equation 5.44 eq:lin-norm-assoc

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• 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 proposition 12.6 Parallelogram law and polarization ¶
  • depends_on theorem 12.14 Closest point in a closed convex set ¶ ↺
• depends_on proposition 5.28 Gram–Schmidt ¶
  • 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 theorem 5.43 The four fundamental subspaces ¶
  • depends_on proposition 12.23 Gram–Schmidt in a Hilbert space ¶
    • depends_on corollary 12.24 Separable spaces have countable orthonormal families ¶
      • depends_on theorem 12.33 Every separable Hilbert space is $\ell^{2}$ ¶ ↺
    • 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 5.42 The adjoint exists, is unique, and is linear ¶
    • depends_on corollary 5.82 Spectral theorem for a normal operator ¶
  • depends_on proposition 5.155 Averaging trick ¶
    • depends_on theorem 101.62 The mixing matrix is unitary ¶ ↺
  • 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 corollary 5.91 The operator norm of a real array ¶
        • depends_on proposition 5.92 The eigenvalues do not control the norm ¶
      • depends_on corollary 5.89 Rayleigh–Ritz ¶ ↺
  • 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 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 proposition A.604 Non-negativity forbids a vanishing overlap ¶ ↺
    • 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 proposition 5.130 An orthogonal transformation is an isometry ¶
  • depends_on theorem 5.133 Isotropic Cartesian tensors of rank at most four ¶
    • depends_on lemma A.742 Isotropic representation ¶
      • depends_on corollary A.743 Two scalar functions instead of a tensor field ¶
        • depends_on lemma A.748 The second-order structure function ¶
        • depends_on lemma A.747 The third-order structure functions ¶
        • depends_on proposition A.746 The Kármán–Howarth equation in classical form ¶
      • depends_on lemma A.747 The third-order structure functions ¶ ↺
      • depends_on lemma A.744 Vanishing of the pressure–velocity correlation ¶
        • depends_on theorem A.745 Kármán–Howarth–Monin relation ¶
    • depends_on lemma A.671 The three products are independent ¶
      • 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 30.26 Isotropic Cartesian tensors of rank four ¶
      • depends_on phenomenon 30.28 Hooke's law ¶
        • depends_on phenomenon 30.32 Poisson contraction ¶
        • depends_on remark 30.78 Anharmonicity: what the quadratic energy also suppresses ¶
    • 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 ¶