definition 4.30 Ring

open in the book · parts/02-mathematical-methods/02-algebraic-structures.tex:762 · p. 66

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 4.28 Distributivity ¶
- depends_on definition 4.3 Algebraic structure ¶ ↺
depends_on definition 4.21 Group ¶
- depends_on definition 4.11 Inverse element ¶
  - depends_on definition 4.9 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 4.18 Semigroup ¶ ↺

Supports

depends_on definition 4.32 Field ¶
- 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 definition 5.124 Ideal ¶
      - depends_on proposition 5.164 The kernel is an ideal ¶
    - depends_on definition 5.127 Lie algebra ¶
      - depends_on definition 24.40 Lie–Poisson bracket ¶
      - depends_on proposition 5.167 Adjoint representation ¶
      - depends_on proposition 5.139 Dimension of the symplectic group ¶
      - 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 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 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 definition 5.103 Sum of subspaces; internal direct sum ¶
      - depends_on corollary 5.108 The sum is the right operation ¶
      - 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 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 definition 12.32 Separable Hilbert space ¶
      - depends_on definition 12.94 Tensor product of Hilbert spaces ¶
      - depends_on definition 10.85 Sobolev space ¶
      - depends_on example 12.9 The sequence space $\ell^{2}$ ¶
      - depends_on example 12.11 The function space $L^{2}$ ¶
      - depends_on lemma A.254 Riemann integral of a continuous curve ¶
      - depends_on proposition 12.4 Cauchy–Schwarz and continuity of the inner product ¶
      - depends_on proposition 12.28 Convergence criterion for orthogonal series ¶
      - depends_on proposition 12.51 The three cases are exclusive and exhaustive ¶
      - depends_on theorem 16.70 The direct method ¶
      - depends_on theorem 12.14 Closest point in a closed convex set ¶
      - … 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 theorem 12.38 Existence and uniqueness of the adjoint ¶
      - depends_on theorem 5.43 The four fundamental subspaces ¶ ↺
      - depends_on theorem 5.80 Simultaneous diagonalization of commuting self-adjoint operators ¶
      - depends_on theorem 5.79 Spectral theorem for a self-adjoint operator ¶
    - depends_on definition 5.32 Levi–Civita symbol; cross product ¶
      - depends_on definition 5.131 Isotropic Cartesian tensor ¶
      - depends_on lemma 5.34 Contraction of two Levi–Civita symbols ¶
      - depends_on lemma 5.33 Determinant through the Levi–Civita symbol ¶
    - 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 5.26 Orthogonal basis ¶
    - depends_on definition 5.144 Unitary representation ¶
      - depends_on lemma 5.150 Invariance of the orthogonal complement ¶
      - 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
  - 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 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 proposition 5.31 Gram criterion ¶
      - depends_on proposition 5.28 Gram–Schmidt ¶
      - depends_on proposition 5.20 Cauchy–Schwarz inequality ¶
    - depends_on definition 5.7 Vector subspace ¶
      - depends_on corollary 5.29 Orthogonal decomposition ¶ ↺
      - depends_on definition 12.13 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 5.163 Faithful, equivalent, invariant ¶ ↺
      - depends_on definition 5.147 Invariant subspace ¶
      - 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 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 theorem 7.100 $C^{1}$ implies differentiable ¶
      - 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 definition 12.88 Reducing subspace ¶
      - depends_on definition 12.49 Resolvent set; spectrum ¶
      - depends_on proposition 12.37 $\mathcal{B}(\mathcal{H})$ is a Banach algebra ¶
      - depends_on proposition 12.36 Boundedness is continuity ¶
      - 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 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 5.111 Symmetric and antisymmetric forms ¶
      - 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 5.76 Diagonalizable operator ¶
      - 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 5.60 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 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 ¶
      - depends_on proposition 5.67 Adjointness of pullback and 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 $\R$ ¶
depends_on definition 4.31 Module ¶
- depends_on definition 4.33 Vector space ¶ ↺

Neighborhood

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Edges

type	direction	node	provenance	where
`depends_on`	→	Commutativity; abelian structure	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:773
`depends_on`	→	Distributivity	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:773
`depends_on`	→	Group	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:773
`depends_on`	→	Semigroup	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:773
`depends_on`	←	Field	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:830
`depends_on`	←	Module	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:814