definition 4.33 Vector space

open in the book · parts/02-mathematical-methods/02-algebraic-structures.tex:838 · p. 67

ground object -- no derivation owed

Rests on

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 4.1 Cartesian product ¶
      - depends_on definition 3.43 Map ¶
  - 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 4.3 Algebraic structure ¶ ↺
    - 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 definition 4.34 Algebra ¶
depends_on definition 5.121 Algebra ¶
- depends_on definition 5.128 Derivation ¶
  - depends_on remark 22.31 The bracket is a Lie algebra structure ¶
    - depends_on remark 22.39 Poisson's theorem does not manufacture new constants ¶
- depends_on definition 5.124 Ideal ¶
  - depends_on proposition 5.164 The kernel is an ideal ¶
    - depends_on corollary 5.168 cor:rep-adjoint-faithful ¶
- depends_on definition 5.127 Lie algebra ¶
  - depends_on definition 24.40 Lie–Poisson bracket ¶
    - depends_on proposition 24.41 The free rigid body is a Lie–Poisson system ¶
      - depends_on remark 24.48 Reduction is what physicists do without saying so ¶
      - depends_on remark 24.42 The intermediate axis ¶
    - depends_on remark 29.27 Euler's equations as a Lie–Poisson system ¶
  - depends_on proposition 5.167 Adjoint representation ¶
    - depends_on corollary 5.168 cor:rep-adjoint-faithful ¶ ↺
  - depends_on proposition 5.139 Dimension of the symplectic group ¶
    - depends_on proposition 24.5 Properties of the symplectic group ¶
      - depends_on remark 24.6 Connectedness, and what it rests on ¶
      - depends_on remark 24.7 Contrast with the orthogonal group ¶
    - depends_on remark 22.21 The count ¶
  - depends_on remark 22.31 The bracket is a Lie algebra structure ¶ ↺
- depends_on definition 5.122 Linear hull ¶
  - depends_on definition 5.124 Ideal ¶ ↺
  - depends_on definition 5.123 Subalgebra ¶
- depends_on definition 5.123 Subalgebra ¶ ↺
- depends_on definition 5.162 Representation of an algebra ¶
  - depends_on definition 5.163 Faithful, equivalent, invariant ¶
    - depends_on corollary 5.168 cor:rep-adjoint-faithful ¶ ↺
    - depends_on proposition 5.164 The kernel is an ideal ¶ ↺
  - depends_on proposition 5.167 Adjoint representation ¶ ↺
  - depends_on proposition 5.164 The kernel is an ideal ¶ ↺
- depends_on proposition 5.125 prop:lin-matrix-algebra ¶
  - depends_on corollary 5.168 cor:rep-adjoint-faithful ¶ ↺
  - depends_on definition 5.162 Representation of an algebra ¶ ↺
  - depends_on definition 9.21 Matrix exponential ¶
    - depends_on proposition 9.22 The exponential and its derivative ¶
      - depends_on theorem 9.23 Solution of a constant-coefficient system ¶
  - depends_on proposition 5.139 Dimension of the symplectic group ¶ ↺
depends_on definition 5.101 External direct sum ¶
- 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 lemma A.250 Decomposition into cyclic subspaces ¶
- depends_on definition 5.103 Sum of subspaces; internal direct sum ¶
  - depends_on corollary 5.108 The sum is the right operation ¶
    - depends_on example 5.109 Two lines in physical space ¶
  - depends_on proposition 5.104 Criterion for a direct sum ¶
  - depends_on proposition 5.105 Grassmann's formula ¶
- depends_on proposition 5.102 Dimensions add ¶
  - depends_on proposition 5.104 Criterion for a direct sum ¶ ↺
depends_on definition 5.18 Inner product ¶
- 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 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 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.453 Uniform bound and uniform Hölder continuity ¶
      - depends_on lemma A.452 Young and Hölder ¶
    - 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 example 12.9 The sequence space $\ell^{2}$ ¶
    - depends_on definition 12.84 External direct sum ¶ ↺
    - 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 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 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 lemma A.257 Integrated form of the equation of motion ¶
      - depends_on proposition A.258 The generator is closed ¶
    - depends_on lemma A.583 Absolutely convergent operator-valued integrals ¶
      - depends_on definition A.585 The Gaussian average of 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 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 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 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 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 ¶
      - 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 ¶
  - … 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 definition 12.58 Projection-valued measure ¶
      - depends_on definition 12.64 Strongly continuous one-parameter unitary group ¶
      - depends_on lemma A.231 Restriction to an invariant closed subspace ¶ ↺
      - depends_on lemma A.230 Sequential characterisation ¶
      - depends_on proposition 12.42 Elementary consequences ¶
      - 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 definition 12.71 Adjoint of a densely defined operator ¶
      - depends_on definition 12.79 Deficiency subspaces and indices ¶
      - depends_on definition 12.72 Symmetric; self-adjoint ¶
      - depends_on proposition 12.73 The adjoint is always closed ¶
    - 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 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 definition 30.79 Maxwell viscoelasticity ¶
      - depends_on proposition 30.74 The maximum shear stress, and its indifference to pressure ¶
      - 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 lemma A.610 Block positivity ¶
      - depends_on lemma A.601 Several variables, complex symmetric matrix ¶
      - depends_on lemma A.600 The real part of an inverse ¶
      - depends_on proposition 5.90 Polar decomposition ¶
      - depends_on proposition 5.94 Principal axes of a real quadratic form ¶
      - 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 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 definition 29.35 The Lagrange top ¶
      - depends_on definition 29.18 Classification of tops ¶
      - depends_on phenomenon 29.31 The intermediate-axis instability ¶
      - depends_on proposition 29.33 Dissipation drives a free body to its largest moment ¶
      - depends_on proposition 29.17 Triangle inequalities ¶
      - depends_on theorem 29.25 Euler's equations ¶
- 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 lemma 5.132 Parity constraint ¶
      - depends_on theorem 5.133 Isotropic Cartesian tensors of rank at most four ¶
  - 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 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 5.32 Levi–Civita symbol; cross product ¶ ↺
      - depends_on proposition 12.23 Gram–Schmidt in a Hilbert space ¶
      - depends_on proposition 5.28 Gram–Schmidt ¶
- 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 theorem 101.62 The mixing matrix is unitary ¶ ↺
- 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
depends_on definition 5.5 Linear combination ¶
- depends_on definition 5.101 External direct sum ¶ ↺
- depends_on definition 5.12 Subspace generated by a set of vectors ¶
  - depends_on corollary 5.108 The sum is the right operation ¶ ↺
  - depends_on definition 5.15 Basis ¶
    - depends_on corollary 9.25 Normal modes of a diagonalizable system ¶
      - depends_on corollary 9.27 Linear stability ¶
      - depends_on proposition 9.29 The planar classification ¶
    - depends_on definition 7.97 Partial derivative; gradient ¶
      - depends_on definition 7.98 Functions of class $C^{1}$ ¶
      - depends_on definition 7.102 Differential of a function ¶
      - depends_on definition 7.119 Envelope of a family ¶
      - depends_on definition A.70 Pullback ¶
      - depends_on definition 32.19 Lyapunov function ¶
      - depends_on definition 30.2 Representative volume element; continuum field ¶
      - depends_on definition 22.29 Poisson bracket ¶
      - depends_on lemma A.73 Differentiation under the integral sign ¶
    - depends_on definition 5.17 Components of a vector ¶
    - depends_on definition 5.16 Dimension ¶
      - depends_on definition 5.142 Dimension of a representation ¶
      - depends_on proposition 5.54 prop:lin-same-dim-isomorphic ¶
    - depends_on definition 5.60 Functional specified on a basis ¶
    - depends_on definition 5.26 Orthogonal basis ¶ ↺
    - depends_on example 5.9 The matrices of a given shape ¶
    - depends_on lemma 5.38 Exchange and completion ¶
      - depends_on proposition 5.61 Annihilator of a subspace ¶
      - depends_on proposition 5.102 Dimensions add ¶ ↺
      - depends_on proposition 5.74 prop:lin-geometric-le-algebraic ¶
      - depends_on proposition 5.105 Grassmann's formula ¶ ↺
      - depends_on theorem 5.40 Rank–nullity ¶
    - depends_on lemma 5.137 The alternating top form is unique up to scale ¶
      - depends_on theorem 5.138 A symplectic transformation has determinant $+1$ ¶
    - depends_on proposition 5.102 Dimensions add ¶ ↺
    - depends_on proposition 5.105 Grassmann's formula ¶ ↺
    - depends_on proposition 5.45 prop:lin-matrix-unique ¶
      - depends_on proposition 5.125 prop:lin-matrix-algebra ¶ ↺
      - depends_on proposition 5.51 prop:lin-matrix-inverse ¶
      - depends_on proposition 5.54 prop:lin-same-dim-isomorphic ¶ ↺
      - depends_on proposition 5.167 Adjoint representation ¶ ↺
    - … 2 more
  - depends_on lemma 5.38 Exchange and completion ¶ ↺
- depends_on definition 5.14 Linear independence ¶
  - depends_on corollary 5.62 A functional that annihilates a set of constraints ¶
  - depends_on definition 5.15 Basis ¶ ↺
  - depends_on lemma 5.38 Exchange and completion ¶ ↺
  - depends_on lemma 13.136 Discrete subgroups of $\R^{f}$ ¶
  - depends_on proposition 5.75 Eigenvectors for distinct eigenvalues are independent ¶
    - depends_on theorem 5.77 Criterion for diagonalizability ¶
      - depends_on proposition 32.39 Stability of a periodic orbit ¶
      - depends_on theorem 32.13 Linear stability ¶
  - depends_on proposition 5.31 Gram criterion ¶
  - depends_on proposition 5.28 Gram–Schmidt ¶ ↺
  - depends_on proposition 5.20 Cauchy–Schwarz inequality ¶
    - depends_on proposition 12.4 Cauchy–Schwarz and continuity of the inner product ¶ ↺
- depends_on definition 5.7 Vector subspace ¶
  - depends_on corollary 5.29 Orthogonal decomposition ¶ ↺
  - depends_on definition 12.13 Convex set ¶
    - depends_on theorem 12.14 Closest point in a closed convex set ¶ ↺
  - depends_on definition 5.12 Subspace generated by a set of vectors ¶ ↺
  - depends_on definition 5.103 Sum of subspaces; internal direct sum ¶ ↺
  - depends_on definition 5.39 Kernel, image, nullity, rank ¶
    - depends_on definition 5.69 Eigenvector, eigenvalue, eigenspace ¶
      - depends_on corollary 9.25 Normal modes of a diagonalizable system ¶ ↺
      - depends_on definition 5.73 Algebraic and geometric multiplicity ¶
      - depends_on definition 9.30 Hyperbolic equilibrium ¶
      - depends_on proposition 5.75 Eigenvectors for distinct eigenvalues are independent ¶ ↺
      - depends_on theorem 5.77 Criterion for diagonalizability ¶ ↺
      - depends_on theorem 5.71 The eigenvalues are the roots of the characteristic polynomial ¶
    - depends_on lemma 5.97 Fitting splitting ¶
      - depends_on theorem 5.98 Primary decomposition ¶
    - depends_on proposition 5.49 Injective, surjective, invertible ¶
      - depends_on lemma 5.97 Fitting splitting ¶ ↺
      - depends_on proposition 5.31 Gram criterion ¶ ↺
      - depends_on proposition 5.2 Jacobi's formula, column form ¶
      - depends_on proposition 5.90 Polar decomposition ¶ ↺
      - depends_on proposition 5.130 An orthogonal transformation is an isometry ¶
      - depends_on theorem 5.71 The eigenvalues are the roots of the characteristic polynomial ¶ ↺
    - depends_on theorem 5.40 Rank–nullity ¶ ↺
  - depends_on definition 5.163 Faithful, equivalent, invariant ¶ ↺
  - depends_on definition 5.147 Invariant subspace ¶
    - depends_on definition 5.148 Irreducible representation ¶
      - depends_on proposition 5.161 Irreducible representations of an abelian group ¶
      - depends_on proposition 5.151 prop:rep-not-totally-reducible ¶ ↺
      - depends_on proposition 5.153 prop:rep-unitary-completely-reducible ¶ ↺
      - depends_on theorem 5.158 Schur's first lemma ¶
      - depends_on theorem 5.160 Schur's second lemma ¶
    - depends_on definition 5.149 Totally reducible representation ¶
      - depends_on proposition 5.151 prop:rep-not-totally-reducible ¶ ↺
      - depends_on proposition 5.153 prop:rep-unitary-completely-reducible ¶ ↺
    - depends_on lemma 5.150 Invariance of the orthogonal complement ¶ ↺
    - depends_on theorem 5.158 Schur's first lemma ¶ ↺
    - depends_on theorem 5.160 Schur's second lemma ¶ ↺
  - depends_on example 5.10 Functions on a set ¶
  - depends_on proposition 5.61 Annihilator of a subspace ¶ ↺
  - depends_on proposition 5.107 The union of two subspaces ¶
    - depends_on corollary 5.108 The sum is the right operation ¶ ↺
    - depends_on example 5.109 Two lines in physical space ¶ ↺
- depends_on example 5.8 The coordinate spaces $\R^{n}$ and $\C^{n}$ ¶
- depends_on proposition 5.107 The union of two subspaces ¶ ↺
depends_on definition 5.37 Linear transformation ¶
- depends_on definition 7.99 Differentiability at a point ¶
  - depends_on definition 7.102 Differential of a function ¶ ↺
  - depends_on lemma A.291 $h$ is differentiable, with the stated derivative ¶
  - depends_on proposition 7.104 Chain rule in several variables ¶
    - depends_on corollary 7.113 Inverse function theorem ¶
      - depends_on definition A.503 The substitution property ¶
      - depends_on lemma 7.116 Functions vanishing on a regular zero set ¶
      - depends_on lemma A.510 Every diffeomorphism factorises locally ¶
      - depends_on remark 7.114 Where the proof is, and why ¶
      - depends_on theorem 7.129 Change of variables in a multiple integral ¶
      - depends_on theorem 7.115 Constant rank ¶
    - depends_on corollary 7.110 Variable limits of integration ¶
    - depends_on corollary A.292 Inverse function theorem ¶
      - depends_on lemma A.549 The action is transitive ¶
      - depends_on lemma A.300 The boundary is well defined, and is a manifold ¶
      - depends_on proposition A.535 Existence of a slice ¶
      - depends_on proposition 22.32 Duality of the two brackets ¶
      - depends_on proposition 22.3 Invertibility of the Legendre map ¶
      - depends_on proposition 13.132 Simultaneous straightening of commuting fields ¶
      - depends_on remark 22.19 Which form exists ¶
      - depends_on theorem 13.63 Constant rank theorem ¶
    - depends_on definition 30.4 Material and spatial descriptions ¶
    - depends_on definition 31.4 Material derivative ¶
      - depends_on definition 31.6 Streamline, pathline, streakline ¶
      - depends_on lemma 31.11 Transport theorem for a material volume ¶
    - depends_on definition 22.26 Lagrange bracket ¶
    - depends_on definition 10.18 Complete integral; envelope ¶
      - depends_on proposition 10.19 An envelope of solutions is a solution ¶
    - depends_on lemma A.510 Every diffeomorphism factorises locally ¶ ↺
    - depends_on lemma A.504 Transitivity ¶
    - depends_on lemma A.306 The standard smooth bump ¶
      - depends_on lemma A.307 Partition of unity on a compact manifold ¶
    - depends_on proposition 7.136 Properties of conservative fields ¶
      - depends_on example 31.15 The stream function of a plane flow ¶
      - depends_on proposition 31.28 Potential flow reduces to Laplace's equation ¶
      - depends_on theorem 16.63 Hilbert's invariant integral ¶
      - depends_on theorem 30.15 Compatibility is sufficient on a simply connected body ¶
    - depends_on proposition 7.120 The envelope touches every member it meets ¶
    - … 9 more
  - depends_on theorem 7.100 $C^{1}$ implies differentiable ¶
    - depends_on lemma A.291 $h$ is differentiable, with the stated derivative ¶ ↺
    - depends_on remark 7.101 Partial derivatives alone do not suffice ¶
    - depends_on theorem 7.112 Implicit function theorem ¶
      - depends_on corollary 7.113 Inverse function theorem ¶ ↺
      - depends_on proposition 7.120 The envelope touches every member it meets ¶ ↺
      - depends_on proposition 7.117 Lagrange multipliers in finitely many variables ¶
      - depends_on remark 7.114 Where the proof is, and why ¶ ↺
    - depends_on theorem A.286 Implicit function theorem ¶
      - depends_on corollary A.292 Inverse function theorem ¶ ↺
      - depends_on corollary A.293 Solving one scalar equation for one coordinate ¶
      - depends_on lemma A.650 Flattening the constraints ¶
      - depends_on remark A.296 Both hypotheses are needed, and the conclusion is local ¶
      - depends_on remark 23.3 What Part II owes this chapter ¶
      - depends_on theorem 23.6 Jacobi ¶
      - depends_on theorem 13.63 Constant rank theorem ¶ ↺
      - depends_on theorem 13.62 Regular value theorem in codimension $k$ ¶
  - depends_on theorem 7.112 Implicit function theorem ¶ ↺
  - depends_on theorem A.286 Implicit function theorem ¶ ↺
- depends_on definition 12.35 Bounded operator; operator norm ¶
  - depends_on definition 12.45 Continuous linear functional; the dual ¶
    - depends_on corollary 12.47 $\mathcal{H}$ is its own dual, antilinearly ¶
      - depends_on definition 12.103 Gelfand triple ¶
    - depends_on definition 12.103 Gelfand triple ¶ ↺
    - depends_on theorem 12.46 Riesz representation ¶ ↺
  - 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 A.589 Cyclic subspaces and the rank of the average ¶
      - depends_on theorem 12.91 Schur's lemma, commutant form ¶
  - depends_on definition 12.49 Resolvent set; spectrum ¶
    - 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 lemma A.240 Spectral mapping for polynomials ¶
      - depends_on proposition A.241 The polynomial calculus is isometric ¶ ↺
    - depends_on lemma A.239 The norm of a self-adjoint operator lies in its spectrum ¶
      - depends_on proposition A.241 The polynomial calculus is isometric ¶ ↺
    - depends_on proposition 12.52 Neumann series; the spectrum is bounded ¶
      - depends_on lemma A.239 The norm of a self-adjoint operator lies in its spectrum ¶ ↺
      - depends_on proposition A.261 Spectral theorem for a unitary operator ¶
      - depends_on proposition 12.53 The resolvent set is open, the resolvent analytic ¶
      - depends_on theorem 12.54 The spectrum is compact and non-empty ¶
  - depends_on proposition 12.37 $\mathcal{B}(\mathcal{H})$ is a Banach algebra ¶
    - depends_on lemma A.240 Spectral mapping for polynomials ¶ ↺
    - depends_on proposition 12.65 Exponential of a bounded self-adjoint operator ¶ ↺
    - depends_on proposition 12.61 Uniqueness of the continuous functional calculus ¶
      - depends_on proposition A.243 Continuous functional calculus ¶
    - depends_on proposition 12.39 Algebra of the adjoint; the $C^{\ast}$ identity ¶ ↺
    - depends_on proposition 12.52 Neumann series; the spectrum is bounded ¶ ↺
    - depends_on theorem 12.38 Existence and uniqueness of the adjoint ¶ ↺
    - depends_on theorem 12.75 The canonical commutation relation admits no bounded solution ¶
      - depends_on corollary 12.76 Position and momentum are unbounded, and cannot be everywhere defined ¶
      - depends_on corollary 25.30 The relation cannot be realized by matrices ¶
      - depends_on definition 12.109 Weyl system ¶
  - depends_on proposition 12.36 Boundedness is continuity ¶
    - depends_on theorem 12.74 Hellinger–Toeplitz ¶
      - depends_on corollary 12.76 Position and momentum are unbounded, and cannot be everywhere defined ¶ ↺
  - depends_on theorem 12.75 The canonical commutation relation admits no bounded solution ¶ ↺
- depends_on definition 12.69 Operator with a domain ¶
  - depends_on definition 12.70 Graph; closed and closable operators ¶
    - depends_on proposition 12.73 The adjoint is always closed ¶ ↺
  - depends_on definition 12.72 Symmetric; self-adjoint ¶ ↺
  - depends_on definition 12.71 Adjoint of a densely defined operator ¶ ↺
- depends_on definition 5.41 Adjoint ¶ ↺
- depends_on definition 5.110 Bilinear map ¶
  - depends_on definition 12.94 Tensor product of Hilbert spaces ¶ ↺
  - depends_on definition 5.113 Non-degenerate form ¶
    - depends_on definition 5.117 Signature ¶
      - depends_on proposition 5.94 Principal axes of a real quadratic form ¶ ↺
    - depends_on definition 5.135 The symplectic group ¶
      - depends_on definition 22.40 Symplectic matrix ¶
      - depends_on definition 24.4 Symplectic group ¶
      - depends_on lemma 22.20 The symplectic condition ¶
      - depends_on proposition 5.139 Dimension of the symplectic group ¶ ↺
      - depends_on proposition 5.136 $\Sp(2n,\R)$ is a subgroup of $\GL(2n,\R)$ ¶
      - depends_on theorem 5.138 A symplectic transformation has determinant $+1$ ¶ ↺
    - depends_on proposition 5.114 prop:lin-bilinear-dual ¶
    - depends_on proposition 5.119 Normal form of a non-degenerate antisymmetric form ¶
      - depends_on definition 5.135 The symplectic group ¶ ↺
      - depends_on definition 24.2 Symplectic vector space ¶
      - depends_on proposition 26.19 Second-class constraints come in pairs ¶
      - depends_on proposition 24.3 Even dimension and the canonical basis ¶
      - depends_on theorem A.68 Darboux ¶
      - depends_on theorem 5.138 A symplectic transformation has determinant $+1$ ¶ ↺
  - depends_on definition 5.111 Symmetric and antisymmetric forms ¶
    - depends_on definition 28.32 The two quadratic forms ¶
    - depends_on proposition 5.112 Symmetric–antisymmetric splitting ¶
    - depends_on proposition 5.119 Normal form of a non-degenerate antisymmetric form ¶ ↺
    - depends_on theorem 5.116 Sylvester's law of inertia ¶
      - depends_on definition 5.117 Signature ¶ ↺
      - depends_on definition 10.6 Type of a second-order operator ¶
      - depends_on proposition 30.30 Stability bounds ¶
      - depends_on proposition 5.94 Principal axes of a real quadratic form ¶ ↺
      - depends_on remark 7.107 What the Hessian is for ¶
      - depends_on theorem 5.86 Simultaneous diagonalization of a definite pencil ¶ ↺
      - depends_on theorem 28.33 Normal modes ¶ ↺
      - depends_on theorem 10.8 The type is a coordinate invariant ¶
  - depends_on lemma 5.137 The alternating top form is unique up to scale ¶ ↺
  - depends_on proposition 5.114 prop:lin-bilinear-dual ¶ ↺
  - depends_on proposition 5.112 Symmetric–antisymmetric splitting ¶ ↺
  - depends_on proposition 5.119 Normal form of a non-degenerate antisymmetric form ¶ ↺
  - depends_on theorem 5.116 Sylvester's law of inertia ¶ ↺
- depends_on definition 5.128 Derivation ¶ ↺
- depends_on definition 5.56 Endomorphism ¶
  - depends_on definition 5.57 Automorphism ¶
  - depends_on definition 5.70 Characteristic polynomial ¶
    - depends_on definition 5.73 Algebraic and geometric multiplicity ¶ ↺
    - depends_on theorem 5.71 The eigenvalues are the roots of the characteristic polynomial ¶ ↺
  - depends_on definition 5.76 Diagonalizable operator ¶
    - depends_on corollary 5.99 Semisimple and nilpotent parts ¶
    - depends_on theorem 5.77 Criterion for diagonalizability ¶ ↺
  - depends_on definition 5.69 Eigenvector, eigenvalue, eigenspace ¶ ↺
- depends_on definition 5.46 Functional ¶
  - depends_on definition 12.45 Continuous linear functional; the dual ¶ ↺
  - depends_on definition 5.59 Dual space ¶
    - depends_on definition 5.66 Pushforward ¶
      - depends_on proposition 5.67 Adjointness of pullback and pushforward ¶
    - depends_on proposition 5.61 Annihilator of a subspace ¶ ↺
    - depends_on proposition 5.114 prop:lin-bilinear-dual ¶ ↺
    - depends_on proposition 5.63 prop:lin-dual-inner-product ¶ ↺
    - depends_on proposition 5.67 Adjointness of pullback and pushforward ¶ ↺
  - depends_on definition 5.60 Functional specified on a basis ¶ ↺
- depends_on definition 5.47 Inverse of a linear transformation ¶
  - depends_on definition 12.49 Resolvent set; spectrum ¶ ↺
  - depends_on proposition 5.48 Existence, uniqueness, and linearity of the inverse ¶
    - depends_on proposition 5.49 Injective, surjective, invertible ¶ ↺
    - depends_on proposition 5.51 prop:lin-matrix-inverse ¶ ↺
    - depends_on proposition 5.136 $\Sp(2n,\R)$ is a subgroup of $\GL(2n,\R)$ ¶ ↺
- depends_on definition 5.53 Isomorphism ¶
  - depends_on definition 5.57 Automorphism ¶ ↺
  - depends_on proposition 5.104 Criterion for a direct sum ¶ ↺
  - depends_on proposition 5.54 prop:lin-same-dim-isomorphic ¶ ↺
  - depends_on theorem 5.160 Schur's second lemma ¶ ↺
- depends_on definition 5.39 Kernel, image, nullity, rank ¶ ↺
- depends_on definition 5.66 Pushforward ¶ ↺
- … 6 more
depends_on definition 5.19 Norm ¶
- depends_on definition 12.35 Bounded operator; operator norm ¶ ↺
- depends_on definition 5.22 Metric associated with a norm ¶
- depends_on definition 5.25 Unit vector ¶
  - depends_on definition 5.27 Orthonormal basis ¶ ↺
- depends_on definition 9.21 Matrix exponential ¶ ↺
- depends_on proposition 12.8 Absolutely convergent series test ¶
  - depends_on proposition 12.37 $\mathcal{B}(\mathcal{H})$ is a Banach algebra ¶ ↺
  - depends_on theorem 12.12 Riesz–Fischer ¶ ↺
depends_on definition 5.3 Scalar ¶
depends_on definition 5.115 Sesquilinear form ¶
depends_on definition 5.4 Vector ¶
depends_on example 5.8 The coordinate spaces $\R^{n}$ and $\C^{n}$ ¶ ↺
depends_on example 5.10 Functions on a set ¶ ↺
… 1 more

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Edges

type	direction	node	provenance	where
`depends_on`	→	Commutativity; abelian structure	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:848
`depends_on`	→	Field	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:848
`depends_on`	→	Module	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:848
`depends_on`	←	Algebra	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:882
`depends_on`	←	Algebra	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:5258
`depends_on`	←	External direct sum	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:4427
`depends_on`	←	Inner product	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:923
`depends_on`	←	Linear combination	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:589
`depends_on`	←	Linear transformation	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:1712
`depends_on`	←	Norm	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:1001
`depends_on`	←	Scalar	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:571
`depends_on`	←	Sesquilinear form	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:4959
`depends_on`	←	Vector	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:578
`depends_on`	←	The coordinate spaces $\R^{n}$ and $\C^{n}$	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:667
`depends_on`	←	Functions on a set	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:721
`depends_on`	←	The matrices of a given shape	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:691