definition 4.32 Field

open in the book · parts/02-mathematical-methods/02-algebraic-structures.tex:817 · p. 67

ground object -- no derivation owed

Rests on

depends_on definition 4.11 Inverse element ¶
- depends_on definition 4.9 Neutral element ¶
  - depends_on definition 4.3 Algebraic structure ¶
    - depends_on definition 4.2 Binary operation ¶
      - depends_on definition 4.1 Cartesian product ¶
      - depends_on definition 3.43 Map ¶
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 4.3 Algebraic structure ¶ ↺
    - depends_on definition 4.2 Binary operation ¶ ↺
depends_on definition 4.30 Ring ¶
- depends_on definition 4.8 Commutativity; abelian structure ¶
  - depends_on definition 4.4 Internal binary operation; magma ¶ ↺
- 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 ¶ ↺

Supports

depends_on definition 4.33 Vector space ¶
- 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 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 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 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 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 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 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 12.98 Two particles in three-dimensional space ¶
      - depends_on proposition 12.95 The tensor inner product is well defined and positive definite ¶
    - depends_on definition 10.85 Sobolev space ¶
      - depends_on definition A.450 The space $W^{1,r}(a,b)$ ¶
      - depends_on definition 10.87 Weak form of an elliptic problem ¶
    - 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 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 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 example 12.104 The Schwartz triple ¶
      - 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 theorem 12.12 Riesz–Fischer ¶
    - depends_on lemma A.254 Riemann integral of a continuous curve ¶
      - depends_on lemma A.255 Smoothed vectors lie in the domain ¶
      - depends_on lemma A.257 Integrated form of the equation of motion ¶
      - depends_on lemma A.583 Absolutely convergent operator-valued integrals ¶
    - 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 lemma A.606 $F$ is entire, and is the overlap in disguise ¶
      - 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 proposition 12.28 Convergence criterion for orthogonal series ¶
      - depends_on theorem 12.30 Completeness, expansion, Parseval ¶
    - 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.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 12.71 Adjoint of a densely defined operator ¶
      - depends_on proposition 12.39 Algebra of the adjoint; the $C^{\ast}$ identity ¶
    - 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 proposition 30.9 Principal strains ¶
      - 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 theorem 28.33 Normal modes ¶
      - depends_on theorem 29.16 Principal axes ¶
  - 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 lemma 5.132 Parity constraint ¶
    - 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 12.17 The complement is always a closed subspace ¶
    - depends_on definition 5.26 Orthogonal basis ¶
      - depends_on definition 5.27 Orthonormal basis ¶
  - depends_on definition 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 definition 7.97 Partial derivative; gradient ¶
      - depends_on definition 5.17 Components of a vector ¶
      - depends_on definition 5.16 Dimension ¶
      - 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 lemma 5.137 The alternating top form is unique up to scale ¶
      - depends_on proposition 5.102 Dimensions add ¶ ↺
      - depends_on proposition 5.105 Grassmann's formula ¶ ↺
      - depends_on proposition 5.45 prop:lin-matrix-unique ¶
      - … 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 5.31 Gram criterion ¶
    - 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 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 lemma 5.97 Fitting splitting ¶
      - depends_on proposition 5.49 Injective, surjective, invertible ¶
      - 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 definition 5.149 Totally reducible representation ¶
      - 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 corollary 5.62 A functional that annihilates a set of constraints ¶ ↺
    - 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 proposition 7.104 Chain rule in several variables ¶
      - depends_on remark 7.103 rem:ana-differential-scalar ¶
    - depends_on lemma A.291 $h$ is differentiable, with the stated derivative ¶
    - depends_on proposition 7.104 Chain rule in several variables ¶ ↺
    - 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 theorem A.286 Implicit function theorem ¶
    - 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 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 definition 12.49 Resolvent set; spectrum ¶
      - depends_on definition 12.50 Point, continuous and residual spectrum ¶
      - depends_on lemma A.240 Spectral mapping for polynomials ¶
      - depends_on lemma A.239 The norm of a self-adjoint operator lies in its spectrum ¶
      - depends_on proposition 12.52 Neumann series; the spectrum is bounded ¶
    - 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 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 proposition 12.36 Boundedness is continuity ¶
      - depends_on theorem 12.74 Hellinger–Toeplitz ¶
    - 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 12.78 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 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 definition 5.135 The symplectic group ¶
      - 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 5.111 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 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 5.61 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
depends_on lemma A.44 $\R$ is an ordered field ¶
depends_on theorem A.35 Completeness of $\R$ ¶

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Edges

type	direction	node	provenance	where
`depends_on`	→	Inverse element	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:830
`depends_on`	→	Monoid	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:830
`depends_on`	→	Ring	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:830
`depends_on`	←	Vector space	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:848
`depends_on`	←	$\R$ is an ordered field	declared	appendices/A-long-proofs.tex:3053
`depends_on`	←	Completeness of $\R$	declared	appendices/A-long-proofs.tex:2884