hcju/proofwikips huggingface dataset:
Definition:Number
There are five main classes of number: :$(1): \quad$ The natural numbers: $\N = \set {0, 1, 2, 3, \ldots}$ :$(2): \quad$ The integers: $\Z = \set {\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots}$ :$(3): \quad$ The rational numbers: $\Q = \set {p / q: p, q \in \Z, q \ne 0}$ :$(4): \quad$ The real numbers: $\R = \set {x: x = \sequence {s_n} }$ where $\sequence {s_n}$ is a Cauchy sequence in $\Q$ :$(5): \quad$ The complex numbers: $\C = \set {a + i b: a, b \in \R, i^2 = -1}$
Definition:Absolute Value
=== Definition 1 === {{:Definition:Absolute Value/Definition 1}} === Definition 2 === {{:Definition:Absolute Value/Definition 2}}
Definition:Remainder
Let $a, b \in \Z$ be integers such that $b \ne 0$. From the Division Theorem, we have that: :$\forall a, b \in \Z, b \ne 0: \exists! q, r \in \Z: a = q b + r, 0 \le r < \left|{b}\right|$ The value $r$ is defined as the '''remainder of $a$ on division by $b$''', or the '''remainder of $\dfrac a b$'''. === Real Arguments === When $x, y \in \R$ the '''remainder''' is still defined: {{:Definition:Remainder/Real}}
Definition:Even Integer
=== Definition 1 === {{:Definition:Even Integer/Definition 1}} === Definition 2 === {{:Definition:Even Integer/Definition 2}} === Definition 3 === {{:Definition:Even Integer/Definition 3}}
Definition:Odd Integer
=== Definition 1 === {{:Definition:Odd Integer/Definition 1}} === Definition 2 === {{:Definition:Odd Integer/Definition 2}} === Definition 3 === {{:Definition:Odd Integer/Definition 3}} === Euclid's Definition ===  :''{{:Definition:Euclid's Definitions - Book VII/7 - Odd Number}}'' {{EuclidDefRefNocat|VII|7|Odd Number}}
Definition:Integer Combination
Let $a, b \in \Z$. An integer $n$ of the form: :$n = p a + q b: p, q \in \Z$ is an '''integer combination''' of $a$ and $b$.
Definition:Common Divisor
=== Integral Domain === {{:Definition:Common Divisor/Integral Domain}} === Integers === The definition is usually applied when the integral domain in question is the set of integers $\Z$, thus: {{:Definition:Common Divisor/Integers}} === Real Numbers === The definition can also be applied when the integral domain in question is the real numbers $\R$, thus: {{:Definition:Common Divisor/Real Numbers}}
Definition:Coprime
=== GCD Domain === {{:Definition:Coprime/GCD Domain}} === Euclidean Domain === {{:Definition:Coprime/Euclidean Domain}} === Integers === {{:Definition:Coprime/Integers}}
Definition:Integers Modulo m
Let $m \in \Z$ be an integer. The '''integers modulo $m$''' are the set of least positive residues of the set of residue classes modulo $m$: :$\Z_m = \set {0, 1, \ldots, m - 1}$
Definition:Prime Number
=== Definition 1 === {{:Definition:Prime Number/Definition 1}} === Definition 2 === {{:Definition:Prime Number/Definition 2}} === Definition 3 === {{:Definition:Prime Number/Definition 3}} === Definition 4 === {{:Definition:Prime Number/Definition 4}} === Definition 5 === {{:Definition:Prime Number/Definition 5}} === Definition 6 === {{:Definition:Prime Number/Definition 6}} === Definition 7 === {{:Definition:Prime Number/Definition 7}} === Euclid's Definition ===  :''{{:Definition:Euclid's Definitions - Book VII/11 - Prime Number}}'' {{EuclidDefRefNocat|VII|11|Prime Number}}
Definition:Composite Number
A '''composite number''' $c$ is a positive integer that has more than two positive divisors. That is, an integer greater than $1$ which is not prime is defined as composite.
Definition:Sylow p-Subgroup
Let $p$ be prime. Let $G$ be a finite group whose order is denoted by $\left\vert{G}\right\vert$. === Definition 1 === {{:Definition:Sylow p-Subgroup/Definition 1}} === Definition 2 === {{:Definition:Sylow p-Subgroup/Definition 2}} === Definition 3 === {{:Definition:Sylow p-Subgroup/Definition 3}}
Definition:Prime Decomposition
Let $n > 1 \in \Z$. From the Fundamental Theorem of Arithmetic, $n$ has a unique factorization of the form: {{begin-eqn}} {{eqn | l = n       | r = \prod_{p_i \mathop \divides n} {p_i}^{k_i}       | c =  }} {{eqn | r = {p_1}^{k_1} {p_2}^{k_2} \cdots {p_r}^{k_r}       | c =  }} {{end-eqn}} where: :$p_1 < p_2 < \cdots < p_r$ are distinct primes :$k_1, k_2, \ldots, k_r$ are (strictly) positive integers. This unique expression is known as the '''prime decomposition of $n$'''.
Definition:Möbius Function
Let $n \in \Z_{>0}$, that is, a strictly positive integer. The '''Möbius function''' is the function $\mu: \Z_{>0} \to \Z_{>0}$ defined as: :$\map \mu n = \begin{cases} 1 & : n = 1 \\ 0 & : \exists p \in \mathbb P: p^2 \divides n\\ \left({-1}\right)^k & : n = p_1 p_2 \ldots p_k: p_i \in \mathbb P \end{cases}$
Definition:Square
=== Geometry === {{:Definition:Quadrilateral/Square}} === Abstract Algebra === {{:Definition:Square/Mapping}}
Definition:Fibonacci Number
The '''Fibonacci numbers''' are a sequence $\sequence {F_n}$ of integers which is formally defined recursively as: :$F_n = \begin {cases} 0 & : n = 0 \\ 1 & : n = 1 \\ F_{n - 1} + F_{n - 2} & : \text {otherwise} \end {cases}$ for all $n \in \Z_{\ge 0}$.
Definition:Cauchy Sequence
=== Metric Space === {{:Definition:Cauchy Sequence/Metric Space}} === Normed Vector Space === {{:Definition:Cauchy Sequence/Normed Vector Space}} === Normed Division Ring === {{:Definition:Cauchy Sequence/Normed Division Ring}}
Definition:Real Number
{{:Definition:Real Number/Number Line Definition}}
Definition:Real Interval
Informally, the set of all real numbers between any two given real numbers $a$ and $b$ is called a '''(real) interval'''.
Definition:Mediant
Let $r, s \in \Q$, i.e. let $r, s$ be rational numbers. Let $r$ and $s$ be expressed as $r = \dfrac a b, s = \dfrac c d$ where $a, b, c, d$ are integers such that $b > 0, d > 0$ (this is always possible by Divided by Positive Element of Field of Quotients). Then the '''mediant''' of $r$ and $s$ is $\dfrac {a + c} {b + d}$.
Definition:Close Packed
Let $\left({S, \preceq}\right)$ be an ordered set. Then $\left({S, \preceq}\right)$ is defined as '''close packed (in itself)'''  between every two elements of $S$ there exists another element of $S$: : $\forall a, b \in S: a \prec b \implies \exists c \in S: a \prec c \prec b$ === Close Packed Subset === {{:Definition:Close Packed/Subset}}
Definition:Floor Function
Let $x$ be a real number. Informally, the '''floor function of $x$''' is the greatest integer less than or equal to $x$. === Definition 1 === {{Definition:Floor Function/Definition 1}} === Definition 2 === {{Definition:Floor Function/Definition 2}} === Definition 3 === {{Definition:Floor Function/Definition 3}}
Definition:Ceiling Function
Let $x$ be a real number. Informally, the '''ceiling function of $x$''' is the smallest integer greater than or equal to $x$. === Definition 1 === {{Definition:Ceiling Function/Definition 1}} === Definition 2 === {{Definition:Ceiling Function/Definition 2}} === Definition 3 === {{Definition:Ceiling Function/Definition 3}}
Definition:Real Function
A '''real function''' is a mapping or function whose domain and codomain are subsets of the set of real numbers $\R$.
Definition:Implicit Function
Consider a (real) function of two independent variables $z = \map f {x, y}$. Let a relation between $x$ and $y$ be expressed in the form $\map f {x, y} = 0$ defined on some interval $\mathbb I$. If there exists a function: :$y = \map g x$ defined on $\mathbb I$ such that: :$\forall x \in \mathbb I: \map f {x, \map g x} = 0$ then the relation $\map f {x, y} = 0$ defines $y$ as an '''implicit function''' of $x$.
Definition:Order of Structure
The '''order''' of an algebraic structure $\struct {S, \circ}$ is the cardinality of its underlying set, and is denoted $\order S$. Thus, for a finite set $S$, the '''order of $\struct {S, \circ}$''' is the number of elements in $S$.
Definition:Word (Abstract Algebra)
Let $\struct {G, \circ}$ be a magma. Let $S \subseteq G$ be a subset. A '''word in $S$''' is the product of a finite number of elements of $S$. The '''set of words in $S$''' is denoted $\map W S$: :$\map W S := \set {s_1 \circ s_2 \circ \cdots \circ s_n: n \in \N_{>0}: s_i \in S, 1 \le i \le n}$
Definition:Truth Table
A '''truth table''' is a tabular array that represents the computation of a truth function, that is, a function of the form: :$f : \mathbb B^k \to \mathbb B$ where: :$k$ is a non-negative integer :$\mathbb B$ is a set of truth values, usually $\set {0, 1}$ or $\set {T, F}$.
Definition:Index of Subgroup
Let $G$ be a group. Let $H$ be a subgroup of $G$. The '''index of $H$ (in $G$)''', denoted $\index G H$, is the cardinality of the left (or right) coset space $G / H$. === Finite Index === {{:Definition:Index of Subgroup/Finite}} === Infinite Index === {{:Definition:Index of Subgroup/Infinite}}
Definition:Parametric Operator
Let $A$ be an index set. Let $\Omega$ be a indexed family of operators $\left({\Omega_{\alpha} }\right)_A = \left\{ {\Omega_\alpha: \alpha \in A}\right\}$ indexed by the index $\alpha$. {{explain|Definition of "operator" in this context.}} Then $\Omega$ is a '''parametric operator''' with '''parameter $\alpha$''' in the '''parameter set $A$'''.
Definition:Order of Group Element
Let $G$ be a group whose identity is $e_G$. Let $x \in G$ be an element of $G$. === Definition 1 === {{:Definition:Order of Group Element/Definition 1}} === Definition 2 === {{:Definition:Order of Group Element/Definition 2}} === Definition 3 === {{:Definition:Order of Group Element/Definition 3}}
Definition:Derangement
A '''derangement''' is a permutation $f: S \to S$ from a set $S$ to itself where $\map f s \ne s$ for any $s \in S$.  If $S$ is finite, the number of '''derangements''' is denoted by $D_n$ where $n = \card S$ (the cardinality of $S$.)
Definition:Hamiltonian Group
A group $G$ is described as '''Hamiltonian'''  $G$ is a non-abelian group such that every subgroup is normal.
Definition:Simple Group
A group $G$ is '''simple'''  it has only $G$ and the trivial group as normal subgroups. That is, if the composition length of $G$ is $1$.
Definition:Group Direct Product
Let $\struct {G, \circ_1}$ and $\struct {H, \circ_2}$ be groups. Let $G \times H: \set {\tuple {g, h}: g \in G, h \in H}$ be their cartesian product. The '''(external) direct product''' of $\struct {G, \circ_1}$ and $\struct {H, \circ_2}$ is the group $\struct {G \times H, \circ}$ where the operation $\circ$ is defined as: :$\tuple {g_1, h_1} \circ \tuple {g_2, h_2} = \tuple {g_1 \circ_1 g_2, h_1 \circ_2 h_2}$ This is usually referred to as the '''group direct product''' of $G$ and $H$.
Definition:Independent Subgroups
Let $G$ be a group whose identity is $e$. Let $\left \langle {H_n} \right \rangle$ be a sequence of subgroups of $G$. === Definition 1 === {{:Definition:Independent Subgroups/Definition 1}} === Definition 2 === {{:Definition:Independent Subgroups/Definition 2}}
Definition:Group Presentation
Let $G$ be a group. A '''presentation for $G$''' is a triple $\tuple {S, R, f}$ where: :$S$ is a set :$R$ is a set of relations on $S$ :$f: \gen {S \mid R} \to G$ is a group isomorphism from the group defined by $\struct {S, R}$.
Definition:Infinite Cyclic Group
=== Definition 1 === {{:Definition:Infinite Cyclic Group/Definition 1}} === Definition 2 === {{:Definition:Infinite Cyclic Group/Definition 2}}
Definition:Free Group
=== Definition 1 === {{:Definition:Free Group/Definition 1}} === Definition 2 === {{:Definition:Free Group/Definition 2}}
Definition:Ring of Integers Modulo m
Let $m \in \Z: m \ge 2$. Let $\Z_m$ be the set of integers modulo $m$. Let $+_m$ and $\times_m$ denote addition modulo $m$ and multiplication modulo $m$ respectively. The algebraic structure $\struct {\Z_m, +_m, \times_m}$ is '''the ring of integers modulo $m$'''.
Definition:Symmetry Mapping
A '''symmetry mapping''' (or just '''symmetry''') of a geometric figure is a bijection from the figure to itself which preserves the distance between points. In other words, it is a self-congruence. Intuitively and informally, a '''symmetry''' is a movement of the figure so that it looks exactly the same after it has been moved.
Definition:Geometric Figure
A '''geometric figure''' is intuitively defined as a set of points and lines in space.  :''{{:Definition:Euclid's Definitions - Book I/14 - Geometric Figure}}'' {{EuclidDefRefNocat|I|14|Geometric Figure}} The boundary may or may not be included in a particular figure. If this is important (and in the study of topology it usually is), then whether it is included or not needs to be specified. === Plane Figure === {{:Definition:Geometric Figure/Plane Figure}} === Three-Dimensional Figure === {{:Definition:Geometric Figure/Three-Dimensional Figure}} === Rectilineal Figure === {{:Definition:Geometric Figure/Rectilineal Figure}}
Definition:Permutation on n Letters
Let $\N_k$ be used to denote the initial segment of natural numbers: : $\N_k = \closedint 1 k = \set {1, 2, 3, \ldots, k}$ A '''permutation on $n$ letters''' is a '''permutation''': : $\pi: \N_n \to \N_n$ The usual symbols for denoting a general '''permutation''' are $\pi$ (not to be confused with the famous circumference over diameter), $\rho$ and $\sigma$.
Definition:Fixed Element of Permutation
Let $S$ be a set. Let $\pi: S \to S$ be a permutation on $S$. Let $x \in S$. $x$ is '''fixed by $\pi$''' : :$\map \pi x = x$
Definition:Cyclic Permutation
Let $S_n$ denote the symmetric group on $n$ letters. Let $\rho \in S_n$ be a permutation on $S$. Then $\rho$ is a '''cyclic permutation of length $k$'''  there exists $k \in \Z: k > 0$ and $i \in \Z$ such that: :$(1): \quad k$ is the smallest such that $\map {\rho^k} i = i$ :$(2): \quad \rho$ fixes each $j$ not in $\set {i, \map \rho i, \ldots, \map {\rho^{k - 1} } i}$.
Definition:Disjoint Permutations
Let $S_n$ denote the symmetric group on $n$ letters. Let $\pi, \rho \in S_n$ both be permutations on $S_n$. Then $\pi$ and $\rho$ are '''disjoint''' : :$(1): \quad i \notin \Fix \pi \implies i \in \Fix \rho$ :$(2): \quad i \notin \Fix \rho \implies i \in \Fix \pi$ That is, each element moved by $\pi$ is fixed by $\rho$ and (equivalently) each element moved by $\rho$ is fixed by $\pi$. That is,  their supports are disjoint sets. We may say that: : $\pi$ is '''disjoint from $\rho$''' : $\rho$ is '''disjoint from $\pi$''' : $\pi$ and $\rho$ are '''(mutually) disjoint'''. Note of course that it is perfectly possible for $i \in \Fix \pi$ and also $i \in \Fix \rho$, that is, there may well be elements fixed by more than one of a pair of '''disjoint permutations'''.
Definition:Group Action
Let $X$ be a set. Let $\struct {G, \circ}$ be a group whose identity is $e$. === Left Group Action === {{:Definition:Group Action/Left Group Action}} === Right Group Action === {{:Definition:Group Action/Right Group Action}} The group $G$ thus '''acts on''' the set $X$. The group $G$ can be referred to as the '''group of transformations''', or a '''transformation group'''.
Definition:Orbit (Group Theory)
Let $G$ be a group acting on a set $X$. === Definition 1 === {{:Definition:Orbit (Group Theory)/Definition 1}} === Definition 2 === {{:Definition:Orbit (Group Theory)/Definition 2}}
Definition:Effective Transformation Group
Let $G$ be a group whose identity is $e$. Let $X$ be a set. Let $\phi: G \times X \to X$ be a group action. Then $G$ is an '''effective transformation group for $\phi$'''  $\phi$ is faithful.
Definition:Transposition
Let $S$ be a set. A '''transposition''' on $S$ is a $2$-cycle. That is, a '''transposition''' is a permutation $\rho$ on $S$ which exchanges, or '''transposes''', exactly two elements of $S$. Thus if $\rho$ is a '''transposition''' which '''transposes''' two elements $r, s \in S$, it follows from the definition of fixed elements that: :$\Fix \rho = S \setminus \set {r, s}$
Definition:R-Algebraic Structure
Let $\struct {R, +_R, \times_R}$ be a ring. Let $\struct {S, \ast_1, \ast_2, \ldots, \ast_n}$ be an algebraic structure with $n$ operations. Let $\circ: R \times S \to S$ be a binary operation. Then $\struct {S, \ast_1, \ast_2, \ldots, \ast_n, \circ}_R$ is an '''$R$-algebraic structure with $n$ operations'''. If the number of operations in $S$ is either understood or general, it is just called an '''$R$-algebraic structure''', and the structure can be denoted $\struct {S, \circ}_R$.
Definition:Module
Let $\struct {R, +_R, \times_R}$ be a ring. Let $\struct {G, +_G}$ be an abelian group. A module over $R$ is an $R$-algebraic structure with one operation $\struct {G, +_G, \circ}_R$ which is both a left module and a right module: === Left Module === {{:Definition:Left Module}} === Right Module === {{:Definition:Right Module}}
Definition:Linear Transformation
A '''linear transformation''' is a homomorphism from one module to another.
Definition:Module on Cartesian Product
Let $\struct {R, +_R, \times_R}$ be a ring. Let $n \in \N_{>0}$. Let $+: R^n \times R^n \to R^n$ be defined as: :$\tuple {\alpha_1, \ldots, \alpha_n} + \tuple {\beta_1, \ldots, \beta_n} = \tuple {\alpha_1 +_R \beta_1, \ldots, \alpha_n +_R \beta_n}$ Let $\times: R \times R^n \to R^n$ be defined as: :$\lambda \times \tuple {\alpha_1, \ldots, \alpha_n} = \tuple {\lambda \times_R \alpha_1, \ldots, \lambda \times_R \alpha_n}$ Then $\struct {R^n, +, \times}_R$ is '''the $R$-module $R^n$'''.
Definition:Trivial Module
Let $\struct {G, +_G}$ be an abelian group whose identity is $e_G$. Let $\struct {R, +_R, \circ_R}$ be a ring. Let $\circ$ be defined as: :$\forall \lambda \in R: \forall x \in G: \lambda \circ x = e_G$ Then $\struct {G, +_G, \circ}_R$ is an $R$-module. Such a module is called a '''trivial module'''.
Definition:Submodule
Let $\struct {R, +, \circ}$ be a ring. Let $\struct {S, +, \circ}_R$ be an $R$-algebraic structure with one operation. Let $T$ be a closed subset of $S$. Let $\struct {T, +_T, \circ_T}_R$ be an $R$-module where: :$+_T$ is the restriction of $+$ to $T \times T$ :$\circ_T$ is the restriction of $\circ$ to $R \times T$. Then $\struct {T, +_T, \circ_T}_R$ is a '''submodule''' of $\struct {S, +, \circ}_R$.
Definition:Null Module
Let $\left({R, +_R, \circ_R}\right)$ be a ring. Let $G$ be the trivial group. Then the $R$-module $\left({G, +_G, \circ}\right)_R$ is known as the '''null module'''.
Definition:P-Subgroup
Let $G$ be a group. Let $H$ be a subgroup of $G$. If $H$ is a $p$-group, then $H$ is a '''$p$-subgroup''' of $G$.  e6bq56il8qbgbcyygojvbcz1jag6lue
Definition:Finitely Generated Module
Let $R$ be a ring. Let $G$ be a module over $R$. Then $G$ is '''finitely generated'''  there is a generator for $G$ which is finite.
Definition:Linear Combination
Let $R$ be a ring. === Linear Combination of Sequence === {{:Definition:Linear Combination/Sequence}} === Linear Combination of Subset === {{:Definition:Linear Combination/Subset}} === Linear Combination of Empty Set === {{:Definition:Linear Combination/Empty Set}}
Definition:Linearly Independent
Let $G$ be an abelian group whose identity is $e$. Let $R$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $\left({G, +_G, \circ}\right)_R$ be a unitary $R$-module. === Sequence === {{:Definition:Linearly Independent/Sequence}} === Set === {{:Definition:Linearly Independent/Set}}
Definition:Basis (Topology)
=== Analytic Basis === {{:Definition:Basis (Topology)/Analytic Basis}} === Synthetic Basis === {{:Definition:Basis (Topology)/Synthetic Basis}}
Definition:Free Module
Let $R$ be a ring with unity. Let $G$ be a unitary $R$-module. Then $G$ is described as '''free''' if there exists a basis of $G$.
Definition:Ordered Basis
Let $R$ be a ring with unity. Let $G$ be a free $R$-module. An '''ordered basis''' of $G$ is a sequence $\left \langle {a_k} \right \rangle_{1 \mathop \le k \mathop \le n}$ of elements of $G$ such that $\left\{{a_1, \ldots, a_n}\right\}$ is a basis of $G$.
Definition:Topological Manifold/Differentiable Manifold
Let $M$ be a second-countable locally Euclidean space of dimension $d$.  Let $\mathscr F$ be a $d$-dimensional differentiable structure on $M$ of class $\mathcal C^k$, where $k \ge 1$. Then $\left({M, \mathscr F}\right)$ is a '''differentiable manifold of class $C^k$ and dimension $d$'''.
Definition:Dimension (Linear Algebra)
=== Module === {{:Definition:Dimension of Module}} === Vector Space === {{:Definition:Dimension of Vector Space}} {{wtd|Infinite dimensional case}}
Definition:Metric Space
A '''metric space''' $M = \struct {A, d}$ is an ordered pair consisting of: :$(1): \quad$ a non-empty set $A$ together with: :$(2): \quad$ a real-valued function $d: A \times A \to \R$ which acts on $A$, satisfying the metric space axioms: {{:Definition:Metric Space Axioms}}
Definition:Topological Space
Let $S$ be a set. Let $\tau$ be a topology on $S$. That is, let $\tau \subseteq \powerset S$ satisfy the open set axioms: {{:Definition:Open Set Axioms}} Then the ordered pair $\struct {S, \tau}$ is called a '''topological space'''. The elements of $\tau$ are called open sets of $\struct {S, \tau}$.
Definition:Topology
{{:Definition:Topology/Definition 1}}
Definition:Smooth Real Function
A real function is '''smooth''' if it is of differentiability class $C^\infty$. That is, if it admits of continuous derivatives of all orders.
Definition:Almost Everywhere
Let $\struct {X, \Sigma, \mu}$ be a measure space. A property $\map P x$ of elements of $X$ is said to hold '''($\mu$-)almost everywhere''' if the set: :$\set {x \in X: \neg \map P x}$ of elements of $X$ such that $P$ does not hold is contained in a $\mu$-null set.
Definition:Regular Value
Let $X$ and $Y$ be smooth manifolds. Let $f: X \to Y$ be a smooth mapping. Then a point $y \in Y$ is called a '''regular value''' of $f$  the pushforward of $f$ at $x$: : $f_* \vert_x: T_x X \to T_y Y$ {{explain|What do all the symbols mean in this context? Presume $\vert$ might mean restriction, but this is not obvious (if so then use $\restriction$); if "pushforward" actually means $f_* \vert_x: T_x X \to T_y Y$ then set up the page to define it, thus doing all the hard work of defining that concept all in one place.}} is surjective for every $x \in f^{-1} \left({y}\right) \subseteq X$.
Definition:Kronecker Delta
Let $\Gamma$ be a set. Let $R$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$. Then $\delta_{\alpha \beta}: \Gamma \times \Gamma \to R$ is the mapping on the cartesian square of $\Gamma$ defined as: :$\forall \tuple {\alpha, \beta} \in \Gamma \times \Gamma: \delta_{\alpha \beta} := \begin{cases} 1_R & : \alpha = \beta \\ 0_R & : \alpha \ne \beta \end{cases}$
Definition:Linear Form
Let $R$ be a commutative ring. Let $\struct {G, +_G, \circ}_R$ be a module over $R$. Let $\phi: \struct {G, +_G, \circ}_R \to \struct {R, +_R, \circ}_R$ be a linear transformation from $G$ to the $R$-module $R$. Then $\phi$ is called a '''linear form on $G$'''.
Definition:Algebraic Dual
Let $R$ be a commutative ring. Let $G$ be a module over $R$. The $R$-module $\map {\LL_R} {G, R}$ of all linear forms on $G$ is usually denoted $G^*$ and is called the '''algebraic dual''' of $G$. === Double Dual === {{:Definition:Algebraic Dual/Double Dual}}
Definition:Ordered Dual Basis
Let $R$ be a commutative ring. Let $\struct {G, +_G, \circ}_R$ be an $n$-dimensional module over $R$. Let $\sequence {a_n}$ be an ordered basis of $G$. Let $G^*$ be the algebraic dual of $G$. Then there is an ordered basis $\sequence {a'_n}$ of $G^*$ satisfying $\forall i, j \in \closedint 1 n: \map {a'_i} {a_j} = \delta_{i j}$. This ordered basis $\sequence {a'_n}$ of $G^*$ is called the '''ordered basis of $G^*$ dual to $\sequence {a_n}$''', or the '''ordered dual basis of $G^*$'''.
Definition:Evaluation Linear Transformation
Let $R$ be a commutative ring. Let $G$ be an $R$-module. Let $G^*$ be the algebraic dual of $G$. Let $G^{**}$ be the algebraic dual of $G^*$. For each $x \in G$, we define the mapping $x^\wedge: G^* \to R$ as: :$\forall t' \in G^*: x^\wedge \left({t'}\right) = t' \left({x}\right)$ Then $x^\wedge \in G^{**}$. The mapping $J: G \to G^{**}$ defined as: :$\forall x \in G: J \left({x}\right) = x^\wedge$ is called the '''evaluation linear transformation from $G$ into $G^{**}$'''. It is usual to denote the mapping $t': G^* \to G$ as follows: :$\forall x \in G, t' \in G^*: \left \langle {x, t'} \right \rangle := t' \left({x}\right)$
Definition:Evaluation Isomorphism
Let $R$ be a commutative ring. Let $G$ be a unitary $R$-module whose dimension is finite. Then the evaluation linear transformation $J: G \to G^{**}$ is called the '''evaluation isomorphism from $G$ to $G^{**}$.