 Non-negativity forbids a vanishing overlap ¶
    - depends_on proposition A.609 The exponent is a quadratic polynomial ¶ ↺

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`	→	Hilbert space	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:115
`depends_on`	→	Cauchy–Schwarz inequality	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:115
`depends_on`	←	Continuity of the norm and of orthogonality	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:146
`depends_on`	←	$F$ is entire, and is the overlap in disguise	declared	appendices/A-long-proofs.tex:29014
`depends_on`	←	Riemann integral of a continuous curve	declared	appendices/A-long-proofs.tex:12667
`depends_on`	←	Absolutely convergent operator-valued integrals	declared	appendices/A-long-proofs.tex:27912
`depends_on`	←	Neither plane waves nor deltas are in $L^{2}$	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2788
`depends_on`	←	Properties of the Wigner function	declared	parts/03-classical-mechanics/08-poisson-quantum-bridge.tex:1902
`proves`	←	ch:10-hilbert-spaces@proof-1	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:118