proposition 5.20 Cauchy–Schwarz inequality

open in the book · parts/02-mathematical-methods/03-linear-algebra-representations.tex:1017 · p. 108

Rests on

depends_on definition 5.14 Linear independence ¶
- 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 4.4 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 5.18 Inner product ¶
- depends_on definition 4.33 Vector space ¶ ↺
proves proof ch:03-linear-algebra-representations@proof-4 ¶

Supports

depends_on proposition 12.4 Cauchy–Schwarz and continuity of the inner product ¶
- 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 12.69 Operator with a domain ¶
      - depends_on definition 12.103 Gelfand triple ¶
      - depends_on definition 12.50 Point, continuous and residual spectrum ¶
      - depends_on definition 12.71 Adjoint of a densely defined operator ¶
      - depends_on theorem 12.30 Completeness, expansion, Parseval ¶
      - depends_on theorem 12.55 The spectrum of a self-adjoint operator is real ¶
    - 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 definition 12.20 Orthogonal projection operator ¶
      - depends_on lemma A.233 Construction of the system ¶
      - 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 proposition 12.21 Characterization of orthogonal projections ¶
      - depends_on theorem 12.46 Riesz representation ¶
- 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 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.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.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 example A.283 Momentum on the line ¶
    - 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 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.

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Edges

type	direction	node	provenance	where
`depends_on`	→	Linear independence	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:1026
`depends_on`	→	Inner product	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:1026
`depends_on`	←	Cauchy–Schwarz and continuity of the inner product	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:115
`proves`	←	ch:03-linear-algebra-representations@proof-4	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:1029