Definition:Galois Group of Field Extension

Let $L / K$ be a field extension. The '''Galois group''' of $L / K$ is the subgroup of the automorphism group of $L$ consisting of field automorphisms that fix $K$ point-wise: :$\Gal {L / K} = \set{\sigma \in \Aut L: \forall k \in K: \map \sigma k = k}$ === As a topological group === The notation $\Gal {L / K}$ is also a shorthand for the topological group: :$\struct {\Gal {L / K}, \tau}$ where $\tau$ is the Krull topology. === Alternative Definition === More generally, we can abandon the condition that $L / K$ be Galois if we choose an algebraic closure $\overline K$ such that $L \subseteq \overline K$ and define: :$\Gal {L / K} = \left\{ {\sigma: L \to \overline K: \sigma}\right.$ is an embedding of $L$ such that $\sigma$ fixes $K$ point-wise$\left.\right\}$ This set will form a group {{iff}} $L / K$ is normal.


Definition:Gradation Compatible with Ring Structure

Let $\left({M, \cdot}\right)$ be a semigroup. Let $\left({R, +, \circ}\right)$ be a ring. Let $(R_n)_{n\in M}$ be an gradation of type $M$ on the additive group of $R$. The gradation is '''compatible with the ring structure''' {{iff}} :$\forall m, n \in M : \forall x \in S_m, y \in S_n: x \circ y \in S_{m \cdot n}$ and so: :$S_m S_n \subseteq S_{m\cdot n}$


Definition:Filtered Algebra

A '''filtered algebra''' is a generalization of the notion of a graded algebra. A '''filtered algebra''' over the field $k$ is an algebra $\left({A_k, \oplus}\right)$ over $k$ which has an increasing sequence $\left\{{0}\right\} \subset F_0 \subset F_1 \subset \cdots \subset F_i \subset \cdots \subset A$ of substructures of $A$ such that: :$\displaystyle A = \bigcup_{i \mathop \in \N} F_i$ and that is compatible with the multiplication in the following sense:   :$\forall m, n \in \N: F_m \cdot F_n \subset F_{n+m}$


Definition:Graded Algebra

Let $R$ be a graded commutative ring with unity. An algebra $A$ over $R$ is a '''graded algebra'''. That is, a '''graded algebra''' is an algebra over a ring where the ring is a graded ring.


Definition:Quadratic Algebra

A '''quadratic algebra''' $A$ is a filtered algebra whose generator consists of degree one elements, with defining relations of degree 2. A '''quadratic algebra''' $A$ is determined by a vector space of generators $V = A_1$ and a subspace of homogeneous quadratic relations $S \subseteq V \times V$. Thus : :$A = T \left({V}\right) / \left \langle {S}\right \rangle$ and inherits its grading from the tensor algebra $T \left({V}\right)$. If the subspace of relations may also contain inhomogeneous degree 2 elements, $S \subseteq k \times V \times \left({V \times V}\right)$, this construction results in a '''filtered quadratic algebra'''. A graded quadratic algebra $A$ as above admits a '''quadratic dual''': the quadratic algebra generated by $V^*$ and with quadratic relations forming the orthogonal complement of $S$ in $V^* \times V^*$. {{Help}} {{MissingLinks}} Category:Definitions/Algebras 9mntgrc4e50kba8fm3dwi3zh9n282ca


Definition:Division Algebra

Let $\left({A_F, \oplus}\right)$ be an algebra over field $F$ such that $A_F$ does not consist solely of the zero vector $\mathbf 0_A$ of $A_F$. === Definition 1 === {{:Definition:Division Algebra/Definition 1}} === Definition 2 === {{:Definition:Division Algebra/Definition 2}}


Definition:Real Algebra

A '''real algebra''' is an algebra over a field where the field in question is the field of real numbers $\R$. Category:Definitions/Algebras ct1zb8hp39391t6npjz6qln9cli5sgw


Definition:Associative Algebra

Let $R$ be a commutative ring. Let $\left({A_R, *}\right)$ be an algebra over $R$. Then $\left({A_R, *}\right)$ is an '''associative algebra''' {{iff}} $*$ is an associative operation. That is: :$\forall a, b, c \in A_R: \left({a * b}\right) * c = a * \left({b * c}\right)$


Definition:Unital Algebra

Let $R$ be a commutative ring. Let $\left({A, *}\right)$ be an algebra over $R$.  Then $\left({A, *}\right)$ is a '''unital algebra''' {{iff}} the algebraic structure $\left({A, \oplus}\right)$ has an identity element. That is: :$\exists 1_A \in A: \forall a \in A: a * 1_A = 1_A * a = a$


Definition:Quadratic Real Algebra

A '''quadratic real algebra''' is a quadratic algebra over the field of real numbers. Category:Definitions/Algebras 65ruwah4xeb28o70pjrtklydsbtu0qr


Definition:Normed Division Algebra

Let $\left({A_F, \oplus}\right)$ be a unitary algebra where $A_F$ is a vector space over a field $F$. Then $\left({A_F, \oplus}\right)$ is a normed divison algebra iff $A_F$ is a normed vector space such that: :$\forall a, b \in A_F: \left \Vert{a \oplus b}\right \Vert = \left \Vert{a}\right \Vert \left \Vert{b}\right \Vert$ where $\left \Vert{a}\right \Vert$ denotes the norm of $a$.


Definition:Unitary Division Algebra

Let $\left({A_F, \oplus}\right)$ be a division algebra. Then $\left({A_F, \oplus}\right)$ is a '''unitary division algebra''' {{iff}} it has an identity element $1_A$ called a '''unit''' for $\oplus$: :$\exists 1_A \in A_R: \forall a \in A_R: a \oplus 1_A = 1_A \oplus a = a$ The unit is usually denoted $1$ when there is no source of confusion with the identity elements of the underlying structures of the algebra.


Definition:Unit of Algebra

Let $R$ be a commutative ring. Let $\left({A_R, \oplus}\right)$ be a unitary algebra over $R$. The '''unit''' of $\left({A_R, \oplus}\right)$, denoted $1_A$, is the identity element of the operation $\oplus$: :$\forall a \in A_R: a \oplus 1_A = 1_A \oplus a = a$ It is sometimes referred to as the '''multiplicative identity''' of $\left({A_R, \oplus}\right)$. It is usually denoted $1$ when there is no source of confusion with the identity elements of the underlying structures of the algebra.


Definition:Multiplicative Inverse

Let $\strut {A_F, \oplus}$ be a unitary algebra whose unit is $1$ and whose zero is $0$. Let $a \in A_F$ such that $a \ne 0$. A '''multiplicative inverse''' of $a$ is an element $b \in A_F$ such that: :$a \oplus b = 1 = b \oplus a$ === Field === {{:Definition:Multiplicative Inverse/Field}} === Multiplicative Inverse of Number === {{:Definition:Multiplicative Inverse/Number}}


Definition:Power-Associative Operation

Let $\circ$ be a binary operation. Then $\circ$ is defined as being '''power-associative on $S$''' {{iff}}: :$\forall x \in S: \paren {x \circ x} \circ x = x \circ \paren {x \circ x}$


Definition:Alternative Operation

Let $\circ$ be a binary operation. Then $\circ$ is defined as being '''alternative on $S$''' {{iff}}: :$\forall T := \set {x, y} \subseteq S: \forall x, y, z \in T: \paren {x \circ y} \circ z = x \circ \paren {y \circ z}$ That is, $\circ$ is associative over any two elements of $S$. For example, for any $x, y \in S$: :$\paren {x \circ y} \circ x = x \circ \paren {y \circ x}$ :$\paren {x \circ x} \circ y = x \circ \paren {x \circ y}$ and so on.


Definition:Multilinear Mapping

Let $\left({R, +_R, \times_R}\right)$ be a commutative ring. Let $\left({A_1, +_1, \circ_1}\right)_R, \left({A_2, +_2, \circ_2}\right)_R, \ldots, \left({A_n, +_n, \circ_n}\right)_R, \left({A_{n+1}, +_{n+1}, \circ_{n+1}}\right)_R$ be $R$-modules. Let $\oplus: A_1 \times A_2 \times \cdots \times A_n \to A_{n+1}$ be a multiary operator with the property that: $\forall \left({a_1, a_2, \ldots, a_n}\right) \in A_1 \times A_2 \times \cdots \times A_n$: * $a_1 \mapsto a_1 \oplus a_2 \oplus \cdots \oplus a_n$ is a linear transformation from $A_1$ to $A_{n+1}$ * $a_2 \mapsto a_1 \oplus a_2 \oplus \cdots \oplus a_n$ is a linear transformation from $A_2$ to $A_{n+1}$ * $\vdots$ * $a_n \mapsto a_1 \oplus a_2 \oplus \cdots \oplus a_n$ is a linear transformation from $A_n$ to $A_{n+1}$ Then $\oplus$ is a '''multilinear mapping'''.


Definition:Associator

Let $\left({A_R, \oplus}\right)$ be an algebra over a ring. Consider the trilinear mapping $\left[{\cdot, \cdot, \cdot}\right]: A_R^3 \to A_R$ defined as: :$\forall a, b, c \in A_R: \left[{a, b, c}\right] := \left({a \oplus b}\right) \oplus c - a \oplus \left({b \oplus c}\right)$ Then $\left[{\cdot, \cdot, \cdot}\right]$ is known as the '''associator''' of $\left({A_R, \oplus}\right)$. It can be considered a measure of how much associativity of $\oplus$ fails in $\left({A_R, \oplus}\right)$. Note that trivially if $\left({A_R, \oplus}\right)$ is an associative algebra, then: :$\forall a, b, c \in A_R: \left[{a, b, c}\right] = \mathbf 0_R$


Definition:Commutator

The '''commutator''' of an algebraic structure can be considered a measure of how commutative the structure is. === Groups === {{:Definition:Commutator/Group}} === Rings === {{:Definition:Commutator/Ring}} === Algebras === {{:Definition:Commutator/Algebra}}


Definition:Commutative Algebra

Let $R$ be a commutative ring. Let $\struct {A_R, \oplus}$ be an algebra over $R$. Then $\struct {A_R, \oplus}$ is a commutative algebra {{iff}} $\oplus$ is a commutative operation. That is: :$\forall a, b \in A_R: a \oplus b = b \oplus a$


Definition:Alternating Bilinear Mapping

Let $\left({A_R, \oplus}\right)$ be an algebra over a ring. By definition, $\oplus$ is a bilinear mapping. Then $\oplus$ is an '''alternating bilinear mapping''' {{iff}}: :$\forall a \in A_R: a \oplus a = 0$ For rings with characteristic other than two, the following condition is equivalent: :$\forall a, b \in A_R: a \oplus b = - b \oplus a$


Definition:Commutator/Group

Let $\struct {G, \circ}$ be a group. Let $g, h \in G$. The '''commutator''' of $g$ and $h$ is the operation: :$\sqbrk {g, h} := g^{-1} \circ h^{-1} \circ g \circ h$


Definition:Commutator/Ring

Let $\struct {R, +, \circ}$ be a ring. Let $a, b \in R$. The '''commutator''' of $a$ and $b$ is the operation: :$\sqbrk {a, b} := a \circ b + \paren {-b \circ a}$ or more compactly: :$\sqbrk {a, b} := a \circ b - b \circ a$


Definition:Commutator/Algebra

Let $\struct {A_R, \oplus}$ be an algebra over a ring. Consider the bilinear mapping $\sqbrk {\, \cdot, \cdot \,}: A_R^2 \to A_R$ defined as: :$\forall a, b \in A_R: \sqbrk {a, b} := a \oplus b - b \oplus a$ Then $\sqbrk {\, \cdot, \cdot \,}$ is known as the '''commutator''' of $\struct {A_R, \oplus}$. Note that trivially if $\struct {A_R, \oplus}$ is a commutative algebra, then: :$\forall a, b \in A_R: \sqbrk {a, b} = \mathbf 0_R$


Definition:Star-Algebra

Let $A = \left({A_F, \oplus}\right)$ be an unitary division algebra. Let $A$ have a mapping $*: A \to A$ such that: :$\forall a \in A: \left({a^*}\right)^* = a$ :$\forall a, b \in A: \left({a \oplus b}\right)^* = b^* \oplus a^*$ Then $A$ is a '''$*$-algebra''' (usually voiced '''star-algebra'''). The mapping $*: A \to A$ is a conjugation on $A$.


Definition:Antihomomorphism

Let $\phi: \struct {S, \circ} \to \struct {T, *}$ be a mapping from one algebraic structure $\struct {S, \circ}$ to another $\struct {T, *}$. Then $\phi$ is an '''antihomomorphism''' {{iff}}: :$\forall x, y \in S: \map \phi {x \circ y} = \map \phi y * \map \phi x$ For structures with more than one operation, $\phi$ may be '''antihomomorphic''' for a subset of those operations.


Definition:Antihomomorphism/Ring Antihomomorphism

Let $\struct {R, +, \circ}$ and $\struct {S, \oplus, *}$ be rings. Then $\phi: R \to S$ is a '''ring antihomomorphism''' {{iff}}: {{begin-eqn}} {{eqn | lo= \forall a, b \in R:       | l = \map \phi {a + b}       | r = \map \phi a \oplus \map \phi b }} {{eqn | lo= \forall a, b \in R:       | l = \map \phi {a \circ b}       | r = \map \phi b * \map \phi a }} {{end-eqn}}


Let $A = \left({A_F, \oplus}\right)$ be an algebra over a field $F$. Let $C: A_F \to A_F$ be a mapping such that: :$\forall a \in A: C \left({C \left({a}\right)}\right) = a$ :$\forall a, b \in A: C \left({a \oplus b}\right) = C \left({b}\right) \oplus C \left({a}\right)$ Then $C$ is called a '''conjugation''' on $A$. === Conjugate === {{:Definition:Conjugation (Abstract Algebra)/Conjugate}}


Definition:Nicely Normed Star-Algebra

Let $A = \left({A_F, \oplus}\right)$ be a star-algebra whose conjugation is denoted $*$. Then $A$ is a '''nicely normed $*$-algebra''' {{iff}}: :$\forall a \in A: a + a^* \in \R$ :$\forall a \in A, a \ne 0: 0 < a \oplus a^* = a^* \oplus a \in \R$ === Real Part === Let $a \in A$ be an element of a '''nicely normed $*$-algebra'''. Then the '''real part''' of $a$ is given by: :$\Re \left({a}\right) = \dfrac {a + a^*} 2$   === Imaginary Part === Let $a \in A$ be an element of a '''nicely normed $*$-algebra'''. Then the '''imaginary part''' of $a$ is given by: :$\Im \left({a}\right) = \dfrac {a - a^*} 2$ === Norm === Let $a \in A$ be an element of a '''nicely normed $*$-algebra'''. Then we can define a norm on $a$ by: :$\left\Vert{a}\right\Vert^2 = a \oplus a^*$


Definition:Real Star-Algebra

Let $A = \left({A_F, \oplus}\right)$ be an star-algebra whose conjugation is denoted $*$. Then $A$ is a '''real $*$-algebra''' {{iff}}: :$\forall a \in A: a^*= a$


Definition:Subalgebra

Let $R$ be a commutative ring. Let $\left({A, *}\right)$ be an algebra over $R$. A '''subalgebra''' of $A$ is a submodule $B \subseteq A$ such that: :$\forall x, y \in B: x * y \in B$ That is, such that $B$ is closed under $*$.


Definition:Unital Subalgebra

Let $R$ be a commutative ring. Let $\left({A_R, *}\right)$ be an unital algebra over $R$ whose unit is $1_A$. Let $\left({B_R, *}\right)$ be a subalgebra of $A_R$. === Definition 1 === {{:Definition:Unital Subalgebra/Definition 1}} === Definition 2 === {{:Definition:Unital Subalgebra/Definition 2}}


Definition:Generator of Field

Let $F$ be a field. Let $S \subseteq F$ be a subset and $K \le F$ a subfield. The '''field generated by $S$''' is the smallest subfield of $F$ containing $S$. The '''subring of $F$ generated by $K \cup S$''', written $K \left[{S}\right]$, is the smallest subring of $F$ containing $K \cup S$. The '''subfield of $F$ generated by $K \cup S$''', written $K \left({S}\right)$, is the smallest subfield of $F$ containing $K \cup S$.


Definition:Generator of Semigroup

Let $\left({S, \circ}\right)$ be a semigroup. Let $\varnothing \subset X \subseteq S$. Let $\left({T, \circ}\right)$ be the smallest subsemigroup of $\left({S, \circ}\right)$ such that $X \subseteq T$. Then: : $X$ is a '''generator'''  of $\left({T, \circ}\right)$ : $X$ '''generates''' $\left({T, \circ}\right)$ : $\left({T, \circ}\right)$ is the '''subsemigroup of $\left({S, \circ}\right)$ generated by $X$'''. This is written $T = \left \langle {X} \right \rangle$.


Definition:Generated Algebraic Substructure

Let $\left({A, \circ}\right)$ be an algebraic structure. Let $G \subseteq A$ be any subset of $A$. The '''algebraic substructure generated''' by $G$ is the smallest substructure of $\left({A, \circ}\right)$ which contains $G$. It is written $\left \langle {G} \right \rangle$.


Definition:Generator of Algebra

Let $\left({A_R, \oplus}\right)$ be an algebra over a ring $R$. Let $S \subseteq A_R$ be a subset of $A_R$. The '''subalgebra generated by $S$''' is the smallest subalgebra $B_R$ of $A_R$ which contains $S$.


Definition:Alternative Algebra

Let $\struct {A_R, \oplus}$ be an algebra over a ring $R$. Then $\struct {A_R, \oplus}$ is an alternative algebra {{iff}} $\oplus$ is an alternative operation. That is: :For all $a, b \in A_R$, the subalgebra generated by $\set {a, b}$ is an associative algebra.


Definition:Power-Associative Algebra

Let $\left({A_R, \oplus}\right)$ be an algebra over a ring $R$. Then $\left({A_R, \oplus}\right)$ is a power-associative algebra {{iff}} $\oplus$ is power-associative. That is: :For all $a \in A_R$, the subalgebra generated by $\left\{{a}\right\}$ is an associative algebra.


Definition:Cayley-Dickson Construction

Let $A = \left({A_F, \oplus}\right)$ be a $*$-algebra. The '''Cayley-Dickson Construction''' on $A$ is the procedure which generates a new algebra $A'$ from $A$ as follows. Let: : $A' = \left({A'_F, \oplus'}\right) = \left({A, \oplus}\right)^2$ where $\left({A, \oplus}\right)^2$ denotes the Cartesian product of $\left({A, \oplus}\right)$ with itself. Then $\oplus'$ and $*'$ are defined on $A'$ as follows: :$\left({a, b}\right) \oplus' \left({c, d}\right) = \left({a \oplus c - d \oplus b^*, a^* \oplus d + c \oplus b}\right)$ :${\left({a, b}\right)^*}' = \left({a^*, -b}\right)$ where: : $\left({a, b}\right), \left({c, d}\right) \in A'$ : $a^*$ is the conjugation of $a \in A$.


Definition:Class Membership

To define membership not only for sets, but also for proper classes, we will extend the membership relation to include specific behaviors with proper classes and sets alike: :$\forall A, B: \paren {A \in B \iff \exists x: \paren {A = x \land x \in B } }$ With this definition, no proper classes is a member of any other class, proper or not.


Definition:Universal Class

The '''universal class''' is the class of which all sets are members. The '''universal class''' is defined most commonly in literature as: :$V = \set {x: x = x}$ where $x$ ranges over all sets. It can be briefly defined as the '''class of all sets'''.


Definition:Real Element in Star-Algebra

Let $A = \left({A_F, \oplus}\right)$ be a $*$-algebra. Let $A' = \left({A_F, \oplus'}\right)$ be constructed from $A$ using the Cayley-Dickson construction. Let $a \in A$ be real. Then $\left({a, 0}\right)$ is defined as real in $A'$. {{SUBPAGENAME}} hiap5z48d0c7np6b1ydn6qgk1gnjqd8


Definition:Octonion

The set of '''octonions''', usually denoted $\Bbb O$, can be defined by using the Cayley-Dickson construction from the quaternions $\Bbb H$ as follows: From Quaternions form Algebra, $\Bbb H$ forms a nicely normed $*$-algebra. Let $a, b \in \Bbb H$. Then $\tuple {a, b} \in \Bbb O$, where: :$\tuple {a, b} \tuple {c, d} = \tuple {a c - d \overline b, \overline a d + c b}$ :$\overline {\tuple {a, b} } = \tuple {\overline a, -b}$ where: :$\overline a$ is the conjugate on $a$ and  :$\overline {\tuple {a, b} }$ is the conjugation operation on $\Bbb O$.


Definition:Quaternion/Construction from Cayley-Dickson Construction

The set of quaternions $\Bbb H$ can be defined by the Cayley-Dickson construction from the set of complex numbers $\C$. From Complex Numbers form Algebra, $\C$ forms a nicely normed $*$-algebra. Let $a, b \in \C$. Then $\left({a, b}\right) \in \Bbb H$, where: :$\left({a, b}\right) \left({c, d}\right) = \left({a c - d \overline b, \overline a d + c b}\right)$ :$\overline {\left({a, b}\right)} = \left({\overline a, -b}\right)$ where: :$\overline a$ is the complex conjugate of $a$ and  :$\overline {\left({a, b}\right)}$ is the conjugation operation on $\Bbb H$. It is clear by direct comparison with the Construction from Complex Pairs that this construction genuinely does generate the Quaternions.


Definition:Complex Number/Construction from Cayley-Dickson Construction

The complex numbers can be defined by the Cayley-Dickson construction from the set of real numbers $\R$. From Real Numbers form Algebra, $\R$ forms a nicely normed $*$-algebra. Let $a, b \in \R$. Then $\left({a, b}\right) \in \C$, where: :$\left({a, b}\right) \left({c, d}\right) = \left({a c - d \overline b, \overline a d + c b}\right)$ :$\overline {\left({a, b}\right)} = \left({\overline a, -b}\right)$ where: :$\overline a$ is the conjugate of $a$ and  :$\overline {\left({a, b}\right)}$ is the conjugation operation on $\C$. From Real Numbers form Algebra, $\overline a = a$ and so the above translate into: :$\left({a, b}\right) \left({c, d}\right) = \left({a c - d b, a d + c b}\right)$ :$\overline {\left({a, b}\right)} = \left({a, -b}\right)$ It is clear by direct comparison with the formal definition that this construction genuinely does generate the complex numbers.


Definition:Monomial

=== Monomial of Polynomial Ring === {{:Definition:Monomial of Polynomial Ring}} === Monomial of Free Commutative Monoid === {{:Definition:Monomial of Free Commutative Monoid}}


Definition:Homogeneous Polynomial

A '''homogeneous polynomial''' is a polynomial whose monomials with nonzero coefficients all have the same total degree.


Definition:Quadratic Form

Let $\mathbb K$ be a field of characteristic $\Char {\mathbb K} \ne 2$. Let $V$ be a vector space over $\mathbb K$. A '''quadratic form''' on $V$ is a mapping $q : V \mapsto \mathbb K$ such that: :$\forall v \in V : \forall \kappa \in \mathbb K : \map q {\kappa v} = \kappa^2 \map q v$ :$b: V \times V \to \mathbb K: \tuple {v, w} \mapsto \map q {v + w} - \map q v - \map q w$ is a bilinear form


Definition:Galois Field

A '''Galois field''' $\struct {\GF, +, \circ}$ is a field such that $\GF$ is a finite set.


Definition:Infinite Field

An infinite field $\struct {F, +, \cdot}$ is a field such that $F$ is a infinite set.


Definition:Subtraction/Ring

Let $\struct {R, +, \circ}$ be a ring. The operation of '''subtraction''' $a - b$ on $R$ is defined as: :$\forall a, b \in R: a - b := a + \paren {-b}$ where $-b$ is the (ring) negative of $b$.


Definition:Zero Divisor/Ring

Let $\struct {R, +, \circ}$ be a ring. A '''zero divisor (in $R$)''' is an element $x \in R$ such that either: :$\exists y \in R^*: x \circ y = 0_R$ or: :$\exists y \in R^*: y \circ x = 0_R$ where $R^*$ is defined as $R \setminus \set {0_R}$. That is, such that $x$ is either a left zero divisor or a right zero divisor. The expression: : '''$x$ is a zero divisor''' can be written: : $x \divides 0_R$


Definition:Zero Divisor/Algebra

Let $\struct {A_R, \oplus}$ be an algebra over a ring $\struct {R, +, \cdot}$. Let the zero vector of $A_R$ be $\mathbf 0_R$. Let $a, b \in A_R$ such that $a \ne \mathbf 0_R$ and $b \ne \mathbf 0_R$. Then $a$ and $b$ are '''zero divisors of $A_R$''' {{iff}}: :$a \oplus b = \mathbf 0_R$


Definition:Multiplicative Group

Let $\struct {F, +, \times}$ be a field. Let $F^* := F \setminus \set 0$ be the set $F$ less its zero. The group $\struct {F^*, \times}$ is known as the '''multiplicative group of $F$'''.


Definition:Multiplicative Identity

Let $\struct {F, +, \times}$ be a field. Then the identity element of the multiplicative group $\struct {F^*, \times}$ of $F$ is called the multiplicative identity of $F$. It is often denoted $e_F$ or $1_F$, or, if there is no danger of ambiguity, $e$ or $1$. {{refactor|Either merge this concept with that of Definition:Unity of Field or enter this statement as a page in its own right.}} Note that the multiplicative identity of $F$ is the unity of the ring that $\struct {F, +, \times}$ is by definition of a field.


Definition:Multiplicative Inverse/Field

Let $\struct {F, +, \times}$ be a field whose zero is $0_F$. Let $a \in F$ such that $a \ne 0_F$. Then the inverse element of $a$ with respect to the $\times$ operator is called the '''multiplicative inverse''' of $F$. It is usually denoted $a^{-1}$ or $\dfrac 1 a$.


Definition:Affine Transformation

Let $\mathcal E$ and $\mathcal F$ be affine spaces with difference spaces $E$ and $F$ respectively. Let $\mathcal L: \mathcal E \to \mathcal F$ be a mapping. Then $\mathcal L$ is an '''affine transformation''' or '''affine mapping''' if there exists a linear transformation $L: E \to F$ such that for every pair of points $p, q \in \mathcal E$: :$\map {\mathcal L} q = \map {\mathcal L} p + \map L {\vec {p q} }$


Definition:Annihilator of Ring

Let $B: R \times \Z$ be a bilinear mapping defined as: :$B: R \times \Z: \tuple {r, n} \mapsto n \cdot r$ where $n \cdot r$ defined as an integral multiple of $r$: :$n \cdot r = r + r + \cdots \paren n \cdots r$ Note the change of order of $r$ and $n$: :$\map B {r, n} = n \cdot r$ Let $D \subseteq R$ be a subring of $R$. Then the '''annihilator''' of $D$ is defined as: :$\map {\mathrm {Ann} } D = \set {n \in \Z: \forall d \in D: n \cdot d = 0_R}$ or, when $D = R$: :$\map {\mathrm {Ann} } R = \set {n \in \Z: \forall r \in R: n \cdot r = 0_R}$ It is seen to be, therefore, the set of all integers whose integral multiples, with respect to the elements of a ring or a field, are all equal to the zero of that ring or field. === Trivial Annihilator === {{:Definition:Trivial Annihilator}}


Definition:Ring Endomorphism

Let $\struct {R, +, \circ}$ be a ring. Let $\phi: R \to R$ be a (ring) homomorphism from $R$ to itself. Then $\phi$ is a ring endomorphism.


Definition:Zero Homomorphism

Let $\struct {R_1, +_1, \circ_1}$ and $\struct {R_2, +_2, \circ_2}$ be rings with zeroes $0_1$ and $0_2$ respectively. Consider the mapping $\zeta: R_1 \to R_2$ defined as: :$\forall r \in R_1: \map \zeta r = 0_2$ Then $\zeta$ is '''the zero homomorphism from $R_1$ to $R_2$'''.


Definition:Identity Automorphism

Let $\struct {R, +, \circ}$ be a ring. Consider the identity mapping $I_R: R \to R$: :$\forall r \in R: \map {I_R} r = r$ Then $I_R$ is '''the identity automorphism on $R$'''.


Definition:Field Monomorphism

Let $\struct {F, +, \circ}$ and $\struct {K, \oplus, *}$ be fields. Let $\phi: F \to K$ be a (field) homomorphism. Then $\phi$ is a field monomorphism {{iff}} $\phi$ is an injection.


Definition:Isomorphism (Abstract Algebra)/F-Isomorphism

Let $R, S$ be rings with unity. Let $F$ be a subfield of both $R$ and $S$. Let $\phi: R \to S$ be an $F$-homomorphism such that $\phi$ is bijective. Then $\phi$ is an '''$F$-isomorphism'''. The relationship between $R$ and $S$ is denoted $R \cong_F S$.


Definition:R-Algebraic Structure Epimorphism

Let $\left({S, \ast_1, \ast_2, \ldots, \ast_n, \circ}\right)_R$ and $\left({T, \odot_1, \odot_2, \ldots, \odot_n, \otimes}\right)_R$ be $R$-algebraic structures. Then $\phi: S \to T$ is an '''$R$-algebraic structure epimorphism''' {{iff}}: : $(1): \quad \phi$ is a surjection : $(2): \quad \forall k: k \in \left[{1 \,.\,.\, n}\right]: \forall x, y \in S: \phi \left({x \ast_k y}\right) = \phi \left({x}\right) \odot_k \phi \left({y}\right)$ : $(3): \quad \forall x \in S: \forall \lambda \in R: \phi \left({\lambda \circ x}\right) = \lambda \otimes \phi \left({x}\right)$ This definition also applies to modules, and also to vector spaces.


Definition:Field Epimorphism

Let $\left({F, +, \circ}\right)$ and $\left({K, \oplus, *}\right)$ be fields. Let $\phi: R \to S$ be a (field) homomorphism. Then $\phi$ is a field epimorphism {{iff}} $\phi$ is a surjection.


Definition:Group Endomorphism

Let $\struct {G, \circ}$ be a group. Let $\phi: G \to G$ be a (group) homomorphism from $G$ to itself. Then $\phi$ is a group endomorphism.


Definition:R-Algebraic Structure Endomorphism

Let $\struct {S, \ast_1, \ast_2, \ldots, \ast_n, \circ}_R$ be an $R$-algebraic structure. Let $\phi: S \to S$ be an $R$-algebraic structure homomorphism from $S$ to itself. Then $\phi$ is an $R$-algebraic structure endomorphism. This definition continues to apply when $S$ is a module, and also when it is a vector space.


Definition:Field Endomorphism

Let $\struct {F, +, \circ}$ be a field. Let $\phi: F \to F$ be a (field) homomorphism from $F$ to itself. Then $\phi$ is a field endomorphism.


Definition:Homogeneous Linear Equations

A '''system of homogeneous linear equations''' is a set of simultaneous linear equations: :$\displaystyle \forall i \in \closedint 1 m: \sum_{j \mathop = 1}^n \alpha_{i j} x_j = \beta_i$ such that all the $\beta_i$ are equal to zero: :$\displaystyle \forall i \in \closedint 1 m : \sum_{j \mathop = 1}^n \alpha_{i j} x_j = 0$ That is: {{begin-eqn}} {{eqn | l = 0       | r = \alpha_{11} x_1 + \alpha_{12} x_2 + \cdots + \alpha_{1n} x_n }} {{eqn | l = 0       | r = \alpha_{21} x_1 + \alpha_{22} x_2 + \cdots + \alpha_{2n} x_n }} {{eqn | o = \cdots}} {{eqn | l = 0       | r = \alpha_{m1} x_1 + \alpha_{m2} x_2 + \cdots + \alpha_{mn} x_n }} {{end-eqn}} === Matrix Representation === {{:Definition:Homogeneous Linear Equations/Matrix Representation}}


Definition:Set/Explicit Set Definition

A (finite) set can be defined by '''explicitly''' specifying ''all'' of its elements between the famous curly brackets, known as '''set braces''': $\set {}$. When a set is defined like this, note that ''all'' and ''only'' the elements in it are listed. This is called '''explicit (set) definition'''. It is possible for a set to contain other sets. For example: :$S = \set {a, \set a}$


Definition:Set/Implicit Set Definition

If the elements in a set have an ''obvious'' pattern to them, we can define the set '''implicitly''' by using an ellipsis ($\ldots$). For example, suppose $S = \set {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}$. A more compact way of defining this set is: :$S = \set {1, 2, \ldots, 10}$ With this notation we are asked to suppose that the numbers count up uniformly, and we can read this definition as: :'''$S$ is the set containing $1$, $2$, and so on, up to $10$.'''


Definition:Set/Definition by Predicate

An object can be specified by means of a '''predicate''', that is, in terms of a property (or properties) that it possesses. Whether an object $x$ possesses a particular (characteristic) property $P$ is either true or false (in Aristotelian logic) and so can be the subject of a propositional function $\map P x$. Hence a set can be specified by means of such a propositional function: :$S = \set {x: \map P x}$ which means: :'''$S$ is the set of all objects which have the property $P$''' or, more formally: :'''$S$ is the set of all $x$ such that $\map P x$ is true.''' In this context, we see that the symbol $:$ is interpreted as '''such that'''.


Definition:Real Number/Operations on Real Numbers

We interpret the following symbols: {{begin-axiom}} {{axiom | n = \text R 1         | lc= Negative         | q = \forall a \in \R         | m = \exists ! \paren {-a} \in \R: a + \paren {-a} = 0 }} {{axiom | n = \text R 2         | lc= Minus         | q = \forall a, b \in \R         | m = a - b = a + \paren {-b} }} {{axiom | n = \text R 3         | lc= Reciprocal         | q = \forall a \in \R \setminus \set 0         | m = \exists ! a^{-1} \in \R: a \times \paren {a^{-1} } = 1 = \paren {a^{-1} } \times a         | rc= it is usual to write $1/a$ or $\dfrac 1 a$ for $a^{-1}$ }} {{axiom | n = \text R 4         | lc= Divided by         | q = \forall a \in \R \setminus \set 0         | m = a \div b = \dfrac a b = a / b = a \times \paren {b^{-1} }         | rc= it is usual to write $1/a$ or $\dfrac 1 a$ for $a^{-1}$ }} {{end-axiom}} The validity of all these operations is justified by Real Numbers form Field.


Definition:Real Number/Real Number Line

From the Cantor-Dedekind Hypothesis, the set of real numbers is isomorphic to any infinite straight line. The '''real number line''' is an arbitrary infinite straight line each of whose points is identified with a real number such that the distance between any two real numbers is consistent with the length of the line between those two points. :800px


Definition:Bounded Above Set

Let $\struct {S, \preceq}$ be an ordered set. A subset $T \subseteq S$ is '''bounded above (in $S$)''' {{iff}} $T$ admits an upper bound (in $S$).


Definition:Bounded Above Mapping

Let $f: S \to T$ be a mapping whose codomain is an ordered set $\struct {T, \preceq}$. Then $f$ is '''bounded above on $S$''' by the upper bound $H$ {{iff}}: :$\forall x \in S: \map f x \preceq H$ That is, {{iff}} $f \sqbrk S = \set {\map f x: x \in S}$ is bounded above by $H$.


Definition:Supremum of Set

Let $\struct {S, \preceq}$ be an ordered set. Let $T \subseteq S$. An element $c \in S$ is the '''supremum of $T$ in $S$''' {{iff}}: :$(1): \quad c$ is an upper bound of $T$ in $S$ :$(2): \quad c \preceq d$ for all upper bounds $d$ of $T$ in $S$. If there exists a '''supremum''' of $T$ (in $S$), we say that: :'''$T$ admits a supremum (in $S$)''' or :'''$T$ has a supremum (in $S$)'''.


{{:Definition:Supremum of Mapping/Real-Valued Function/Definition 1}}


Definition:Maximal/Ordered Set

{{:Definition:Maximal/Ordered Set/Definition 1}}


Definition:Maximum Value of Real Function/Absolute

Let $f: \R \to \R$ be a real function. Let $f$ be bounded above by a supremum $B$. It may or may not be the case that $\exists x \in \R: \map f x = B$. If such a value exists, it is called the '''maximum''' of $f$ on $S$, and that this '''maximum''' is '''attained at $x$'''.


Definition:Bounded Below Set

Let $\struct {S, \preceq}$ be an ordered set. A subset $T \subseteq S$ is '''bounded below (in $S$)''' {{iff}} $T$ admits a lower bound (in $S$).


Definition:Bounded Below Mapping

Let $f: S \to T$ be a mapping whose codomain is an ordered set $\struct {T, \preceq}$. Then $f$ is said to be '''bounded below (in $T$)''' by the lower bound $L$ {{iff}}: :$\forall x \in S: L \preceq \map f x$ That is, iff $f \sqbrk S = \set {\map f x: x \in S}$ is bounded below by $L$.


Definition:Upper Bound of Set

Let $\struct {S, \preceq}$ be an ordered set. Let $T$ be a subset of $S$. An '''upper bound for $T$ (in $S$)''' is an element $M \in S$ such that: :$\forall t \in T: t \preceq M$ That is, $M$ succeeds every element of $T$.


Definition:Upper Bound of Mapping

Let $f: S \to T$ be a mapping whose codomain is an ordered set $\struct {T, \preceq}$. Let $f$ be bounded above in $T$ by $H \in T$. Then $H$ is an '''upper bound of $f$'''.


Definition:Lower Bound of Set

Let $\struct {S, \preceq}$ be an ordered set. Let $T$ be a subset of $S$. A '''lower bound for $T$ (in $S$)''' is an element $m \in S$ such that: :$\forall t \in T: m \preceq t$ That is, $m$ precedes every element of $T$.


Definition:Lower Bound of Mapping

Let $f: S \to T$ be a mapping whose codomain is an ordered set $\struct {T, \preceq}$. Let $f$ be bounded below in $T$ by $H \in T$. Then $H$ is a '''lower bound of $f$'''.


Definition:Infimum of Set

Let $\struct {S, \preceq}$ be an ordered set. Let $T \subseteq S$. An element $c \in S$ is the '''infimum of $T$ in $S$''' {{iff}}: :$(1): \quad c$ is a lower bound of $T$ in $S$ :$(2): \quad d \preceq c$ for all lower bounds $d$ of $T$ in $S$. If there exists an '''infimum''' of $T$ (in $S$), we say that '''$T$ admits an infimum (in $S$)'''.


Definition:Infimum of Mapping

Let $S$ be a set. Let $\struct {T, \preceq}$ be an ordered set. Let $f: S \to T$ be a mapping from $S$ to $T$. Let $f \sqbrk S$, the image of $f$, admit an infimum. Then the '''infimum''' of $f$ (on $S$) is defined by: :$\displaystyle \inf_{x \mathop \in S} \map f x = \inf f \sqbrk S$ === Real-Valued Function === {{:Definition:Infimum of Mapping/Real-Valued Function}}


Definition:Bounded Ordered Set

Let $\struct {S, \preceq}$ be an ordered set. Let $T \subseteq S$ be both bounded below and bounded above in $S$. Then $T$ is '''bounded in $S$'''.


Definition:Bounded Mapping

Let $\struct {T, \preceq}$ be an ordered set. Let $f: S \to T$ be a mapping. Let the image of $f$ be bounded. Then $f$ is '''bounded'''. That is, $f$ is '''bounded''' {{iff}} it is both bounded above and bounded below.


Definition:Minimal/Ordered Set

{{:Definition:Minimal/Ordered Set/Definition 1}}


Definition:Minimum Value of Real Function/Absolute

Let $f: \R \to \R$ be a real function. Let $f$ be bounded below by an infimum $B$. It may or may not be the case that $\exists x \in \R: \map f x = B$. If such a value exists, it is called the '''minimum value''' of $f$ on $S$, and this '''minimum''' is '''attained at $x$'''.


Definition:Minimal/Set

Let $S$ be a set. Let $\powerset S$ be the power set of $S$. Let $\mathcal T \subseteq \powerset S$ be a subset of $\powerset S$. Let $\struct {\mathcal T, \subseteq}$ be the ordered set formed on $\mathcal T$ by $\subseteq$ considered as an ordering. Then $T \in \mathcal T$ is a '''minimal set''' of $\mathcal T$ {{iff}} $T$ is a minimal element of $\struct {\mathcal T, \subseteq}$. That is: :$\forall X \in \mathcal T: X \subseteq T \implies X = T$


Definition:Continuous Real Function/Interval

=== Open Interval === This is a straightforward application of continuity on a set. {{Definition:Continuous Real Function/Open Interval}} === Closed Interval === {{Definition:Continuous Real Function/Closed Interval}} === Half Open Intervals === Similar definitions apply to half open intervals: {{Definition:Continuous Real Function/Half Open Interval}}


Definition:Derivative/Complex Function

The definition for a complex function is similar to that for real functions. === At a Point === {{:Definition:Derivative/Complex Function/Point}} === On an Open Set === {{:Definition:Derivative/Complex Function/Open Set}}


Let $I$ be an open real interval. Let $f: I \to \R$ be a real function defined on $I$. Let $\xi \in I$ be a point in $I$. Let $f$ be differentiable at the point $\xi$. ==== Definition 1 ==== {{:Definition:Derivative/Real Function/Derivative at Point/Definition 1}} ==== Definition 2 ==== {{:Definition:Derivative/Real Function/Derivative at Point/Definition 2}}


Definition:Derivative/Real Function/Derivative on Interval

Let $I\subset\R$ be an open interval. Let $f : I \to \R$ be a real function. Let $f$ be differentiable on the interval $I$. Then the '''derivative of $f$''' is the real function  $f': I \to \R$ whose value at each point $x \in I$ is the derivative $f' \left({x}\right)$: :$\displaystyle \forall x \in I: f' \left({x}\right) := \lim_{h \mathop \to 0} \frac {f \left({x + h}\right) - f \left({x}\right)} h$


Definition:Derivative/Real Function

=== At a Point === {{:Definition:Derivative/Real Function/Derivative at Point}} === On an Open Interval === {{:Definition:Derivative/Real Function/Derivative on Interval}} === With Respect To === {{:Definition:Derivative/Real Function/With Respect To}}


Definition:Derivative/Higher Derivatives/Order of Derivative

The '''order''' of a derivative is the '''number of times it has been differentiated'''. For example: :a first derivative is of '''first order''', or '''order $1$''' :a second derivative is of '''second order''', or '''order $2$''' and so on.


=== Second Derivative === {{:Definition:Derivative/Higher Derivatives/Second Derivative}} === Third Derivative === {{:Definition:Derivative/Higher Derivatives/Third Derivative}} === Higher Order Derivatives === Higher order derivatives are defined in similar ways: {{:Definition:Derivative/Higher Derivatives/Higher Order}} === First Derivative === If derivatives of various orders are being discussed, then what has been described here as the derivative is frequently referred to as the '''first derivative''': {{:Definition:Derivative/Real Function/Derivative on Interval}} == Order of Derivative == {{:Definition:Derivative/Higher Derivatives/Order of Derivative}}


Definition:Derivative/Real Function/With Respect To

Let $f$ be a real function which is differentiable on an open interval $I$. Let $f$ be defined as an equation: $y = \map f x$. Then the '''derivative of $y$ with respect to $x$''' is defined as: :$\displaystyle y^\prime = \lim_{h \mathop \to 0} \frac {\map f {x + h} - \map f x} h = D_x \, \map f x$ This is frequently abbreviated as '''derivative of $y$''' '''WRT''' or '''w.r.t.''' '''$x$''', and often pronounced something like '''wurt'''. We introduce the quantity $\delta y = \map f {x + \delta x} - \map f x$. This is often referred to as '''the small change in $y$ consequent on the small change in $x$'''. Hence the motivation behind the popular and commonly-seen notation: :$\displaystyle \dfrac {\d y} {\d x} := \lim_{\delta x \mathop \to 0} \dfrac {\map f {x + \delta x} - \map f x} {\delta x} = \lim_{\delta x \mathop \to 0} \dfrac {\delta y} {\delta x}$ Hence the notation $\map {f^\prime} x = \dfrac {\d y} {\d x}$. This notation is useful and powerful, and emphasizes the concept of a derivative as being the limit of a ratio of very small changes. However, it has the disadvantage that the variable $x$ is used ambiguously: both as the point at which the derivative is calculated and as the variable with respect to which the derivation is done. For practical applications, however, this is not usually a problem.


Definition:O Notation/Big-O Notation

'''Big-O notation''' occurs in a variety of contexts. === Sequences === {{:Definition:O Notation/Big-O Notation/Sequence}} === Real Analysis === {{:Definition:O Notation/Big-O Notation/Real/Infinity}} === Complex Analysis === {{:Definition:O Notation/Big-O Notation/Complex/Infinity}} === General Definition === {{:Definition:O Notation/Big-O Notation/General Definition/Infinity}}


Definition:O Notation/Little-O Notation

'''Little-O notation''' occurs in a variety of contexts. === Sequences === Let $\sequence {a_n}$ and $\sequence {b_n}$ be sequences of real or complex numbers. {{:Definition:O Notation/Little-O Notation/Sequence/Definition 1}} === Real Functions === Let $f$ and $g$ be real-valued or complex-valued functions on a subset of $\R$ containing all sufficiently large real numbers. {{:Definition:O Notation/Little-O Notation/Real Functions/Definition 1}} === Point Estimate === {{:Definition:O Notation/Little-O Notation/Real Point}} === Complex Functions === {{:Definition:O Notation/Little-O Notation/Complex Functions}} === Complex Point Estimate === {{:Definition:O Notation/Little-O Notation/Complex Point}} === General Definition for point estimates === {{:Definition:O Notation/Little-O Notation/General Definition/Point}}


Definition:Differential

=== Real Function === {{:Definition:Differential/Real Function/Point}} === Real-Valued Function === {{:Definition:Differential/Real-Valued Function/Point}} === Vector-Valued Function === {{:Definition:Differential/Vector-Valued Function/Point}} === Manifolds === {{:Definition:Differential/Manifolds}} === Functional === {{:Definition:Differential/Functional}}


Definition:Strictly Monotone/Sequence

Let $\struct {S, \preceq}$ be a totally ordered set. Then a sequence $\sequence {a_k}_{k \mathop \in A}$ of elements of $S$ is '''strictly monotone''' {{iff}} it is either strictly increasing or strictly decreasing.


Definition:Increasing/Mapping

Let $\struct {S, \preceq_1}$ and $\struct {T, \preceq_2}$ be ordered sets. Let $\phi: S \to T$ be a mapping. Then $\phi$ is '''increasing''' {{iff}}: :$\forall x, y \in S: x \preceq_1 y \implies \map \phi x \preceq_2 \map \phi y$ Note that this definition also holds if $S = T$.


Definition:Increasing/Real Function

Let $f$ be a real function. Then $f$ is '''increasing''' {{iff}}: :$x \le y \implies \map f x \le \map f y$


Definition:Strictly Increasing/Mapping

Let $\left({S, \preceq_1}\right)$ and $\left({T, \preceq_2}\right)$ be ordered sets. Let $\phi: \left({S, \preceq_1}\right) \to \left({T, \preceq_2}\right)$ be a mapping. Then $\phi$ is '''strictly increasing''' if: :$\forall x, y \in S: x \prec_1 y \implies \phi \left({x}\right) \prec_2 \phi \left({y}\right)$ Note that this definition also holds if $S = T$.


Definition:Strictly Increasing/Real Function

Let $f$ be a real function. Then $f$ is '''strictly increasing''' {{iff}}: : $x < y \implies f \left({x}\right) < f \left({y}\right)$


Definition:Strictly Decreasing/Mapping

Let $\left({S, \preceq_1}\right)$ and $\left({T, \preceq_2}\right)$ be ordered sets. Let $\phi: \left({S, \preceq_1}\right) \to \left({T, \preceq_2}\right)$ be a mapping. Then $\phi$ is '''strictly decreasing''' {{iff}}: :$\forall x, y \in S: x \prec_1 y \implies \phi \left({y}\right) \prec_2 \phi \left({x}\right)$ Note that this definition also holds if $S = T$.


Definition:Strictly Decreasing/Real Function

Let $f$ be a real function. Then $f$ is '''strictly decreasing''' {{iff}}: :$x < y \implies \map f y < \map f x$


Definition:Decreasing/Mapping

Let $\struct {S, \preceq_1}$ and $\struct {T, \preceq_2}$ be ordered sets. Let $\phi: \struct {S, \preceq_1} \to \struct {T, \preceq_2}$ be a mapping. Then $\phi$ is '''decreasing''' {{iff}}: :$\forall x, y \in S: x \preceq_1 y \implies \map \phi y \preceq_2 \map \phi x$ Note that this definition also holds if $S = T$.


Definition:Decreasing/Real Function

Let $f$ be a real function. Then $f$ is '''decreasing''' {{iff}}: :$x \le y \implies \map f y \le \map f x$.


Definition:Monotone (Order Theory)/Mapping

Let $\left({S, \preceq_1}\right)$ and $\left({T, \preceq_2}\right)$ be ordered sets. Let $\phi: \left({S, \preceq_1}\right) \to \left({T, \preceq_2}\right)$ be a mapping. Then $\phi$ is '''monotone''' {{iff}} it is either increasing or decreasing. Note that this definition also holds if $S = T$.


Definition:Monotone (Order Theory)/Real Function

This definition continues to hold when $S = T = \R$. Thus, let $f$ be a real function. Then $f$ is '''monotone''' {{iff}} it is either increasing or decreasing.


Definition:Strictly Monotone/Mapping

Let $\left({S, \preceq_1}\right)$ and $\left({T, \preceq_2}\right)$ be ordered sets. Let $\phi: \left({S, \preceq_1}\right) \to \left({T, \preceq_2}\right)$ be a mapping. Then $\phi$ is '''strictly monotone''' {{iff}} it is either strictly increasing or strictly decreasing. Note that this definition also holds if $S = T$.


Definition:Strictly Monotone/Real Function

Let $f: S \to \R$ be a real function, where $S \subseteq \R$. Then $f$ is '''strictly monotone''' {{iff}} it is either strictly increasing or strictly decreasing.


Definition:Analytic Function/Real Numbers

Let $f$ be a real function which is smooth on the open interval $\openint a b$. Let $\xi \in \openint a b$. Let $\openint c d \subseteq \openint a b$ be an open interval such that: :$(1): \quad \xi \in \openint c d$ :$(2): \quad \displaystyle \forall x \in \openint c d: \map f x = \sum_{n \mathop = 0}^\infty \frac {\paren {x - \xi}^n} {n!} \map {f^{\paren n} } x$ Then $f$ is described as being '''analytic''' at the point $\xi$. That is, a function is analytic at a point if it equals its Taylor series expansion in some interval containing that point.


Definition:Analytic Function/Complex Plane

Let $U \subset \C$ be an open set. Let $f : U \to \C$ be a complex function. Then $f$ is '''analytic''' in $U$ {{iff}} for every $z_0 \in U$ there exists a sequence $(a_n) : \N \to \C$ such that the series: :$\displaystyle \sum_{n\mathop = 0}^\infty a_n(z-z_0)^n$ converges to $f(z)$ in a neighborhood of $z_0$ in $U$.


Definition:Tangent/Geometry

=== Tangent Line === {{:Definition:Tangent/Geometry/Tangent Line}} === Tangent Circles === {{:Definition:Tangent/Geometry/Tangent Circles}}


Definition:Cosecant/Analysis

==== Real Function ==== {{:Definition:Cosecant/Real Function}} ==== Complex Function ==== {{:Definition:Cosecant/Complex Function}}


Definition:Propositional Expansion/Universal Quantifier

Let $\forall$ be the universal quantifier. What $\forall x: \map P x$ means is: :$\mathbf X_1$ has property $P$, and $\mathbf X_2$ has property $P$, and $\mathbf X_3$ has property $P$, and ... This translates into propositional logic as: :$\map P {\mathbf X_1} \land \map P {\mathbf X_2} \land \map P {\mathbf X_3} \land \ldots$ This expression of $\forall x$ as a conjunction is known as the '''propositional expansion''' of $\forall x$.  The '''propositional expansion''' for the universal quantifier can exist in actuality only when the number of objects in the universe is finite. If the universe is infinite, then the '''propositional expansion''' can exist only conceptually, and the universal quantifier cannot be eliminated.


Definition:Propositional Expansion/Existential Quantifier

Let $\exists$ be the existential quantifier. What $\exists x: \map P x$ means is: :At least one of $\mathbf X_1, \mathbf X_2, \mathbf X_3, \ldots$ has property $P$. This means: :Either $\mathbf X_1$ has property $P$, or $\mathbf X_2$ has property $P$, or $\mathbf X_3$ has property $P$, or ... This translates into propositional logic as: :$\map P {\mathbf X_1} \lor \map P {\mathbf X_2} \lor \map P {\mathbf X_3} \lor \ldots$ This expression of $\exists x$ as a disjunction is known as the '''propositional expansion''' of $\exists x$. The propositional expansion for the existential quantifier can exist in actuality only when the number of objects in the universe is finite. If the universe is infinite, then the propositional expansion can exist only conceptually, and the existential quantifier cannot be eliminated.


Definition:Inverse Statement

The '''inverse''' of the conditional: : $p \implies q$ is the statement: :$\neg p \implies \neg q$


Definition:Non-reflexive Relation

$\mathcal R$ is '''non-reflexive''' {{iff}} it is neither reflexive nor antireflexive.


Definition:Non-symmetric Relation

$\RR$ is '''non-symmetric''' {{iff}} it is neither symmetric nor asymmetric.


Definition:Non-transitive Relation

$\RR$ is '''non-transitive''' {{iff}} it is neither transitive nor antitransitive.


Definition:Restriction/Operation

Let $\left({S, \circ}\right)$ be an algebraic structure, and let $T \subseteq S$. The '''restriction of $\circ$ to $T \times T$''' is denoted $\circ {\restriction_T}$, and is defined as: :$\forall t_1, t_2 \in T: t_1 \mathbin{\circ {\restriction_T}} t_2 = t_1 \circ t_2$ The notation $\circ {\restriction_T}$ is generally used only if it is necessary to emphasise that $\circ {\restriction_T}$ is strictly different from $\circ$ (through having a different domain). When no confusion is likely to result, $\circ$ is generally used for both. Thus in this context, $\left({T, \circ {\restriction_T}}\right)$ and $\left({T, \circ}\right)$ mean the same thing.


Definition:Restriction/Relation

Let $\RR$ be a relation on $S \times T$. Let $X \subseteq S$, $Y \subseteq T$. The '''restriction of $\RR$ to $X \times Y$''' is the relation on $X \times Y$ defined as: :$\RR {\restriction_{X \times Y} }: = \RR \cap \paren {X \times Y}$ If $Y = T$, then we simply call this the '''restriction of $\RR$ to $X$''', and denote it as $\RR {\restriction_X}$.


Definition:Restriction/Mapping

Let $f: S \to T$ be a mapping. Let $X \subseteq S$. Let $f \sqbrk X \subseteq Y \subseteq T$. The '''restriction of $f$ to $X \times Y$''' is the mapping $f {\restriction_{X \times Y} }: X \to Y$ defined as: :$f {\restriction_{X \times Y} } = f \cap \paren {X \times Y}$ If $Y = T$, then we simply call this the '''restriction of $f$ to $X$''', and denote it as $f {\restriction_X}$.


Definition:Identity (Abstract Algebra)/Left Identity

An element $e_L \in S$ is called a '''left identity''' {{iff}}: :$\forall x \in S: e_L \circ x = x$


Definition:Identity (Abstract Algebra)/Right Identity

An element $e_R \in S$ is called a '''right identity''' {{iff}}: :$\forall x \in S: x \circ e_R = x$


Definition:Identity (Abstract Algebra)/Two-Sided Identity

An element $e \in S$ is called an '''identity (element)''' {{iff}} it is both a left identity and a right identity: :$\forall x \in S: x \circ e = x = e \circ x$ In Identity is Unique it is established that an identity element, if it exists, is unique within $\struct {S, \circ}$. Thus it is justified to refer to it as '''the''' identity (of a given algebraic structure). This identity is often denoted $e_S$, or $e$ if it is clearly understood what structure is being discussed.


Definition:Inverse (Abstract Algebra)/Left Inverse

An element $x_L \in S$ is called a '''left inverse''' of $x$ {{iff}}: :$x_L \circ x = e_S$


Definition:Inverse (Abstract Algebra)/Right Inverse

An element $x_R \in S$ is called a '''right inverse''' of $x$ {{iff}}: :$x \circ x_R = e_S$


Definition:Inverse (Abstract Algebra)/Inverse

The element $y$ is an '''inverse of $x$''' {{iff}}: :$y \circ x = e_S = x \circ y$ that is, {{iff}} $y$ is both: :a left inverse of $x$ and: :a right inverse of $x$.


Definition:Idempotence/Mapping

Let $f: S \to S$ be a mapping. Then $f$ is '''idempotent''' {{iff}}: :$\forall x \in S: f \left({f \left({x}\right)}\right) = f \left({x}\right)$ That is, {{iff}} applying the same mapping a second time to an argument gives the same result as applying it once. And of course, that means the same as applying it as many times as you want. The condition for '''idempotence''' can also be written: :$f \circ f = f$ where $\circ$ denotes composition of mappings.


Definition:Idempotence/Operation

Let $\circ: S \times S \to S$ be a binary operation on $S$. If ''all'' the elements of $S$ are idempotent under $\circ$, then the term can be applied to the operation itself: The binary operation $\circ$ is '''idempotent''' {{iff}}: : $\forall x \in S: x \circ x = x$


Definition:Idempotence/Element

Let $\circ: S \times S \to S$ be a binary operation on $S$. Let $x \in S$ have the property that $x \circ x = x$. Then $x \in S$ is described as '''idempotent under the operation $\circ$'''.


Definition:Unique

Suppose $A$ and $B$ are two objects whose definition is in terms of a given set of properties. If it can be demonstrated that, in order for both $A$ and $B$ to fulfil those properties, it is necessary for $A$ to be equal to $B$, then $A$ (and indeed $B$) is '''unique'''. Equivalently, there is '''one and only one''', or '''exactly one''', such object. Thus, intuitively, an object is '''unique''' if there is precisely one such object. === Unique Existential Quantifier === In the language of predicate logic, '''uniqueness''' can be defined as follows: Let $\map P x$ be a propositional function and let $x$ and $y$ be objects.  {{:Definition:Existential Quantifier/Unique}}


Definition:Paradox

A '''paradox''' is a statement or group of statements that leads to one of the following: * a contradiction * a situation which defies intuition * a result that is merely "puzzling".


Definition:Regular Representations/Left Regular Representation

The mapping $\lambda_a: S \to S$ is defined as: :$\forall x \in S: \map {\lambda_a} x = a \circ x$ This is known as the '''left regular representation of $\struct {S, \circ}$ with respect to $a$'''.


Definition:Regular Representations/Right Regular Representation

The mapping $\rho_a: S \to S$ is defined as: :$\forall x \in S: \map {\rho_a} x = x \circ a$ This is known as the '''right regular representation of $\struct {S, \circ}$ with respect to $a$'''.


Definition:Centralizer/Group Element

Let $\struct {G, \circ}$ be a group. Let $a \in \struct {G, \circ}$. The '''centralizer of $a$ (in $G$)''' is defined as: :$\map {C_G} a = \set {x \in G: x \circ a = a \circ x}$ That is, the centralizer of $a$ is the set of elements of $G$ which commute with $a$.


Definition:Centralizer/Subgroup

Let $\struct {G, \circ}$ be a group. Let $H \le \struct {G, \circ}$. The '''centralizer of $H$ (in $G$)''' is the set of elements of $G$ which commute with all $h \in H$: :$\map {C_G} H = \set {g \in G: \forall h \in H: g \circ h = h \circ g}$


Definition:Centralizer/Ring Subset

Let $S$ be a subset of a ring $\struct {R, +, \circ}$. The '''centralizer of $S$ in $R$''' is defined as: :$\map {C_R} S = \set {x \in R: \forall s \in S: s \circ x = x \circ s}$ That is, the '''centralizer of $S$''' is the set of elements of $R$ which commute with all elements of $S$.


Definition:Center (Abstract Algebra)

=== Group === {{:Definition:Center (Abstract Algebra)/Group}} === Ring === {{:Definition:Center (Abstract Algebra)/Ring}}


Let $\struct {G, \circ}$ be a group. === Definition 1 === {{:Definition:Conjugate (Group Theory)/Element/Definition 1}} === Definition 2 === {{:Definition:Conjugate (Group Theory)/Element/Definition 2}}


Definition:Conjugate (Group Theory)/Subset

Let $S \subseteq G, a \in G$. Then the '''$G$-conjugate of $S$ by $a$''' is: :$S^a := \set {y \in G: \exists x \in S: y = a \circ x \circ a^{-1} } = a \circ S \circ a^{-1}$ That is, $S^a$ is the set of all elements of $G$ that are the conjugates of elements of $S$ by $a$. When $G$ is the only group under consideration, we usually just refer to the '''conjugate of $S$ by $a$'''.


Definition:Algebra (Abstract Algebra)

In the context of abstract algebra, in particular ring theory and linear algebra, the following varieties of '''algebra''' exist: * Definition:Boolean Algebra * Definition:Algebra over Ring: an $R$-module $G_R$ over a commutative ring $R$ with a bilinear mapping $\oplus: G^2 \to G$. * Definition:Algebra over Field: a vector space $G_F$ over a field $F$ with a bilinear mapping $\oplus: G^2 \to G$. * Definition:Real Algebra: an algebra over a field where the field in question is the field of real numbers $\R$. * Definition:Division Algebra: an algebra over a field $\left({A_F, \oplus}\right)$ such that $\forall a, b \in A_F, b \ne \mathbf 0_A: \exists_1 x \in A_F, y \in A_F: a = b \oplus x, a = y \oplus b$. * Definition:Associative Algebra: an algebra over a ring in which the bilinear mapping $\oplus$ is associative.  * Definition:Unitary Algebra, also known as a Unital Algebra: an algebra over a ring $\left({A_R, \oplus}\right)$ in which there exists an identity element, that is, a '''unit''', usually denoted $1$, for $\oplus$. * Definition:Unitary Division Algebra: a division algebra $\left({A_F, \oplus}\right)$ in which there exists an identity element, that is, a '''unit''', usually denoted $1$, for $\oplus$. * Definition:Graded Algebra: an algebra over a ring where the ring has a gradation, that is, is a graded ring. * Definition:Filtered Algebra: an algebra over a field which has a sequence of subalgebras which constitute a gradation. * Definition:Quadratic Algebra: a filtered algebra whose generator consists of degree one elements, with defining relations of degree 2. * Definition:Lie Algebra


Definition:Ideal of Ring/Left Ideal

$J$ is a '''left ideal of $R$''' {{iff}}: :$\forall j \in J: \forall r \in R: r \circ j \in J$ that is, {{iff}}: :$\forall r \in R: r \circ J \subseteq J$


Definition:Ideal of Ring/Right Ideal

$J$ is a '''right ideal of $R$''' {{iff}}: :$\forall j \in J: \forall r \in R: j \circ r \in J$ that is, {{iff}}: :$\forall r \in R: J \circ r \subseteq J$


Definition:Divisor (Algebra)/Ring with Unity

Let $\struct {R, +, \circ}$ be an ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $x, y \in D$. We define the term '''$x$ divides $y$ in $R$''' as follows: :$x \mathrel {\divides_R} y \iff \exists t \in R: y = t \circ x$ When no ambiguity results, the subscript is usually dropped, and '''$x$ divides $y$ in $R$''' is just written $x \divides y$.


Definition:Divisor (Algebra)/Integer

Let $\struct {\Z, +, \times}$ be the ring of integers. Let $x, y \in \Z$. Then '''$x$ divides $y$''' is defined as: :$x \divides y \iff \exists t \in \Z: y = t \times x$


Definition:Divisor (Algebra)/Factorization

Let $x, y \in D$ where $\struct {D, +, \times}$ is an integral domain. Let $x$ be a divisor of $y$. Then by definition it is possible to find some $t \in D$ such that $y = t \times x$. The act of breaking down such a $y$ into the product $t \circ x$ is called '''factorization'''.


Definition:Divisor (Algebra)/Integer/Aliquot Part

An '''aliquot part''' of an integer $n$ is a divisor of $n$ which is strictly less than $n$.


Definition:Divisor (Algebra)/Integer/Aliquant Part

An '''aliquant part''' of an integer $n$ is a positive integer which is less than $n$ but is not a divisor of $n$.


Definition:Common Divisor/Integral Domain

Let $\struct {D, +, \times}$ be an integral domain. Let $S \subseteq D$ be a finite subset of $D$. Let $c \in D$ such that $c$ divides all the elements of $S$, that is: :$\forall x \in S: c \divides x$ Then $c$ is a '''common divisor''' (or '''common factor''') of all the elements in $S$.


Definition:Common Divisor/Integers

Let $S$ be a finite set of integers, that is: :$S = \set {x_1, x_2, \ldots, x_n: \forall k \in \N^*_n: x_k \in \Z}$ Let $c \in \Z$ such that $c$ divides all the elements of $S$, that is: :$\forall x \in S: c \divides x$ Then $c$ is a '''common divisor''' (or '''common factor''') of all the elements in $S$.


Definition:Greatest Common Divisor/Integral Domain

Let $\struct {D, +, \times}$ be an integral domain whose zero is $0$. Let $a, b \in D: a \ne 0 \lor b \ne 0$. Let $d \divides a$ denote that $d$ is a divisor of $a$. Let $d \in D$ have the following properties: : $(1): \quad d \divides a \land d \divides b$ : $(2): \quad c \divides a \land c \divides b \implies c \divides d$ Then $d$ is called '''a greatest common divisor of $a$ and $b$''' (abbreviated '''GCD''' or '''gcd''') and denoted $\gcd \set {a, b}$. That is, in the integral domain $D$, $d$ is the GCD of $a$ and $b$ {{iff}}: : $d$ is a common divisor of $a$ and $b$ : Any other common divisor of $a$ and $b$ also divides $d$. We see that, trivially: :$\gcd \set {a, b} = \gcd \set {b, a}$ so the set notation is justified.


Let $a, b \in \Z: a \ne 0 \lor b \ne 0$. === Definition 1 === {{:Definition:Greatest Common Divisor/Integers/Definition 1}} === Definition 2 === {{:Definition:Greatest Common Divisor/Integers/Definition 2}} This is denoted $\gcd \set {a, b}$.


Definition:Compact Space/Topology

=== Definition 1 === {{:Definition:Compact Space/Topology/Definition 1}} === Definition 2 === {{:Definition:Compact Space/Topology/Definition 2}} === Definition 3 === {{:Definition:Compact Space/Topology/Definition 3}} === Definition 4 === {{:Definition:Compact Space/Topology/Definition 4}} === Definition 5 === {{:Definition:Compact Space/Topology/Definition 5}}


Definition:Compact Space/Real Analysis

Let $\R$ be the real number space considered as a topological space under the Euclidean topology. Let $H \subseteq \R$. Then $H$ is '''compact in $\R$''' {{iff}} $H$ is closed and bounded.


Definition:Ordered Tree

Let $T$ be a rooted tree whose root is $v$. Let $O$ be an orchard whose trees are subtrees of $v$. {{explain|Cannot find a reference to an "orchard" in this context -- probably means "ordered forest".}} Then $T$ is an '''ordered tree'''. Category:Definitions/Rooted Trees ogux298rvf516io4ev39g5353mzk3xb


Definition:Closed Ball

{{:Definition:Closed Ball/Metric Space}}


Definition:Average Value of Function

Let $f$ be an integrable function on some closed interval $\closedint a b$. The '''average value of $f$''' (or '''mean value of $f$''') on $\closedint a b$ is defined as: :$\displaystyle \frac 1 {b - a} \int_a^b \map f x \rd x$


Definition:Unit Circle

The '''unit circle''' is a circle with radius equal to $1$.


Definition:Cantor Set/Limit of Intersections

Define, for $n \in \N$, subsequently: :$\map k n := \dfrac {3^n - 1} 2$ :$\displaystyle A_n := \bigcup_{i \mathop = 1}^{\map k n} \openint {\frac {2 i - 1} {3^n} } {\frac {2 i} {3^n} }$ Since $3^n$ is always odd, $\map k n$ is always an integer, and hence the union will always be perfectly defined. Consider the closed interval $\closedint 0 1 \subset \R$. Define: :$\CC_n := \closedint 0 1 \setminus A_n$ The '''Cantor set''' $\CC$ is defined as: :$\displaystyle \CC = \bigcap_{n \mathop = 1}^\infty \CC_n$


Definition:Ternary Notation

'''Ternary notation''' is the technique of expressing numbers in base $3$. That is, every number $x \in \R$ is expressed in the form: :$\displaystyle x = \sum_{j \mathop \in \Z} r_j 3^j$ where: :$\forall j \in \Z: r_j \in \set {0, 1, 2}$


Definition:Cantor Set/Ternary Representation

Consider the closed interval $\closedint 0 1 \subset \R$. The '''Cantor set''' $\CC$ consists of all the points in $\closedint 0 1$ which can be expressed in base $3$ without using the digit $1$. From Representation of Ternary Expansions, if any number has two different ternary representations, for example: :$\dfrac 1 3 = 0.10000 \ldots = 0.02222$ then at most one of these can be written without any $1$'s in it. Therefore this representation of points of $\CC$ is unique.


Definition:Finite Path

Let $G$ be a graph. A '''finite path''' in $G$ is a path consisting of a finite number of edges.


Definition:Cantor Space

Let $\CC$ be the Cantor set. Let $\tau_d$ be the Euclidean topology on $\R$. Then since $\CC \subseteq \R$, we can endow $\CC$ with the subspace topology $\tau_\CC$. The topological space $\struct {\CC, \tau_\CC}$ is referred to as the '''Cantor space'''.


Definition:Hyperbolic Sine

The '''hyperbolic sine''' function is defined on the complex numbers as: :$\sinh: \C \to \C$: :$\forall z \in \C: \sinh z := \dfrac {e^z - e^{-z} } 2$


Definition:Hyperbolic Cosine

The '''hyperbolic cosine''' function is defined on the complex numbers as: :$\cosh: \C \to \C$: :$\forall z \in \C: \cosh z := \dfrac {e^z + e^{-z} } 2$


Definition:Norm

A '''norm''' is a measure which describes a sense of the '''size''' or '''length''' of a mathematical object. In its various contexts: === Ring === {{:Definition:Norm/Ring}} === Division Ring === {{:Definition:Norm/Division Ring}} === Vector Space === {{:Definition:Norm/Vector Space}} === Algebra === {{:Definition:Norm/Algebra}} === Unital Algebra === Let $R$ be a division ring with norm $\norm {\,\cdot\,}_R$. {{:Definition:Norm/Unital Algebra}}


Definition:Hyperbolic Tangent

{{:Definition:Hyperbolic Tangent/Definition 1}}


Definition:Domain (Category Theory)

Let $f: X \to Y$ be a morphism. Then the '''domain''' of $f$ is defined to be the object $X$. This is usually denoted $X = \Dom f$ or $X = \map D f$. Category:Definitions/Morphisms mv70ukx8v5i1q4bf5wdjcx8hdzuqnll


Definition:Codomain (Category Theory)

Let $f: X \to Y$ be a morphism. Then the '''codomain''' of $f$ is defined to be the object $Y$. This is usually denoted $Y = \Cdm f$. Category:Definitions/Morphisms 6ezwuussj083nlrkxcn5m7hvdirbrji


Definition:Codomain (Set Theory)/Relation

The '''codomain''' of a relation $\RR \subseteq S \times T$ is the set $T$. It can be denoted $\Cdm \RR$.


Definition:Codomain (Set Theory)/Mapping

Let $f: S \to T$ be a mapping. The '''codomain''' of $f$ is the set $T$. It is denoted on {{ProofWiki}} by $\Cdm f$.


Definition:Domain (Set Theory)/Relation

Let $\RR \subseteq S \times T$ be a relation. The '''domain''' of $\RR$ is defined and denoted as: :$\Dom \RR := \set {s \in S: \exists t \in T: \tuple {s, t} \in \RR}$ That is, it is the same as what is defined here as the preimage of $\RR$.


Definition:Domain (Set Theory)/Mapping

Let $f: S \to T$ be a mapping. The '''domain''' of $f$ is $S$, and can be denoted $\Dom f$. In the context of mappings, the '''domain''' and the preimage of a mapping are the same set.


Definition:Domain (Set Theory)/Binary Operation

Let $\circ: S \times S \to T$ be a binary operation. The '''domain''' of $\circ$ is the set $S$ and can be denoted $\Dom \circ$.


Definition:Finite Sequence

A '''finite sequence''' is a sequence whose domain is finite. === Length of Sequence === {{transclude:Definition:Length of Sequence|increase=1|section=tc}}


Definition:Operation/Binary Operation

A binary operation is the special case of an operation where the operation has exactly two operands. A '''binary operation''' is a mapping $\circ$ from the Cartesian product of two sets $S \times T$ to a universe $\mathbb U$: :$\circ: S \times T \to \mathbb U: \map \circ {s, t} = y \in \mathbb U$ If $S = T$, then $\circ$ can be referred to as a '''binary operation on''' $S$.


Definition:Locus

A '''locus''' is a set of points which satisfy a particular condition. Such points usually form a curve or a surface, depending on the context.


Definition:Vertical Tangent Line

Let $P = \left({c, f \left({c}\right)}\right)$ be a point on the graph of a real function $f$. The vertical line $x = c$ is a vertical tangent line to the graph of $f$ at $P$ {{iff}} any of the following hold: : $(1): \quad f$ is right continuous at $c$ and $\displaystyle \lim_{x \mathop \to c ^+} f' \left({x}\right) = +\infty$ : $(2): \quad f$ is right continuous at $c$ and $\displaystyle \lim_{x \mathop \to c ^+} f' \left({x}\right) = -\infty$ : $(3): \quad f$ is left continuous at $c$ and $\displaystyle \lim_{x \mathop \to c ^-} f' \left({x}\right) = +\infty$ : $(4): \quad f$ is left continuous at $c$ and $\displaystyle \lim_{x \mathop \to c ^-} f' \left({x}\right) = -\infty$


Definition:Left Zero

An element $z_L \in S$ is called a '''left zero element''' (or just '''left zero''') {{iff}}: :$\forall x \in S: z_L \circ x = z_L$


Definition:Right Zero

An element $z_R \in S$ is called a '''right zero element''' (or just '''right zero''') {{iff}}: :$\forall x \in S: x \circ z_R = z_R$


Definition:Divergent Improper Integral

An improper integral of a real function $f$ is said to '''diverge''' if any of the following hold: : $(1): \quad f$ is continuous on $\left[{a \,.\,.\, +\infty}\right)$ and the limit $\displaystyle \lim_{b \mathop \to +\infty} \int_a^b f \left({x}\right) \ \mathrm d x$ does not exist : $(2): \quad f$ is continuous on $\left({-\infty  \,.\,.\, b}\right]$ and the limit $\displaystyle \lim_{a \mathop \to -\infty} \int_a^b f \left({x}\right) \ \mathrm d x$ does not exist : $(3): \quad f$ is continuous on $\left[{a \,.\,.\, b}\right)$, has an infinite discontinuity at $b$, and the limit $\displaystyle \lim_{c \mathop \to b^-} \int_a^c f \left({x}\right) \ \mathrm dx$ does not exist : $(4): \quad f$ is continuous on $\left({a \,.\,.\, b}\right]$, has an infinite discontinuity at $a$, and the limit $\displaystyle \lim_{c \mathop \to a^+} \int_c^b f \left({x}\right) \ \mathrm dx$ does not exist.


In the expression $\displaystyle \int_a^b \map f x \rd x$, the values $a$ and $b$ are called the '''limits of integration'''. If there is no danger of confusing the concept with limit of a function or of a sequence, just '''limits'''. Thus $\displaystyle \int_a^b \map f x \rd x$ can be voiced: : '''The integral of (the function) $f$ of $x$ {{WRT|Integration}} $x$ (evaluated) between the limits (of integration) $a$ and $b$.''' More compactly (and usually), it is voiced: : '''The integral of $f$ of $x$ {{WRT|Integration}} $x$ between $a$ and $b$''' or: : '''The integral of $f$ of $x$ dee $x$ from $a$ to $b$''' === Lower Limit === {{:Definition:Definite Integral/Limits of Integration/Lower Limit}} === Upper Limit === {{:Definition:Definite Integral/Limits of Integration/Upper Limit}} == Also known as == The interval defined by the '''limits of integration''' can be referred to as the '''range of integration'''. Some sources refer to it as the '''interval of integration'''. == Also see == From the Fundamental Theorem of Calculus, we have that: :$\displaystyle \int_a^b \map f x \rd x = \map F b - \map F a$ where $F$ is a primitive of $f$, that is: :$\map f x = \dfrac \d {\d x} \map F x$ Then $\map F b - \map F a$ is usually written: :$\bigintlimits {\map F x} {x \mathop = a} {x \mathop = b} := \map F b - \map F a$ or, when there is no chance of ambiguity as to the independent variable: :$\bigintlimits {\map F x} a b := \map F b - \map F a$ Some sources use: :$\Big.{\map F x}\Big|_a^b := \map F b - \map F a$ but this is not recommended, as it is not so clear exactly where the expression being evaluated actually starts.


Definition:Cofinal Subset

Let $\left({S, \mathcal R}\right)$ be a relational structure, that is, a set $S$ endowed with a binary relation $\mathcal R$. Let $T \subseteq S$ be a subset of $S$. Then $T$ is a '''cofinal subset of $S$ with respect to $\mathcal R$''' {{iff}}: :$\forall x \in S: \exists t \in T: x \mathop {\mathcal R} t$


Definition:Net (Preordered Set)

Let $X$ be a nonempty set. Let $\left({\Lambda, \precsim}\right)$ be a preordered set. Let $F: \Lambda \to X$ be a mapping. Then $f$ is referred to as a '''net'''.


Definition:Inner Limit

Let $\left({\mathcal X, \tau}\right)$ be a Hausdorff topological space. Let $\left \langle {C_n}\right \rangle_{n \in \N}$ be a sequence of sets in $\mathcal X$. The '''inner limit''' of $\left \langle {C_n}\right \rangle_{n \in \N}$ is defined as: : $\displaystyle \liminf_{n \to\infty} \ C_n := \left\{{x : \exists N \text{ cofinite set of }\N, \exists x_v \in C_v \left({v \in N}\right) \text{ such that } x_v \to x}\right\}$ where $x_v \to x$ denotes convergence in the topology $\tau$.


Definition:Horizontal Asymptote

The horizontal line $y = L$ is a '''horizontal asymptote''' of the graph of a real function $f$ if either of the following limits exist: :$\displaystyle \lim_{x \mathop \to +\infty} \map f x = L_1$ :$\displaystyle \lim_{x \mathop \to -\infty} \map f x = L_2$


Definition:Strict Well-Ordering

=== Definition 1 === {{:Definition:Strict Well-Ordering/Definition 1}} === Definition 2 === {{:Definition:Strict Well-Ordering/Definition 2}}


Definition:Foundational Relation

Let $\left({A, \mathcal R}\right)$ be a relational structure where $A$ is either a proper class or a set. Then $\mathcal R$ is a '''foundational relation on $A$''' {{iff}} every non-empty subset of $A$ has an $\mathcal R$-minimal element.


Definition:Coset/Right Coset

The '''right coset of $y$ modulo $H$''', or '''right coset of $H$ by $y$''', is: :$H y = \set {x \in G: \exists h \in H: x = h y}$ This is the equivalence class defined by right congruence modulo $H$. That is, it is the subset product with singleton: :$H y = H \set y$


Definition:Coset Space/Right Coset Space

The '''right coset space (of $G$ modulo $H$)''' is the quotient set of $G$ by right congruence modulo $H$, denoted $G / H^r$. It is the set of all the right cosets of $H$ in $G$.


Definition:Coset/Left Coset

The '''left coset of $x$ modulo $H$''', or '''left coset of $H$ by $x$''', is: :$x H = \set {y \in G: \exists h \in H: y = x h}$ This is the equivalence class defined by left congruence modulo $H$. That is, it is the subset product with singleton: :$x H = \set x H$


Definition:Coset Space/Left Coset Space

The '''left coset space (of $G$ modulo $H$)''' is the quotient set of $G$ by left congruence modulo $H$, denoted $G / H^l$. It is the set of all the left cosets of $H$ in $G$.


Definition:Congruence Modulo Subgroup/Right Congruence

:$\mathcal R^r_H = \set {\tuple {x, y} \in G \times G: x y^{-1} \in H}$ This is called '''right congruence modulo $H$'''.


Definition:Congruence Modulo Subgroup/Left Congruence

:$\mathcal R^l_H := \set {\tuple {x, y} \in G \times G: x^{-1} y \in H}$ This is called '''left congruence modulo $H$'''.


Definition:Transversal (Group Theory)/Left Transversal

$S$ is a '''left transversal for $H$ in $G$''' {{iff}} every left coset of $H$ contains '''exactly one''' element of $S$.


Definition:Transversal (Group Theory)/Right Transversal

$S$ is a '''right transversal for $H$ in $G$''' {{iff}} every right coset of $H$ contains '''exactly one''' element of $S$.


Definition:Transitive Class

Let $A$ denote a class, which can be either a set or a proper class. Then $A$ is '''transitive''' {{iff}} every element of $A$ is also a subclass of $A$. That is, $A$ is '''transitive''' {{iff}}: :$x \in A \implies x \subseteq A$ or: :$\forall x: \forall y: \paren {x \in y \land y \in A \implies x \in A}$


Definition:Epsilon Relation

In the language of set theory $\in$, the membership primitive, is neither a class nor a set, but a primitive predicate. To simplify formulations, it is useful to introduce a class which behaves identically to the standard membership relation $\in$ for sets. This class, denoted $\Epsilon$, will be referred to as the '''epsilon relation'''. In class-builder notation: :$\Epsilon := \left\{{ \left({ x , y }\right) : x \in y }\right\}$ Thus, explicitly, $\Epsilon$ is a relation, taking arguments from ordered pairs of sets $x$ and $y$. It consists of precisely those ordered pairs $\left({ x , y }\right)$ satisfying $x \in y$. The behavior is thus seen to be identical to regular membership with sets. It is '''not''' the same as class membership, because $x$ and $y$ must be set variables. === Restriction of Epsilon Relation === {{:Definition:Epsilon Relation/Restriction}}


Definition:Point-to-Set Distance

Let $\mathcal X$ be a normed space. The '''point-to-set distance''' on $\mathcal X$ is a mapping: :$d: \mathcal X \times \mathcal P \left({\mathcal X}\right) \to \left[{0 \,.\,.\, \infty}\right)$ where $P \left({\mathcal X}\right)$ is the power set of $\mathcal X$. This mapping is defined as: :$d \left({x, C}\right) := \inf \left\{{\left\|{x - y}\right\|: y \in C}\right\}$ with the convention that for any $x \in \mathcal X$: :$d \left({x, \varnothing}\right) = +\infty$ or, to be more mathematically rigorous: :$\forall C \in \mathcal P \left({\mathcal X}\right) \setminus \left\{{\varnothing}\right\}: d \left({x, \varnothing}\right) > d \left({x, C}\right)$ Category:Definitions/Normed Spaces 9okxhvq12y3hx8b7toz44iwi4jn1cj6


Definition:Minkowski Sum

Let $V$ be a vector space. Let $A, B$ be two subsets of $V$. Then the '''Minkowski sum''' of $A$ and $B$, denoted as $A + B$, is defined as: :$A + B := \set {a + b: a \in A, b \in B}$ where the operation $+$ is vector addition. The '''Minkowski sum''' is therefore a relation in the power set of $V$ in the sense that it is a mapping: :$+: \powerset V^2 \to \powerset V$ {{NamedforDef|Hermann Minkowski|cat = Minkowski}} Category:Definitions/Linear Algebra 7vwxhez7dklqrfg3es0op1k4cg19psl


Definition:Well-Defined/Mapping

Let $f: S \to T$ be a mapping. Let $\RR$ be an equivalence relation on $S$. Let $S / \RR$ be the quotient set determined by $\RR$. Let $\phi: S / \RR \to T$ be a mapping such that: :$\map \phi {\eqclass x \RR} = \map f x$ Then $\phi: S / \RR \to T$ is '''well-defined''' {{iff}}: :$\forall \tuple {x, y} \in \RR: \map f x = \map f y$


Definition:Well-Defined/Relation

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Definition:Well-Defined/Operation

Let $\struct {S, \circ}$ be an algebraic structure. Let $\RR$ be a congruence for $\circ$. Let $\circ_\RR$ be the operation induced on $S / \RR$ by $\circ$. Let $\struct {S / \RR, \circ_\RR}$ be the quotient structure defined by $\RR$, where $\circ_\RR$ is defined as: :$\eqclass x \RR \circ_\RR \eqclass y \RR = \eqclass {x \circ y} \RR$ Then $\circ_\RR$ is '''well-defined (on $S / \RR$)''' {{iff}}: :$x, x' \in \eqclass x \RR, y, y' \in \eqclass y \RR \implies x \circ y = x' \circ y'$


Definition:Topological Vector Space

Let $V$ be a vector space over a topological field $K$. Let $\tau$ be a topology on $V$. Then $\left({V, \tau}\right)$ is called a '''topological vector space''' {{iff}}: {{begin-axiom}} {{axiom | n = 1         | t = $\tau$ is a Hausdorff topology }} {{axiom | n = 2         | t = $+: V \times V \to V$ is continuous with respect to $\tau$ }} {{axiom | n = 3         | t = $\cdot: K \times V \to V$ is continuous with respect to $\tau$ }} {{end-axiom}} Category:Definitions/Vector Spaces Category:Definitions/Topology p891asntxlcepkrmxk66n6z83g66t2u


Definition:Star Shaped Set

Let $V$ be a vector space over a field $K$. Let $W \subseteq V$ be a subset of $V$. Then $W$ is called a '''star shaped set''' {{iff}}: :$\forall x \in W: -x \in W$ where $-x$ is the negative of $x$.


Definition:Absorbent Set

Let $V$ be a vector space over a field $K$. Let $W \subseteq V$ be a subset of $V$. Let $a \in K$. Let the set $a \cdot W$ be defined as: : $a \cdot W := \left\{{a \cdot y: y \in W} \right\}$ Then $W$ is an '''absorbent set in $V$''' {{iff}}: : $\displaystyle \bigcup_{a \mathop \in K} a \cdot W = V$ which symbolically can be represented as: : $K \cdot W = V$


Definition:Closure (Abstract Algebra)/Algebraic Structure

Let $\struct {S, \circ}$ be an algebraic structure. Then $S$ has the property of '''closure under $\circ$''' {{iff}}: :$\forall \tuple {x, y} \in S \times S: x \circ y \in S$ $S$ is said to be '''closed under $\circ$''', or just that '''$\struct {S, \circ}$ is closed'''.


Definition:Closure (Abstract Algebra)/Scalar Product

Let $\struct {S, \circ}_R$ be an $R$-algebraic structure over a ring $R$. Let $T \subseteq S$ such that $\forall \lambda \in R: \forall x \in T: \lambda \circ x \in T$. Then $T$ is '''closed for scalar product'''. If $T$ is also closed for operations on $S$, then it is called a '''closed subset''' of $S$.


Definition:Directed Set

Let $\struct {S, \precsim}$ be a preordered set. Then $\struct {S, \precsim}$ is a '''directed set''' {{iff}} every pair of elements of $S$ has an upper bound in $S$: :$\forall x, y \in S: \exists z \in S: x \precsim z$ and $y \precsim z$


Definition:Epigraph

Let $f: S \to \overline \R$ be an extended real-valued function. The '''epigraph''' of $f$ is the set: :$\map {\operatorname {epi} } f := \set {\tuple {x, \alpha} \in S \times \R: \map f x \le \alpha}$


Definition:Lower Level Set

Let $f: S \to \overline \R$ be an extended real-valued function. Let $\alpha \in \R$. The '''$\alpha$-lower level set of $f$''' is the set: :$\displaystyle \operatorname{lev} \limits_{\mathop \le \alpha} f := \left\{ {x \in S: f \left({x}\right) \le \alpha}\right\}$


Definition:Lower Semicontinuous

Let $f: S \to \R \cup \left\{{-\infty, \infty}\right\}$ be an extended real valued function. Let $S$ be endowed with a topology $\tau$.  Then $f$ is '''lower semicontinuous''' at $\bar x \in S$ {{iff}}: : $\displaystyle \liminf_{x \mathop \to \bar x} f \left({x}\right) = f \left({\bar x}\right)$ where $\displaystyle \liminf_{x \mathop \to \bar x} f \left({x}\right)$ stands for the lower limit of $f$ at $\bar x$. === Lower Semicontinuous on Subset === {{:Definition:Lower Semicontinuous/Subset}}


Definition:Ordinal Class

The '''Ordinal Class''' is defined as the class of all ordinals: :$\operatorname{On} = \{ x : x$ is an ordinal $\}$


Definition:Trivial Norm

{{#switch: 3|1 = =|2 = ==|3 = ===|4 = ====|5 = =====|6 = ======|#default = }} Division Ring {{#switch: 3|1 = =|2 = ==|3 = ===|4 = ====|5 = =====|6 = ======|#default = }} {{:Definition:Trivial Norm/Division Ring}} {{#switch: 3|1 = =|2 = ==|3 = ===|4 = ====|5 = =====|6 = ======|#default = }} Vector Space {{#switch: 3|1 = =|2 = ==|3 = ===|4 = ====|5 = =====|6 = ======|#default = }} {{:Definition:Trivial Norm/Vector Space}}


Definition:Lower Limit (Topological Space)

Let $\struct {S, \tau}$ be a topological space. Let $f: S \to \R \cup \set {-\infty, \infty}$ be an extended real-valued function. The '''lower limit''' of $f$ at some $x_0 \in S$ is defined as: :$\displaystyle \liminf_{x \mathop \to x_0} \map f x := \sup_{V \mathop \in \map \mho {x_0} } \set {\inf_{x \mathop \in V} \map f x}$ where $\map \mho {x_0}$ stands for the set of open neighborhoods of $x_0$.


Definition:Outer Limit

Let $\left({\mathcal X, \tau}\right)$ be a Hausdorff topological space. Let $\left \langle {C_n}\right \rangle_{n \in \N}$ be a sequence of sets in $\mathcal X$. The '''outer limit''' of $\left \langle {C_n}\right \rangle_{n \in \N}$ is defined as: : $\displaystyle \limsup_{n \to\infty} \ C_n := \left\{{x : \exists N \text{ cofinal set of }\N, \exists x_v \in C_v \left({v \in N}\right) \text{ such that } x_v \to x}\right\}$ where $x_v \to x$ denotes convergence in the topology $\tau$.


Definition:Painlevé-Kuratowski Convergence

Let $T = \struct {S, \tau}$ be a Hausdorff topological space. Let $\sequence {C_n}_{n \mathop \in \N}$ be a sequence of sets in $T$. Let $\sequence {C_n}_{n \mathop \in \N}$ be such that: :$\displaystyle \liminf_n C_n = \limsup_n C_n = C$ where: :$\displaystyle \liminf_n C_n$ denotes the inner limit of $\sequence {C_n}_{n \mathop \in \N}$ :$\displaystyle \limsup_n C_n$ denotes the outer limit of $\sequence {C_n}_{n \mathop \in \N}$ Then $\sequence {C_n}_{n \mathop \in \N}$ is said to be '''convergent in the sense of Painlevé-Kuratowski'''. It can be denoted as: :$C_n \overset K \to C$ or: :$\operatorname {K-lim} \limits_{n \mathop \to \infty} C_n = C$ or simply: :$\displaystyle \lim_n C_n = C$ {{NamedforDef|Paul Painlevé|name2 = Kazimierz Kuratowski|cat = Painlevé|cat2 = Kuratowski}} Category:Definitions/Hausdorff Spaces Category:Definitions/Convergence a3gln6xeea3jcurnylyvfie89koyxxa


Definition:Ambiguity

An '''ambiguity''' is a statement which has more than one distinct meaning. Thus a statement is '''ambiguous''' if, without extraneous clarification, it can be interpreted in more than one way.


Definition:Classical Probability Model

Let $\EE$ be an experiment The '''classical probability model''' on $\EE$ is a mathematical model that defines the probability space $\struct {\Omega, \Sigma, \Pr}$ of $\EE$ as follows: :$(1) \quad$ All outcomes of $\EE$ are equally likely :$(2) \quad$ There are a finite number of outcomes. Then: :$\map \Pr {\text {event occurring} } := \dfrac {\paren {\text {number of outcomes favorable to event} } } {\paren {\text {total number of outcomes possible} } }$ or formally: :$\map \Pr \omega := \dfrac {\card \Sigma} {\card \Omega}$ where: :$\card {\, \cdot \,}$ denotes the cardinality of a set :$\omega$ denotes an event: $\omega \in \Omega$ :$\Sigma$ denotes the event space: $\Sigma \subseteq \Omega$ :$\Omega$ denotes the sample space.


Definition:Equivocation

An '''equivocation''' is a shift in meaning of an ambiguous term. If an argument's persuasive force depends on utilizing an '''equivocation''' it is fallacious. To make such an argument is to commit the '''fallacy of equivocation'''.


Definition:Relative Frequency Model

The '''relative frequency model''' is a mathematical model that defines the probability of an event occurring as follows: :$\map \Pr {\text {event occurring} } := \dfrac {\paren {\text {observed number of times event has occurred in the past} } } {\paren {\text {observed number of times event has occurred or not occurred} } }$ That is, the probability of an event happening is defined as the '''relative frequency''' of a finite number of events of a particular type in some finite reference class of events.  Symbolically: :$\map \Pr \omega := \dfrac {f_{\omega} } n$ where: :$\omega$ is an elementary event :$f_{\omega}$ is how many times $\omega$ occurred :$n$ is the number of trials observed.


Definition:Relative Frequency

Let $S$ be a sample or a finite population. Let $\omega$ be a qualitative variable, or a class of a quantitative variable. The '''relative frequency''' of $\omega$ is defined as: :$\map {\operatorname {RF} } \omega := \dfrac {f_\omega} n$ where: :$f_\omega$ is the (absolute) frequency of $\omega$ :$n$ is the number of individuals in $S$.


Definition:Sorgenfrey Line

Let $\R$ be the set of real numbers Let $\BB$ be the set: :$\BB = \set {\hointr a b: a, b \in \R}$ where $\hointr a b$ is the half-open interval $\set {x \in \R: a \le x < b}$. Then $\BB$ is the basis for a topology $\tau$ on $\R$. The topological space $T = \struct {\R, \tau}$ is referred to as the '''Sorgenfrey line'''.


Definition:Sequence of Distinct Terms

A '''sequence of distinct terms of $S$''' is an injection from a subset of $\N$ into $S$. Thus a sequence $\sequence {a_k}_{k \mathop \in A}$ is a '''sequence of distinct terms''' {{iff}}: :$\forall j, k \in A: j \ne k \implies a_j \ne a_k$


Definition:Categorical Statement

Let $S$ and $P$ be predicates. A '''categorical statement''' is a statement that can be expressed in one of the following ways in natural language: {{begin-axiom}} {{axiom | n  = A         | lc = Universal Affirmative:         | t  = Every $S$ is $P$ }} {{axiom | n  = E         | lc = Universal Negative:         | t  = No $S$ is $P$ }} {{axiom | n  = I         | lc = Particular Affirmative:         | t  = Some $S$ is $P$ }} {{axiom | n  = O         | lc = Particular Negative:         | t  = Some $S$ is not $P$ }} {{end-axiom}}


Definition:Topological Group

Let $\left({G, \odot}\right)$ be a group. On its underlying set $G$, let $\left({G, \tau}\right)$ be a topological space. === Definition 1 === {{:Definition:Topological Group/Definition 1}} === Definition 2 === {{:Definition:Topological Group/Definition 2}}


Definition:Inversion Mapping/Topology

Let $T = \left({G, \circ, \tau}\right)$ be a topological group. Let $\phi: G \to G$ be the mapping defined as: : $\forall x \in G: \phi \left({x}\right) = x^{-1}$ Then $\phi$ is the '''inversion mapping''' of $T$.


Definition:Semi-Inner Product

Let $\C$ be the field of complex numbers. Let $\F$ be a subfield of $\C$. Let $V$ be a vector space over $\F$ A '''semi-inner product''' is a mapping $\innerprod \cdot \cdot: V \times V \to \mathbb F$ that satisfies the following properties: {{begin-axiom}} {{axiom | n = 1         | lc= Conjugate Symmetry         | q = \forall x, y \in V         | m = \quad \innerprod x y = \overline {\innerprod y x} }} {{axiom | n = 2         | lc= Sesquilinearity         | q = \forall x, y, z \in V, \forall a \in \mathbb F         | m = \quad \innerprod {a x + y} z = a \innerprod x z + \innerprod y z }} {{axiom | n = 3         | lc= Non-Negative Definiteness         | q = \forall x \in V         | m = \quad \innerprod x x \in \R_{\ge 0} }} {{end-axiom}} {{refactor|This section merits a separate page: "Semi-Inner Product in Real Number Field", perhaps.}} If $\mathbb F$ is a subfield of the field of real numbers $\R$, it follows from Complex Number equals Conjugate iff Wholly Real that $\overline {\innerprod y x} = \innerprod y x$ for all $x, y \in V$. Then $(1)$ above may be replaced by: {{begin-axiom}} {{axiom | n = 1^\prime         | lc= Symmetry         | q = \forall x, y \in V         | m = \innerprod x y = \innerprod y x }} {{end-axiom}} === Semi-Inner Product Space === {{:Definition:Semi-Inner Product Space}}


Definition:Semi-Inner Product Space

A '''semi-inner product space''' is a vector space together with an associated semi-inner product.


Definition:Inner Product Norm

Let $V$ be an inner product space over a subfield $\Bbb F$ of $\C$. Let $\left \langle{\cdot, \cdot}\right \rangle$ be the inner product of $V$. Then the '''inner product norm''' on $V$ is the mapping $\left\Vert{\cdot}\right\Vert: V \to \R_{\ge 0}$ given by :$\left\Vert{x}\right\Vert := \left\langle{x,x}\right\rangle^{1/2}$.


Definition:Metric Induced by Norm

Let $V$ be a normed vector space. Let $\norm{\,\cdot\,}$ be the norm of $V$. Then the '''induced metric''' or the '''metric induced by $\norm{\,\cdot\,}$''' is the map $d: V \times V \to \R_{\ge 0}$ defined as: :$d \left({x, y}\right) = \left\Vert{x - y}\right\Vert$


Definition:Hilbert Space

Let $V$ be an inner product space over $\Bbb F \in \set {\R, \C}$. Let $d: V \times V \to \R_{\ge 0}$ be the metric induced by the inner product norm $\norm {\,\cdot\,}_V$. If $\struct {V, d}$ is a complete metric space, $V$ is said to be a '''Hilbert space'''.


Definition:Orthogonal (Hilbert Space)

Let $H$ be a Hilbert space. Let $f, g \in H$. Let $\innerprod f g = 0$, where $\innerprod \cdot \cdot$ denotes the inner product. Then $f$ and $g$ are defined as being '''orthogonal''': :$f \perp g$ === Sets === Let $A, B \subseteq H$. Then $A$ and $B$ are defined as '''orthogonal''' {{iff}}: :$\forall a \in A, b \in B: a \perp b$ This is denoted by $A \perp B$.  When $A, B$ are closed linear subspaces of $H$, they are called '''orthogonal subspaces'''. If $A$ consists of only one element $a$, also $a \perp B$ is encountered.


Definition:Convex Set (Vector Space)

Let $V$ be a vector space over $\R$ or $\C$. A subset $A \subseteq V$ is said to be '''convex''' {{iff}}: :$\forall x, y \in A: \forall t \in \closedint 0 1: t x + \paren {1 - t} y \in A$ === Line Segment === {{:Definition:Convex Set (Vector Space)/Line Segment}}


Definition:Distance/Points

=== Metric Space === Let $\struct {X, d}$ be a metric space. {{:Definition:Metric Space/Distance Function}} === Normed Vector Space === {{:Definition:Distance/Points/Normed Vector Space}} === Real Numbers === {{:Definition:Distance/Points/Real Numbers}} === Complex Numbers === {{:Definition:Distance/Points/Complex Numbers}}


Definition:Distance/Sets

=== Real Numbers === {{:Definition:Distance/Sets/Real Numbers}} === Metric Spaces === {{:Definition:Distance/Sets/Metric Spaces}}


Definition:Outer Automorphism

Let $G$ be a group. Let $\phi: G \to G$ be an automorphism which is not an inner automorphism. Then $\phi$ is an '''outer automorphism'''.


Definition:Group of Outer Automorphisms

Let $G$ be a group. Let $\Aut G$ be the automorphism group of $G$. Let $\Inn G$ be the inner automorphism group of $G$. Let $\dfrac {\Aut G} {\Inn G}$ be the quotient group of $\Aut G$ by $\Inn G$. Then $\dfrac {\Aut G} {\Inn G}$ is called the '''group of outer automorphisms''' of $G$. This group $\dfrac {\Aut G} {\Inn G}$ is often denoted $\Out G$.


Definition:Characteristic Subgroup

Let $G$ be a group. Let $H$ be a subgroup such that: :$\forall \phi \in \Aut G: \phi \sqbrk H = H$ where $\Aut G$ is the automorphism group of $G$. Then $H$ is  '''characteristic (in $G$)''', or '''a characteristic subgroup of $G$'''.


Definition:Projection (Analytic Geometry)

Let $M$ and $N$ be distinct lines through the origin in the plane. The '''projection on $M$ along $N$''' is the mapping $\pr_{M, N}$ such that: :$\forall x \in \R^2: \map {\pr_{M, N} } x =$ the intersection of $M$ with the line through $x$ parallel to $N$.


Definition:Orthogonal Projection

Let $H$ be a Hilbert space. Let $K$ be a closed linear subspace of $H$. Then the '''orthogonal projection''' on $K$ is the map $P_K: H \to H$ defined by :$k = \map {P_K} h \iff k \in K$ and $\map d {h, k} = \map d {h, K}$ where the latter $d$ signifies distance to a set. {{refactor|Extract the 3 statements below and put them into their own pages}} That $P_K$ is well-defined follows from Unique Point of Minimal Distance. The name orthogonal projection stems from the fact that $\paren {h - \map {P_K} h} \perp K$. This and other properties of $P_K$ are collected in Properties of Orthogonal Projection.


Definition:Normal Series

Let $G$ be a group whose identity is $e$. A '''normal series''' for $G$ is a sequence of (normal) subgroups of $G$: :$\set e = G_0 \lhd G_1 \lhd \cdots \lhd G_n = G$ where $G_{i - 1} \lhd G_i$ denotes that $G_{i - 1}$ is a proper normal subgroup of $G_i$.


Definition:Refinement of Normal Series

Let $G$ be a group whose identity is $e$. Let $\sequence {G_i}_{i \mathop \in \closedint 0 n}$ be a normal series for $G$: :$\sequence {G_i}_{i \mathop \in \closedint 0 n} = \tuple {\set e = G_0 \lhd G_1 \lhd \cdots \lhd G_{n - 1} \lhd G_n = G}$ Let $\sequence {H_j}_{j \mathop \in \closedint 0 m}$ be another normal series for $G$: :$\sequence {H_j}_{j \mathop \in \closedint 0 m} = \tuple {\set e = H_0 \lhd H_1 \lhd \cdots \lhd H_{m - 1} \lhd H_m = G}$ such that $\sequence {G_i}_{i \mathop \in \closedint 0 n} \subseteq \sequence {H_j}_{j \mathop \in \closedint 0 m}$ Then $\sequence {H_j}_{j \mathop \in \closedint 0 m}$ is a '''refinement''' of $\sequence {G_i}_{i \mathop \in \closedint 0 n}$. That is, a '''refinement''' of a normal series is a normal series which contains all the (normal) subgroups of the original normal series, and may contain more. === Proper Refinement === {{:Definition:Refinement of Normal Series/Proper Refinement}}


Definition:Composition Series

Let $G$ be a finite group. A '''composition series for $G$''' is a normal series for $G$ which has no proper refinement. === Composition Length === {{:Definition:Composition Series/Composition Length}} === Composition Factor === {{:Definition:Composition Series/Composition Factor}}


Definition:Contradictory Statements

Two statements $p$ and $q$ are said to be '''contradictory''' iff: : whenever $p$ is true, $q$ is false. and: : whenever $q$ is true, $p$ is false.


Definition:Contrary Statements

Two statements are said to be '''contrary''' if they can both be false, but they cannot both be true.


Definition:Subcontrary Statements

Two statements are said to be '''subcontrary''' if they can both be true, but they cannot both be false.


Definition:Closed Linear Span

Let $H$ be a Hilbert space, and let $A \subseteq H$ be a subset. Then the '''closed linear span''' of $A$, denoted $\vee A$, is defined in the following ways: :$(1): \qquad \displaystyle \vee A = \bigcap \Bbb M$, where $\Bbb M$ consists of all closed linear subspace $M$ of $H$ with $A \subseteq M$ :$(2): \qquad \vee A$ is the smallest closed linear subspace $M$ of $H$ with $A \subseteq M$ :$(3): \qquad \vee A = \map \cl {\map {\operatorname {span} } A}$, where $\cl$ denotes closure, and $\operatorname {span}$ denotes linear span.


Definition:Zero Subspace

Let $V$ be a vector space with zero vector $\mathbf 0$. Then the set $(\mathbf 0) := \left\{{\mathbf 0}\right\}$ is called the '''zero subspace''' of $V$. This name is appropriate as $(\mathbf 0)$ is in fact a subspace of $V$, as proved in Zero Subspace is Subspace. {{Explain|I'm not sure about the nomenclature, but this deserves a name. Feel free to adapt if you have sources to back it up.}} Category:Definitions/Vector Spaces o6xb6n4gd224dpk0cmgqrc2o7idp9ro


Definition:Length of Group

Let $G$ be a finite group. The '''length''' of $G$ is the length of a composition series for $G$. That is, the '''length''' of $G$ is the number of factors in a composition series for $G$ (not including $G$ itself). The '''length''' of $G$ can be denoted $\map l G$. By the Jordan-Hölder Theorem, all composition series for $G$ have the same length. Therefore, the '''length''' of a finite group $G$ is well-defined.


Definition:Bounded Linear Functional

Let $H$ be a Hilbert space, and let $L$ be a linear functional on $H$. Then $L$ is said to be a '''bounded linear functional''' iff :$\exists c > 0: \forall h \in H: \left|{Lh}\right| \le c \left\|{h}\right\|$ In view of Continuity of Linear Functionals, a linear functional on a Hilbert space is bounded if and only if it is continuous.


Definition:Orthonormal Subset

Let $\struct {V, \innerprod \cdot \cdot}$ be an inner product space. Let $S \subseteq V$ be a subset of $V$. Then $S$ is an '''orthonormal subset''' {{iff}}: :$(1): \quad \forall u \in S: \norm u = 1$ where $\norm {\, \cdot \,}$ is the inner product norm. :$(2): \quad S$ is an orthogonal set: ::$\forall u, v \in S: u \ne v \implies \innerprod u v = 0$


Definition:Basis (Hilbert Space)

Let $H$ be a Hilbert space. A '''basis for $H$''' is a maximal orthonormal subset of $H$. Thus, $B$ is a '''basis''' for $H$ {{iff}} for all orthonormal subsets $B'$ of $H$: :$B \subseteq B' \implies B = B'$


Definition:Linear Span

Let $V$ be a vector space over $K$. Let $A \subseteq V$ be a subset of $V$. Then the '''linear span''' of $A$, denoted $\operatorname{span} A$ or $\map {\operatorname{span} } A$, is the set: :$\displaystyle \set {\sum_{k \mathop = 1}^n \alpha_k f_k: n \in \N_{\ge 1}, \alpha_i \in K, f_i \in A}$ The '''linear span''' can be interpreted as the set of all linear combinations (of finite length) of these vectors. === Definition for $\R^n$ === In $\R^n$ (where $n \in \N_{>0}$), above definition translates to: :$\displaystyle \map {\operatorname{span} } {\mathbf v_1, \mathbf v_2, \dotsc, \mathbf v_k} = \set {\sum_{i \mathop = 1}^k \ c_i \ \mathbf v_i: c_i \in \R, \mathbf v_i \in \R^n, 1 \le i \le k}$


Definition:Prime Group

Let $G$ be a group. Then $G$ is a '''prime group''' {{iff}} the order of $G$ is a prime number.


Definition:Solvable Group

Let $G$ be a finite group. Then $G$ is a '''solvable group''' {{iff}} it has a composition series in which each factor is a cyclic group.


Definition:Bayesian Probability Model

The '''Bayesian probability model''' is a mathematical model that defines the probability of an event occurring as a degree of belief. That is, that probability is defined as the degree of which it is rational to believe certain statements based on intuition, experience, judgment, or opinion. {{DefinitionWanted}} {{NamedforDef|Thomas Bayes|cat = Bayes}}


Definition:Amphiboly

An '''amphiboly''' is an ambiguity resulting from poor syntax.


Definition:Permutation on n Letters/Two-Row Notation

Let $\pi$ be a permutation on $n$ letters. The '''two-row notation''' for $\pi$ is written as two rows of elements of $\N_n$, as follows: :$\pi = \begin{pmatrix} 1 & 2 & 3 & \ldots & n \\ \map \pi 1 & \map \pi 2 & \map \pi 3 & \ldots & \map \pi n \end{pmatrix}$ The bottom row contains the effect of $\pi$ on the corresponding entries in the top row.


Definition:Inductive Argument

An '''inductive argument''' is a form of argument in which, if all the premises are true, the conclusion is ''probably true'', but might not be.  Such lines of reasoning are ubiquitous in everyday life and in most human endeavors. However, '''inductive arguments''' are only conjectures in the field of mathematics. Such arguments are not truth preserving and therefore they are not proofs.


Definition:Syllogism

A '''syllogism''' is an argument with exactly two premises and one conclusion.