definition 5.18 Inner product

open in the book · parts/02-mathematical-methods/03-linear-algebra-representations.tex:884 · p. 107

ground object -- no derivation owed

Rests on

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 4.2 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 4.3 Algebraic structure ¶ ↺
  - depends_on definition 4.20 Monoid ¶
    - depends_on definition 4.9 Neutral element ¶ ↺
    - depends_on definition 4.18 Semigroup ¶
      - depends_on definition 4.6 Associativity ¶
      - depends_on definition 4.4 Internal binary operation; magma ¶ ↺
  - depends_on definition 4.30 Ring ¶
    - depends_on definition 4.8 Commutativity; abelian structure ¶ ↺
    - depends_on definition 4.28 Distributivity ¶
      - depends_on definition 4.3 Algebraic structure ¶ ↺
    - depends_on definition 4.21 Group ¶
      - depends_on definition 4.11 Inverse element ¶ ↺
      - depends_on definition 4.20 Monoid ¶ ↺
    - 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 corollary 5.29 Orthogonal decomposition ¶
- depends_on proposition 5.153 prop:rep-unitary-completely-reducible ¶
  - depends_on theorem 101.62 The mixing matrix is unitary ¶
    - depends_on proposition 101.63 Parameter counting, and why three generations permit $CP$ violation ¶
    - depends_on theorem 103.11 GIM suppression at one loop ¶
      - depends_on phenomenon 103.42 $CP$ violation in charm decays ¶
- depends_on theorem 5.43 The four fundamental subspaces ¶
depends_on definition 7.96 Plane ¶
depends_on definition 12.2 Hilbert space ¶
- depends_on definition A.278 Countably Hilbert nuclear space ¶
  - depends_on theorem A.281 Nuclear spaces embed by Hilbert–Schmidt maps; quoted ¶
    - depends_on proposition A.282 The fibre maps are continuous on $\Phi$ ¶
- depends_on definition 12.32 Separable Hilbert space ¶
  - depends_on lemma A.250 Decomposition into cyclic subspaces ¶
    - depends_on proposition A.280 Direct-integral form of the spectral theorem ¶
      - depends_on proposition A.282 The fibre maps are continuous on $\Phi$ ¶ ↺
- depends_on definition 12.94 Tensor product of Hilbert spaces ¶
  - depends_on definition 12.99 Product and entangled vectors ¶
    - depends_on example 12.100 Entangled vectors exist ¶
  - depends_on example 12.98 Two particles in three-dimensional space ¶
  - 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 definition 10.85 Sobolev space ¶
  - depends_on definition A.450 The space $W^{1,r}(a,b)$ ¶
    - depends_on definition A.462 The weighted space and the energy space ¶
      - depends_on lemma A.463 Poincaré inequality; $B_{K}$ is an inner product ¶
    - depends_on lemma A.453 Uniform bound and uniform Hölder continuity ¶
      - depends_on theorem A.455 Compact embedding of $W^{1,r}(a,b)$ into the continuous functions ¶
    - depends_on lemma A.452 Young and Hölder ¶
      - depends_on lemma A.489 Courant–Lebesgue ¶
      - depends_on lemma A.463 Poincaré inequality; $B_{K}$ is an inner product ¶ ↺
      - depends_on lemma A.453 Uniform bound and uniform Hölder continuity ¶ ↺
  - depends_on definition 10.87 Weak form of an elliptic problem ¶
    - depends_on corollary 10.89 Existence for the elliptic problem ¶
    - depends_on proposition 10.91 Dirichlet principle ¶
    - depends_on theorem 10.88 Lax–Milgram ¶
      - depends_on corollary 10.89 Existence for the elliptic problem ¶ ↺
      - depends_on proposition 10.91 Dirichlet principle ¶ ↺
- depends_on example 12.9 The sequence space $\ell^{2}$ ¶
  - depends_on definition 12.84 External direct sum ¶
    - depends_on proposition 12.85 The direct sum is a Hilbert space ¶
      - depends_on proposition 12.87 Expansion in an orthogonal decomposition ¶
  - depends_on proposition 12.10 $\ell^{2}$ is complete ¶
    - depends_on proposition 12.85 The direct sum is a Hilbert space ¶ ↺
    - depends_on theorem 12.33 Every separable Hilbert space is $\ell^{2}$ ¶
- depends_on example 12.11 The function space $L^{2}$ ¶
  - 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 corollary A.275 Momentum on $[0,\infty)$: no extension ¶
  - depends_on example 12.56 Multiplication by the coordinate: spectrum without eigenvectors ¶
  - depends_on example 12.25 Orthogonal polynomials ¶
  - depends_on example 12.110 The Schrödinger system ¶
    - depends_on proposition A.592 The vacuum of the Schrödinger system ¶
    - depends_on theorem A.579 Stone–von Neumann ¶
  - 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 example 12.98 Two particles in three-dimensional space ¶ ↺
  - 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 12.12 Riesz–Fischer ¶
    - depends_on lemma A.464 $H_{E}$ is a Hilbert space ¶
- 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 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 theorem 12.18 Projection theorem ¶
  - 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 lemma A.254 Riemann integral of a continuous curve ¶ ↺
  - 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 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 12.28 Convergence criterion for orthogonal series ¶
  - 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 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 proposition 12.51 The three cases are exclusive and exhaustive ¶
- depends_on theorem 16.70 The direct method ¶
  - depends_on lemma 16.72 Convexity implies weak lower semicontinuity ¶
    - depends_on theorem A.448 Tonelli ¶
    - depends_on theorem A.456 Convexity in $q$ implies weak lower semicontinuity ¶
    - depends_on theorem 16.73 Tonelli ¶
  - depends_on theorem A.477 Douglas; Radó ¶
  - depends_on theorem A.448 Tonelli ¶ ↺
  - depends_on theorem 16.81 Douglas; Radó ¶
  - depends_on theorem 16.73 Tonelli ¶ ↺
- depends_on theorem 12.14 Closest point in a closed convex set ¶
  - depends_on theorem 12.18 Projection theorem ¶ ↺
- … 1 more
depends_on definition 12.94 Tensor product of Hilbert spaces ¶ ↺
depends_on definition 5.41 Adjoint ¶
- depends_on proposition 5.42 The adjoint exists, is unique, and is linear ¶
  - depends_on corollary 5.82 Spectral theorem for a normal operator ¶
- 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 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 definition 12.58 Projection-valued measure ¶
      - depends_on definition 12.60 Functional calculus ¶
      - depends_on lemma A.247 Integration against a projection-valued measure ¶
      - 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 12.64 Strongly continuous one-parameter unitary group ¶
      - depends_on definition A.580 Weyl operator ¶
      - depends_on definition 12.109 Weyl system ¶
      - depends_on lemma A.255 Smoothed vectors lie in the domain ¶ ↺
      - depends_on lemma A.581 Composition law ¶
      - depends_on lemma A.582 Joint strong continuity ¶
      - depends_on proposition A.256 Density ¶ ↺
      - depends_on proposition 12.65 Exponential of a bounded self-adjoint operator ¶
      - depends_on proposition 12.67 The generator is symmetric, and generates the motion ¶
      - depends_on theorem A.253 Stone ¶
      - depends_on theorem 12.66 Stone ¶
      - depends_on theorem 25.34 Stone–von Neumann ¶
    - depends_on lemma A.231 Restriction to an invariant closed subspace ¶
      - depends_on lemma A.233 Construction of the system ¶
    - depends_on lemma A.230 Sequential characterisation ¶
      - depends_on lemma A.232 Attainment ¶
      - depends_on lemma A.231 Restriction to an invariant closed subspace ¶ ↺
      - depends_on lemma A.234 The eigenvalues tend to zero, with finite multiplicity ¶
    - depends_on proposition 12.42 Elementary consequences ¶
      - depends_on lemma A.232 Attainment ¶ ↺
      - depends_on proposition 12.43 Norm of a self-adjoint operator ¶
    - 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 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 12.71 Adjoint of a densely defined operator ¶
    - 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 12.72 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 proposition 12.73 The adjoint is always closed ¶
      - depends_on definition 12.78 Essential self-adjointness ¶ ↺
      - depends_on lemma A.260 Cayley transform of a self-adjoint 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 A.243 Continuous functional calculus ¶
    - 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 theorem 5.43 The four fundamental subspaces ¶ ↺
- depends_on theorem 5.80 Simultaneous diagonalization of commuting self-adjoint operators ¶
  - depends_on corollary 5.82 Spectral theorem for a normal operator ¶ ↺
- depends_on theorem 5.79 Spectral theorem for a self-adjoint operator ¶
  - depends_on definition 30.20 Pressure and deviatoric stress ¶
    - depends_on definition 30.31 The engineering moduli ¶
      - depends_on phenomenon 30.70 Hertzian contact ¶
      - depends_on phenomenon 30.60 Coulomb's law of torsion ¶
      - depends_on proposition 30.33 Conversion among the moduli ¶
      - depends_on proposition 30.51 Moment and curvature ¶
      - depends_on remark A.676 The third cubic constant is measured, and it is not small ¶
      - depends_on theorem A.695 Boussinesq's point-force solution, quoted ¶
    - depends_on definition 30.79 Maxwell viscoelasticity ¶
      - depends_on proposition 30.80 A Maxwell material is a solid or a fluid depending on how long you watch ¶
    - depends_on proposition 30.74 The maximum shear stress, and its indifference to pressure ¶
      - depends_on phenomenon 30.73 Yield and brittle fracture ¶
      - depends_on remark A.705 The dangerous stress is below the surface ¶
      - depends_on remark 30.75 Two criteria, one uncited ¶
    - depends_on remark 30.75 Two criteria, one uncited ¶ ↺
  - depends_on proposition 30.9 Principal strains ¶
    - depends_on example 30.10 Simple shear and pure shear are the same strain ¶
  - depends_on theorem 5.80 Simultaneous diagonalization of commuting self-adjoint operators ¶ ↺
  - depends_on theorem 5.84 Spectral theorem for a real symmetric operator ¶
    - depends_on corollary 5.91 The operator norm of a real array ¶
      - depends_on proposition 5.92 The eigenvalues do not control the norm ¶
    - depends_on lemma A.610 Block positivity ¶
      - depends_on proposition A.611 The constraint on $\Gamma$ ¶ ↺
    - depends_on lemma A.601 Several variables, complex symmetric matrix ¶
      - depends_on proposition A.612 The dictionary between $A$ and $\Gamma$ ¶
      - depends_on proposition A.602 The Wigner function of a Gaussian is a positive Gaussian ¶
    - depends_on lemma A.600 The real part of an inverse ¶
      - depends_on lemma A.601 Several variables, complex symmetric matrix ¶ ↺
      - depends_on proposition A.612 The dictionary between $A$ and $\Gamma$ ¶ ↺
    - depends_on proposition 5.90 Polar decomposition ¶
    - depends_on proposition 5.94 Principal axes of a real quadratic form ¶
      - depends_on proposition 5.95 The focal polar equation of a conic ¶
    - depends_on proposition 5.92 The eigenvalues do not control the norm ¶ ↺
    - depends_on theorem 5.86 Simultaneous diagonalization of a definite pencil ¶
      - depends_on corollary 5.89 Rayleigh–Ritz ¶
      - depends_on proposition 5.88 Rayleigh quotient of a definite pencil ¶
  - depends_on theorem 28.33 Normal modes ¶
    - depends_on example 28.39 The linear triatomic molecule ¶
    - depends_on proposition 28.40 Modes and dispersion of the loaded string ¶
    - depends_on remark 28.34 Part II owes the simultaneous diagonalization of two quadratic forms ¶
  - depends_on theorem 29.16 Principal axes ¶
    - depends_on definition 29.19 Inertia ellipsoid ¶
      - depends_on theorem 29.29 Poinsot's construction ¶
    - depends_on definition 29.35 The Lagrange top ¶
    - depends_on definition 29.18 Classification of tops ¶
      - depends_on proposition 29.28 Free rotation of a symmetric body ¶
    - depends_on phenomenon 29.31 The intermediate-axis instability ¶
      - depends_on remark 29.32 What the linearization is entitled to conclude ¶
    - depends_on proposition 29.33 Dissipation drives a free body to its largest moment ¶
    - depends_on proposition 29.17 Triangle inequalities ¶
      - depends_on proposition 29.22 Perpendicular-axis theorem ¶
    - depends_on theorem 29.25 Euler's equations ¶
      - depends_on phenomenon 29.46 A fast spinning top does not fall ¶
      - depends_on phenomenon 29.31 The intermediate-axis instability ¶ ↺
      - depends_on proposition 29.26 The two integrals of torque-free motion ¶
      - depends_on proposition 29.28 Free rotation of a symmetric body ¶ ↺
      - depends_on proposition 24.41 The free rigid body is a Lie–Poisson system ¶
depends_on definition 5.32 Levi–Civita symbol; cross product ¶
- depends_on definition 5.131 Isotropic Cartesian tensor ¶
  - depends_on definition A.672 The octahedral rotation group ¶
    - depends_on proposition A.673 Cubic tensors of rank four ¶
      - depends_on theorem A.675 Three constants for a cubic crystal, two for an isotropic solid ¶
  - depends_on lemma 5.132 Parity constraint ¶
    - depends_on theorem 5.133 Isotropic Cartesian tensors of rank at most four ¶
      - depends_on lemma A.742 Isotropic representation ¶
      - depends_on lemma A.671 The three products are independent ¶
      - depends_on lemma 30.26 Isotropic Cartesian tensors of rank four ¶
      - depends_on remark A.677 The lower ranks, and why an isotropic solid is not piezoelectric ¶
      - depends_on remark 30.27 Two results this chapter borrows from Part II ¶
      - depends_on remark 13.7 What the classification is used for ¶
- depends_on lemma 5.34 Contraction of two Levi–Civita symbols ¶
  - depends_on proposition 5.35 The identities of the vector algebra of $\R^{3}$ ¶
- depends_on lemma 5.33 Determinant through the Levi–Civita symbol ¶
  - depends_on proposition 5.35 The identities of the vector algebra of $\R^{3}$ ¶ ↺
  - depends_on theorem 5.133 Isotropic Cartesian tensors of rank at most four ¶ ↺
depends_on definition 5.24 Orthogonal vectors ¶
- depends_on corollary 12.5 Continuity of the norm and of orthogonality ¶ ↺
- depends_on definition 12.16 Orthogonal complement ¶
  - depends_on definition 12.29 Orthonormal basis ¶
    - depends_on definition 12.86 Internal orthogonal decomposition ¶
      - depends_on proposition 12.87 Expansion in an orthogonal decomposition ¶ ↺
  - depends_on definition 12.86 Internal orthogonal decomposition ¶ ↺
  - depends_on proposition 12.17 The complement is always a closed subspace ¶ ↺
- depends_on definition 5.26 Orthogonal basis ¶
  - depends_on definition 5.27 Orthonormal basis ¶
    - depends_on definition 12.26 Orthonormal system; Fourier coefficients ¶
      - depends_on definition 12.29 Orthonormal basis ¶ ↺
      - depends_on proposition 12.27 Best approximation and Bessel's inequality ¶
      - depends_on proposition 12.28 Convergence criterion for orthogonal series ¶ ↺
    - depends_on definition 5.32 Levi–Civita symbol; cross product ¶ ↺
    - depends_on proposition 12.23 Gram–Schmidt in a Hilbert space ¶
      - depends_on corollary 12.24 Separable spaces have countable orthonormal families ¶
      - depends_on example 12.25 Orthogonal polynomials ¶ ↺
      - depends_on proposition 12.95 The tensor inner product is well defined and positive definite ¶ ↺
    - depends_on proposition 5.28 Gram–Schmidt ¶
      - depends_on corollary 5.29 Orthogonal decomposition ¶ ↺
      - depends_on proposition 12.23 Gram–Schmidt in a Hilbert space ¶ ↺
      - depends_on proposition 5.42 The adjoint exists, is unique, and is linear ¶ ↺
      - depends_on proposition 5.155 Averaging trick ¶
      - depends_on theorem 5.86 Simultaneous diagonalization of a definite pencil ¶ ↺
      - depends_on theorem 5.84 Spectral theorem for a real symmetric operator ¶ ↺
depends_on definition 5.144 Unitary representation ¶
- depends_on lemma 5.150 Invariance of the orthogonal complement ¶
  - depends_on proposition 5.151 prop:rep-not-totally-reducible ¶
  - depends_on proposition 5.153 prop:rep-unitary-completely-reducible ¶ ↺
- depends_on proposition 5.155 Averaging trick ¶ ↺
depends_on example 12.9 The sequence space $\ell^{2}$ ¶ ↺
depends_on example 12.11 The function space $L^{2}$ ¶ ↺
depends_on proposition 12.6 Parallelogram law and polarization ¶
- depends_on theorem 12.14 Closest point in a closed convex set ¶ ↺
depends_on proposition 5.63 prop:lin-dual-inner-product ¶
… 5 more

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Edges

type	direction	node	provenance	where
`depends_on`	→	Vector space	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:923
`depends_on`	←	Orthogonal decomposition	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:1363
`depends_on`	←	Plane	declared	parts/02-mathematical-methods/05-real-analysis.tex:2891
`depends_on`	←	Hilbert space	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:74
`depends_on`	←	Tensor product of Hilbert spaces	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:2596
`depends_on`	←	Adjoint	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:1875
`depends_on`	←	Levi–Civita symbol; cross product	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:1489
`depends_on`	←	Orthogonal vectors	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:1248
`depends_on`	←	Unitary representation	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:6233
`depends_on`	←	The sequence space $\ell^{2}$	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:245
`depends_on`	←	The function space $L^{2}$	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:286
`depends_on`	←	Parallelogram law and polarization	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:168
`depends_on`	←	prop:lin-dual-inner-product	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:2794
`depends_on`	←	Gram criterion	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:1424
`depends_on`	←	Gram–Schmidt	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:1312
`depends_on`	←	Cauchy–Schwarz inequality	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:1026
`depends_on`	←	Averaging trick	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:6603
`depends_on`	←	An orthogonal transformation is an isometry	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:5658