Definition:Module

Let $\struct {R, +_R, \times_R}$ be a ring. Let $\struct {G, +_G}$ be an abelian group. A module over $R$ is an $R$-algebraic structure with one operation $\struct {G, +_G, \circ}_R$ which is both a left module and a right module: === Left Module === {{:Definition:Left Module}} === Right Module === {{:Definition:Right Module}}


Definition:Linear Transformation

A '''linear transformation''' is a homomorphism from one module to another.


Definition:Module on Cartesian Product

Let $\struct {R, +_R, \times_R}$ be a ring. Let $n \in \N_{>0}$. Let $+: R^n \times R^n \to R^n$ be defined as: :$\tuple {\alpha_1, \ldots, \alpha_n} + \tuple {\beta_1, \ldots, \beta_n} = \tuple {\alpha_1 +_R \beta_1, \ldots, \alpha_n +_R \beta_n}$ Let $\times: R \times R^n \to R^n$ be defined as: :$\lambda \times \tuple {\alpha_1, \ldots, \alpha_n} = \tuple {\lambda \times_R \alpha_1, \ldots, \lambda \times_R \alpha_n}$ Then $\struct {R^n, +, \times}_R$ is '''the $R$-module $R^n$'''.


Definition:Trivial Module

Let $\struct {G, +_G}$ be an abelian group whose identity is $e_G$. Let $\struct {R, +_R, \circ_R}$ be a ring. Let $\circ$ be defined as: :$\forall \lambda \in R: \forall x \in G: \lambda \circ x = e_G$ Then $\struct {G, +_G, \circ}_R$ is an $R$-module. Such a module is called a '''trivial module'''.


Definition:Submodule

Let $\struct {R, +, \circ}$ be a ring. Let $\struct {S, +, \circ}_R$ be an $R$-algebraic structure with one operation. Let $T$ be a closed subset of $S$. Let $\struct {T, +_T, \circ_T}_R$ be an $R$-module where: :$+_T$ is the restriction of $+$ to $T \times T$ :$\circ_T$ is the restriction of $\circ$ to $R \times T$. Then $\struct {T, +_T, \circ_T}_R$ is a '''submodule''' of $\struct {S, +, \circ}_R$.


Definition:Null Module

Let $\left({R, +_R, \circ_R}\right)$ be a ring. Let $G$ be the trivial group. Then the $R$-module $\left({G, +_G, \circ}\right)_R$ is known as the '''null module'''.


Definition:P-Subgroup

Let $G$ be a group. Let $H$ be a subgroup of $G$. If $H$ is a $p$-group, then $H$ is a '''$p$-subgroup''' of $G$. {{SUBPAGENAME}} e6bq56il8qbgbcyygojvbcz1jag6lue


Definition:Finitely Generated Module

Let $R$ be a ring. Let $G$ be a module over $R$. Then $G$ is '''finitely generated''' {{iff}} there is a generator for $G$ which is finite.


Definition:Linear Combination

Let $R$ be a ring. === Linear Combination of Sequence === {{:Definition:Linear Combination/Sequence}} === Linear Combination of Subset === {{:Definition:Linear Combination/Subset}} === Linear Combination of Empty Set === {{:Definition:Linear Combination/Empty Set}}


Definition:Linearly Independent

Let $G$ be an abelian group whose identity is $e$. Let $R$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $\left({G, +_G, \circ}\right)_R$ be a unitary $R$-module. === Sequence === {{:Definition:Linearly Independent/Sequence}} === Set === {{:Definition:Linearly Independent/Set}}


Definition:Basis (Topology)

=== Analytic Basis === {{:Definition:Basis (Topology)/Analytic Basis}} === Synthetic Basis === {{:Definition:Basis (Topology)/Synthetic Basis}}


Definition:Free Module

Let $R$ be a ring with unity. Let $G$ be a unitary $R$-module. Then $G$ is described as '''free''' if there exists a basis of $G$.


Definition:Ordered Basis

Let $R$ be a ring with unity. Let $G$ be a free $R$-module. An '''ordered basis''' of $G$ is a sequence $\left \langle {a_k} \right \rangle_{1 \mathop \le k \mathop \le n}$ of elements of $G$ such that $\left\{{a_1, \ldots, a_n}\right\}$ is a basis of $G$.


Definition:Topological Manifold/Differentiable Manifold

Let $M$ be a second-countable locally Euclidean space of dimension $d$.  Let $\mathscr F$ be a $d$-dimensional differentiable structure on $M$ of class $\mathcal C^k$, where $k \ge 1$. Then $\left({M, \mathscr F}\right)$ is a '''differentiable manifold of class $C^k$ and dimension $d$'''.


Definition:Dimension (Linear Algebra)

=== Module === {{:Definition:Dimension of Module}} === Vector Space === {{:Definition:Dimension of Vector Space}} {{wtd|Infinite dimensional case}}


Definition:Metric Space

A '''metric space''' $M = \struct {A, d}$ is an ordered pair consisting of: :$(1): \quad$ a non-empty set $A$ together with: :$(2): \quad$ a real-valued function $d: A \times A \to \R$ which acts on $A$, satisfying the metric space axioms: {{:Definition:Metric Space Axioms}}


Definition:Topological Space

Let $S$ be a set. Let $\tau$ be a topology on $S$. That is, let $\tau \subseteq \powerset S$ satisfy the open set axioms: {{:Definition:Open Set Axioms}} Then the ordered pair $\struct {S, \tau}$ is called a '''topological space'''. The elements of $\tau$ are called open sets of $\struct {S, \tau}$.


Definition:Topology

{{:Definition:Topology/Definition 1}}


Definition:Smooth Real Function

A real function is '''smooth''' if it is of differentiability class $C^\infty$. That is, if it admits of continuous derivatives of all orders.


Definition:Almost Everywhere

Let $\struct {X, \Sigma, \mu}$ be a measure space. A property $\map P x$ of elements of $X$ is said to hold '''($\mu$-)almost everywhere''' if the set: :$\set {x \in X: \neg \map P x}$ of elements of $X$ such that $P$ does not hold is contained in a $\mu$-null set.


Definition:Regular Value

Let $X$ and $Y$ be smooth manifolds. Let $f: X \to Y$ be a smooth mapping. Then a point $y \in Y$ is called a '''regular value''' of $f$ {{iff}} the pushforward of $f$ at $x$: : $f_* \vert_x: T_x X \to T_y Y$ {{explain|What do all the symbols mean in this context? Presume $\vert$ might mean restriction, but this is not obvious (if so then use $\restriction$); if "pushforward" actually means $f_* \vert_x: T_x X \to T_y Y$ then set up the page to define it, thus doing all the hard work of defining that concept all in one place.}} is surjective for every $x \in f^{-1} \left({y}\right) \subseteq X$.


Definition:Kronecker Delta

Let $\Gamma$ be a set. Let $R$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$. Then $\delta_{\alpha \beta}: \Gamma \times \Gamma \to R$ is the mapping on the cartesian square of $\Gamma$ defined as: :$\forall \tuple {\alpha, \beta} \in \Gamma \times \Gamma: \delta_{\alpha \beta} := \begin{cases} 1_R & : \alpha = \beta \\ 0_R & : \alpha \ne \beta \end{cases}$


Definition:Linear Form

Let $R$ be a commutative ring. Let $\struct {G, +_G, \circ}_R$ be a module over $R$. Let $\phi: \struct {G, +_G, \circ}_R \to \struct {R, +_R, \circ}_R$ be a linear transformation from $G$ to the $R$-module $R$. Then $\phi$ is called a '''linear form on $G$'''.


Definition:Algebraic Dual

Let $R$ be a commutative ring. Let $G$ be a module over $R$. The $R$-module $\map {\LL_R} {G, R}$ of all linear forms on $G$ is usually denoted $G^*$ and is called the '''algebraic dual''' of $G$. === Double Dual === {{:Definition:Algebraic Dual/Double Dual}}


Definition:Ordered Dual Basis

Let $R$ be a commutative ring. Let $\struct {G, +_G, \circ}_R$ be an $n$-dimensional module over $R$. Let $\sequence {a_n}$ be an ordered basis of $G$. Let $G^*$ be the algebraic dual of $G$. Then there is an ordered basis $\sequence {a'_n}$ of $G^*$ satisfying $\forall i, j \in \closedint 1 n: \map {a'_i} {a_j} = \delta_{i j}$. This ordered basis $\sequence {a'_n}$ of $G^*$ is called the '''ordered basis of $G^*$ dual to $\sequence {a_n}$''', or the '''ordered dual basis of $G^*$'''.


Definition:Evaluation Linear Transformation

Let $R$ be a commutative ring. Let $G$ be an $R$-module. Let $G^*$ be the algebraic dual of $G$. Let $G^{**}$ be the algebraic dual of $G^*$. For each $x \in G$, we define the mapping $x^\wedge: G^* \to R$ as: :$\forall t' \in G^*: x^\wedge \left({t'}\right) = t' \left({x}\right)$ Then $x^\wedge \in G^{**}$. The mapping $J: G \to G^{**}$ defined as: :$\forall x \in G: J \left({x}\right) = x^\wedge$ is called the '''evaluation linear transformation from $G$ into $G^{**}$'''. It is usual to denote the mapping $t': G^* \to G$ as follows: :$\forall x \in G, t' \in G^*: \left \langle {x, t'} \right \rangle := t' \left({x}\right)$


Definition:Evaluation Isomorphism

Let $R$ be a commutative ring. Let $G$ be a unitary $R$-module whose dimension is finite. Then the evaluation linear transformation $J: G \to G^{**}$ is called the '''evaluation isomorphism from $G$ to $G^{**}$.


Definition:Annihilator

Let $R$ be a commutative ring. Let $M$ and $N$ be modules over $R$. Let $B : M \times N \to R$ be a bilinear mapping. The '''annihilator of $D \subseteq M$''', denoted $\operatorname{Ann}_N \left({D}\right)$ is the set: :$\left\{{n \in N : \forall d \in D: B \left({d, n}\right) = 0}\right\}$


Definition:Transpose of Linear Transformation

Let $R$ be a commutative ring. Let $G$ and $H$ be $R$-modules. Let $G^*$ and $H^*$ be the algebraic duals of $G$ and $H$ respectively. Let $\mathcal L_R \left({G, H}\right)$ be the set of all linear transformations from $G$ to $H$. Let $u \in \mathcal L_R \left({G, H}\right)$. The '''transpose''' of $u$ is the mapping $u^t: H^* \to G^*$ defined as: :$\forall y' \in H^*: u^t \left({y'}\right) = y' \circ u$ where $y' \circ u$ is the composition of $y'$ and $u$.


Definition:Vector Space

{{:Definition:Vector Space/Definition 1}}


Definition:Vector Quantity

A '''vector quantity''' is a is a real-world concept that needs for its model a mathematical object with more than one component to specify it. Formally, a '''vector quantity''' is an element of a vector space, often the real vector space $\R^n$. The usual intellectual frame of reference is to interpret a '''vector quantity''' as having: :A magnitude :A direction.


Definition:Vector Space of All Mappings

Let $\struct {K, +, \circ}$ be a division ring. Let $\struct {G, +_G, \circ}_K$ be a $K$-vector space. Let $S$ be a set. Let $G^S$ be the set of all mappings from $S$ to $G$. Then $\struct {G^S, +_G', \circ}_K$ is a $K$-vector space, where: :$+_G'$ is the operation induced on $G^S$ by $+_G$ :$\forall \lambda \in K: \forall f \in G^S: \forall x \in S: \map {\paren {\lambda \circ f} } x = \lambda \circ \paren {\map f x}$ This is the $K$-vector space $G^S$ of all mappings from $S$ to $G$.


Definition:Real Vector Space

Let $\R$ be the set of real numbers. Then the $\R$-module $\R^n$ is called the '''real ($n$-dimensional) vector space'''.


Definition:Vector Subspace

Let $K$ be a division ring. Let $\struct {S, +, \circ}_K$ be a $K$-algebraic structure with one operation. Let $T$ be a closed subset of $S$. Let $\struct {T, +_T, \circ_T}_K$ be a $K$-vector space where: :$+_T$ is the restriction of $+$ to $T \times T$ and :$\circ_T$ is the restriction of $\circ$ to $K \times T$. Then $\struct {T, +_T, \circ_T}_K$ is a '''(vector) subspace''' of $\struct {S, +, \circ}_K$.


Definition:Rank (Linear Algebra)

=== Linear Transformation === {{:Definition:Rank/Linear Transformation}} === Matrix === {{:Definition:Rank/Matrix}} {{expand|These definitions can be proved compatible (by viewing a matrix as a lin. transform., and maybe vice versa); such is a typical PW entry. Also, cf. Definition:Finite Rank Operator}} {{SUBPAGENAME}} e3nadl00pqmmtky0opygemoaw128urn


Definition:Nullity

=== Linear Transformation === {{:Definition:Nullity/Linear Transformation}} === Matrix === {{:Definition:Nullity/Matrix}} Category:Definitions/Linear Algebra lc760v5s457b9y1705qimrz55m6xrua


Definition:Coordinate System/Coordinate

Let $\sequence {a_n}$ be a coordinate system of a unitary $R$-module $G$. Let $\displaystyle x \in G: x = \sum_{k \mathop = 1}^n \lambda_k a_k$. The scalars $\lambda_1, \lambda_2, \ldots, \lambda_n$ can be referred to as the '''coordinates of $x$ relative to $\sequence {a_n}$'''.


Definition:Real Number Plane

The points on the plane are in one-to-one correspondence with the (real) Cartesian plane $\R^2$. So from the definition of an ordered $n$-tuple, the general element of $\R^2$ can be defined as an ordered couple $\tuple {x_1, x_2}$ where $x_1, x_2 \in \R$, or, conventionally, $\tuple {x, y}$. Thus, we can identify the elements of $\R^2$ with points in the plane and refer to the point ''as'' its coordinates. Thus we can refer to $\R^2$ ''as'' '''the plane'''.


Definition:Point

{{EuclidDefinition|book = I|def = 1|name = Point}} This is interpreted to mean that a '''point''' is something that can not be divided into anything smaller. That is, it is an object in space which has zero size and is therefore the smallest thing that can be imagined. The only property possessed by a '''point''' is position.


Definition:Angle

Given two intersecting lines or line segments, the amount of rotation about the intersection required to bring one into correspondence with the other is called the '''angle''' between them.


Definition:Right Angle

A '''right angle''' is an angle that is equal to half of a straight angle.


Definition:Straight Angle

A '''straight angle''' is defined to be the angle formed by the two parts of a straight line from a point on that line.


Definition:Zero Angle

The '''zero angle''' is an angle the measure of which is $0$ regardless of the unit of measurement.


Definition:Acute Angle

An '''acute angle''' is an angle which has a measure between that of a right angle and that of a zero angle.


Definition:Obtuse Angle

An '''obtuse angle''' is an angle which has a measurement between those of a right angle and a straight angle.


Definition:Full Angle

A '''full angle''' is an angle equivalent to one full rotation.


Definition:Angle/Unit/Degree

The '''degree (of arc)''' is a measurement of plane angles, symbolized by $\degrees$. {{begin-eqn}} {{eqn | o =        | r = 1       | c = '''degree''' }} {{eqn | r = 60       | c = minutes }} {{eqn | r = 60 \times 60 = 3600       | c = seconds }} {{eqn | r = \dfrac 1 {360}       | c = full angle (by definition) }} {{end-eqn}}


Definition:Reflex Angle

A '''reflex angle''' is an angle which has a measure between that of a straight angle and that of a full angle.


Definition:Angle/Unit/Radian

The '''radian''' is a measure of plane angles symbolized either by the word $\radians$ or without any unit. '''Radians''' are pure numbers, as they are ratios of lengths. The addition of $\radians$ is merely for clarification. $1 \radians$ is the angle subtended at the center of a circle by an arc whose length is equal to the radius: :360px


Definition:Line

{{EuclidSaid}} :''{{Definition:Euclid's Definitions - Book I/2 - Line}}'' {{EuclidDefRefNocat|I|2|Line}} This can be interpreted to mean that a line is a construct that has no thickness. This mathematical abstraction can not of course be actualised in reality because however thin you make your line, it will have some finite width. It can be considered as a continuous succession of points.


Definition:Circle

{{EuclidSaid}} :''{{:Definition:Euclid's Definitions - Book I/15 - Circle}}'' {{EuclidDefRefNocat|I|15|Circle}} :300px


Definition:Boundary (Geometry)

{{EuclidDefinition|book = I|def = 13|name = Boundary}} For example, the endpoints of a line segment are its boundaries. === Containment === {{:Definition:Boundary (Geometry)/Containment}}


Definition:Surface (Geometry)

{{EuclidDefinition|book = I|def = 5|name = Surface}} and: :''{{:Definition:Euclid's Definitions - Book XI/2 - Extremity of Solid}}'' {{EuclidDefRefNocat|XI|2|Extremity of Solid}}


Definition:Linear Measure

'''Linear measure''' is the means of measurement of physical displacement. === Dimension === {{:Definition:Linear Measure/Dimension}} === Units === {{:Definition:Linear Measure/Units}} === Length === {{:Definition:Linear Measure/Length}} === Breadth === {{:Definition:Linear Measure/Breadth}} === Depth === {{:Definition:Linear Measure/Depth}} === Height === {{:Definition:Linear Measure/Height}} === Thickness === {{:Definition:Linear Measure/Thickness}} === Distance === {{:Definition:Linear Measure/Distance}}


Definition:Segment of Circle

:300px {{EuclidSaid}} :''{{:Definition:Euclid's Definitions - Book III/6 - Segment of Circle}}'' {{EuclidDefRefNocat|III|6|Segment of Circle}} === Base === {{:Definition:Segment of Circle/Base}} === Angle of a Segment === {{:Definition:Segment of Circle/Angle of Segment}} === Angle in a Segment === {{:Definition:Segment of Circle/Angle in Segment}} === Similar Segments === {{:Definition:Segment of Circle/Similar}}


Definition:Diameter of Bounded Metric Subspace

Let $M = \struct {A, d}$ be a metric space. Let $S \subseteq A$ be bounded in $M$. Then the '''diameter''' of $S$ is defined as: :$\map {\operatorname {diam} } S := \sup \set {\map d {x, y}: x, y \in S}$ That is, by the definition of the supremum, $\map {\operatorname {diam} } S$ is the smallest real number $D$ such that any two points of $S$ are at most a distance $D$ apart.


Definition:Region

=== Metric Space === {{:Definition:Region/Metric Space}} === Complex === {{:Definition:Region/Complex}} === Region in the Plane === The usual usage of '''region''' is in the real number plane or complex plane. {{:Definition:Region/Plane}} ==== Interior ==== The boundary of a region separates its '''interior''' from the '''exterior'''. The '''interior''' consists of the points of the plane which are the elements of the region. Such points are called '''interior points''' of the region. It is "usual" that the interior is the "smaller bit" which is visually apparently on the ''inside'' as it appears on the page or screen, but this is of course not necessarily the case. Also see the definition of interior and boundary from a topological perspective. ==== Bounded ==== A region in the the plane is '''bounded''' if there is a circle in the plane which encloses it. Also see the definition of bounded in the context of a metric space.


Definition:Cartesian Coordinate System

A '''Cartesian coordinate system''' is a coordinate system in which the position of a point is determined by its relation to a set of perpendicular straight lines. These straight lines are referred to as coordinate axes.


Definition:Axis

An '''axis''' is the name used for a general infinite straight line which is particularly significant in some particular way in the study of linear transformations of a real vector space. === Coordinate Axes === {{:Definition:Axis/Coordinate Axes}}


Definition:Analytic Geometry

'''Analytic geometry''' is the study of geometry by algebraic manipulation of systems of ordered pairs of variables representing points in Cartesian space.


Definition:Parallel (Geometry)

=== Lines === {{:Definition:Parallel (Geometry)/Lines}} === Planes === {{:Definition:Parallel (Geometry)/Planes}} === Line Parallel to Plane === {{:Definition:Parallel (Geometry)/Line to Plane}}


Definition:Geometry

'''Geometry''' is a branch of mathematics which studies such matters as form, position, dimension and various other properties of ordinary space. It has been suggested that '''geometry''' can be divided into $3$ main branches: :'''Metrical geometry''', that is to say, what is understood as '''geometry''' proper :'''Projective geometry''' :'''Analytic geometry'''


Definition:Homogeneous (Analytic Geometry)

A straight line or plane is '''homogeneous''' if it contains the origin.


Definition:Matrix

Let $S$ be a set. Let $m, n \in \Z_{>0}$ be strictly positive integers. An '''$m \times n$ matrix over $S$''' (said '''$m$ times $n$''' or '''$m$ by $n$''') is a mapping from the cartesian product of two integer intervals $\closedint 1 m \times \closedint 1 n$ into $S$.


Definition:Matrix Space

Let $m, n \in \Z_{>0}$ be (strictly) positive integers. Let $S$ be a set. The '''$m \times n$ matrix space over $S$''' is defined as the set of all $m \times n$ matrices over $S$, and is denoted $\map {\MM_S} {m, n}$.


Definition:Matrix Entrywise Addition

Let $\mathbf A$ and $\mathbf B$ be matrices of numbers. Let the orders of $\mathbf A$ and $\mathbf B$ both be $m \times n$. Then the '''matrix entrywise sum of $\mathbf A$ and $\mathbf B$''' is written $\mathbf A + \mathbf B$, and is defined as follows: Let $\mathbf A + \mathbf B = \mathbf C = \sqbrk c_{m n}$. Then: :$\forall i \in \closedint 1 m, j \in \closedint 1 n: c_{i j} = a_{i j} + b_{i j}$ Thus $\mathbf C = \sqbrk c_{m n}$ is the $m \times n$ matrix whose entries are made by performing the adding corresponding entries of $\mathbf A$ and $\mathbf B$. That is, the '''matrix entrywise sum of $\mathbf A$ and $\mathbf B$''' is the '''Hadamard product''' of $\mathbf A$ and $\mathbf B$ with respect to addition of numbers. This operation is called '''matrix entrywise addition'''.


Definition:Zero Matrix/General Monoid

Let $\struct {S, \circ}$ be a monoid whose identity is $e$. Let $\map {\MM_S} {m, n}$ be an $m \times n$ matrix space over $S$. The '''zero matrix of $\map {\MM_S} {m, n}$''', denoted $\mathbf e$, is the $m \times n$ matrix whose elements are all $e$, and can be written $\sqbrk e_{m n}$.


Definition:Negative Matrix

Let $\GF$ denote one of the standard number systems. Let $\map \MM {m, n}$ be a $m \times n$ matrix space over $\GF$. Let $\mathbf A = \sqbrk a_{m n}$ be an element of $\map \MM {m, n}$. Then the '''negative (matrix) of $\mathbf A$''' is denoted and defined as: :$-\mathbf A := \sqbrk {-a}_{m n}$ {{begin-eqn}} {{eqn | l = -\mathbf A       | r = -1 \mathbf A       | c =  }} {{eqn | r = \sqbrk {-a}_{m n}       | c =  }} {{end-eqn}} where: :$-1 \mathbf A$ denotes the matrix scalar product of $-1$ with $\mathbf A$ :$-a$ is the negative of $a$.


Definition:Matrix Scalar Product

Let $\GF$ denote one of the standard number systems. Let $\map \MM {m, n}$ be the $m \times n$ matrix space over $\GF$. Let $\mathbf A = \sqbrk a_{m n} \in \map \MM {m, n}$. Let $\lambda \in \GF$ be any element of $\Bbb F$. The operation of '''scalar multiplication of $\mathbf A$ by $\lambda$''' is defined as follows. Let $\lambda \mathbf A = \mathbf C$. Then: :$\forall i \in \closedint 1 m, j \in \closedint 1 n: c_{i j} = \lambda a_{i j}$ $\lambda \mathbf A$ is the '''scalar product of $\lambda$ and $\mathbf A$'''. Thus $\mathbf C = \sqbrk c_{m n}$ is the $m \times n$ matrix composed of the product of $\lambda$ with the corresponding elements of $\mathbf A$.


Definition:Matrix Product (Conventional)

Let $\struct {R, +, \circ}$ be a ring. Let $\mathbf A = \sqbrk a_{m n}$ be an $m \times n$ matrix over $R$. Let $\mathbf B = \sqbrk b_{n p}$ be an $n \times p$ matrix over $R$. Then the '''matrix product of $\mathbf A$ and $\mathbf B$''' is written $\mathbf A \mathbf B$ and is defined as follows. Let $\mathbf A \mathbf B = \mathbf C = \sqbrk c_{m p}$. Then: :$\displaystyle \forall i \in \closedint 1 m, j \in \closedint 1 p: c_{i j} = \sum_{k \mathop = 1}^n a_{i k} \circ b_{k j}$ Thus $\sqbrk c_{m p}$ is the $m \times p$ matrix where each entry $c_{i j}$ is built by forming the (ring) product of each entry in the $i$'th row of $\mathbf A$ with the corresponding entry in the $j$'th column of $\mathbf B$ and adding up all those products. This operation is called '''matrix multiplication''', and $\mathbf C$ is the '''matrix product''' of $\mathbf A$ with $\mathbf B$.


Definition:Relative Matrix

Let $\left({R, +, \circ}\right)$ be a ring with unity. Let $G$ be a free $R$-module of finite dimension $n>0$ Let $H$ be a free $R$-module of finite dimension $m>0$ Let $\left \langle {a_n} \right \rangle$ be an ordered basis of $G$. Let $\left \langle {b_m} \right \rangle$ be an ordered basis of $H$. Let $u : G \to H$ be a linear transformation. The '''matrix of $u$ relative to $\left \langle {a_n} \right \rangle$ and $\left \langle {b_m} \right \rangle$''' is the $m \times n$ matrix $\left[{\alpha}\right]_{m n}$ where: :$\displaystyle \forall \left({i, j}\right) \in \left[{1 \,.\,.\, m}\right] \times \left[{1 \,.\,.\, n}\right]:u \left({a_j}\right) = \sum_{i \mathop = 1}^m \alpha_{i j} \circ b_i$ That is, the matrix whose columns are the coordinate vectors of the image of the basis elements of $\mathcal A$ relative to the basis $\mathcal B$. The matrix of such a linear transformation $u$ relative to the ordered bases $\left \langle {a_n} \right \rangle$ and $\left \langle {b_m} \right \rangle$ is denoted: : $\left[{u; \left \langle {b_m} \right \rangle, \left \langle {a_n} \right \rangle}\right]$ If $u$ is an automorphism on an $n$-dimensional module $G$, we can write $\left[{u; \left \langle {a_n} \right \rangle, \left \langle {a_n} \right \rangle}\right]$ as $\left[{u; \left \langle {a_n} \right \rangle}\right]$.


Definition:Invertible Matrix

Let $\struct {R, +, \circ}$ be a ring with unity. Let $n \in \Z_{>0}$ be a (strictly) positive integer. Let $\mathbf A$ be an element of the ring of square matrices $\struct {\map {\MM_R} n, +, \times}$. Then $\mathbf A$ is '''invertible''' {{iff}}: :$\exists \mathbf B \in \struct {\map {\MM_R} n, +, \times}: \mathbf A \mathbf B = \mathbf I_n = \mathbf B \mathbf A$ where $\mathbf I_n$ denotes the unit matrix of order $n$. Such a $\mathbf B$ is the inverse of $\mathbf A$. It is usually denoted $\mathbf A^{-1}$.


Definition:Change of Basis Matrix

Let $R$ be a ring with unity. Let $G$ be a finite-dimensional free $R$-module. Let $A = \left \langle {a_n} \right \rangle$ and $B = \left \langle {b_n} \right \rangle$ be ordered bases of $G$. === Definition 1 === {{:Definition:Change of Basis Matrix/Definition 1}} === Definition 2 === {{:Definition:Change of Basis Matrix/Definition 2}}


Definition:Matrix Equivalence

Let $R$ be a ring with unity. Let $\mathbf A, \mathbf B$ be $m \times n$ matrices over $R$. Let there exist: :an invertible square matrix $\mathbf P$ of order $n$ over $R$ :an invertible square matrix $\mathbf Q$ of order $m$ over $R$ such that $\mathbf B = \mathbf Q^{-1} \mathbf A \mathbf P$. Then $\mathbf A$ and $\mathbf B$ are '''equivalent''', and we can write $\mathbf A \equiv \mathbf B$. Thus, from Matrix Corresponding to Change of Basis under Linear Transformation, two matrices are '''equivalent''' {{iff}} they are the matrices of the same linear transformation, relative to (possibly) different ordered bases.


Definition:Matrix Similarity

Let $R$ be a ring with unity. Let $n\geq1$ be a natural number. Let $\mathbf A, \mathbf B$ be square matrices of order $n$ over $R$. Let there exist an invertible square matrix $\mathbf P$ of order $n$ over $R$ such that $\mathbf B = \mathbf P^{-1} \mathbf A \mathbf P$. Then $\mathbf A$ and $\mathbf B$ are '''similar''', and we can write $\mathbf A \sim \mathbf B$. Thus, from the corollary to Matrix Corresponding to Change of Basis under Linear Transformation, two matrices are similar {{iff}} they are the matrices of the same linear operator, relative to (possibly) different ordered bases.


Definition:Transpose of Matrix

Let $\mathbf A = \sqbrk \alpha_{m n}$ be an $m \times n$ matrix over a set. Then the '''transpose''' of $\mathbf A$ is denoted $\mathbf A^\intercal$ and is defined as: :$\mathbf A^\intercal = \sqbrk \beta_{n m}: \forall i \in \closedint 1 n, j \in \closedint 1 m: \beta_{i j} = \alpha_{j i}$


Definition:Linear Equation

A '''linear equation''' is an equation in the form: :$b = a_1 x_1 + a_2 x_2 + \cdots + a_n x_n$ where all of $a_1, \ldots, a_n, x_1, \ldots x_n, b$ are elements of a given field.


Definition:Simultaneous Equations

A '''system of simultaneous equations''' is a set of equations: :$\forall i \in \set {1, 2, \ldots, m} : \map {f_i} {x_1, x_2, \ldots x_n} = \beta_i$ That is: {{begin-eqn}} {{eqn | l = \beta_1       | r = \map {f_1} {x_1, x_2, \ldots x_n} }} {{eqn | l = \beta_2       | r = \map {f_2} {x_1, x_2, \ldots x_n} }} {{eqn | o = \cdots}} {{eqn | l = \beta_m       | r = \map {f_m} {x_1, x_2, \ldots x_n} }} {{end-eqn}}


Definition:Principal Ideal of Ring

Let $\struct {R, +, \circ}$ be a ring with unity. Let $a \in R$. We define: :$\ideal a = \displaystyle \set {\sum_{i \mathop = 1}^n r_i \circ a \circ s_i: n \in \N, r_i, s_i \in R}$ The ideal $\ideal a$ is called the '''principal ideal of $R$ generated by $a$'''.


Definition:Principal Ideal Domain

A '''principal ideal domain''' is an integral domain in which every ideal is a principal ideal.


Definition:Maximal Ideal of Ring

Let $R$ be a ring. An ideal $J$ of $R$ is '''maximal''' {{iff}}: :$(1): \quad J \subsetneq R$ :$(2): \quad$ There is no ideal $K$ of $R$ such that $J \subsetneq K \subsetneq R$. That is, {{iff}} $J$ is a maximal element of the set of all proper ideals of $R$ ordered by inclusion.


Definition:Iff

The logical connective '''iff''' is a convenient shorthand for '''if and only if'''.


Definition:Characteristic of Ring

Let $\struct {R, +, \circ}$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$. === Definition 1 === {{:Definition:Characteristic of Ring/Definition 1}} === Definition 2 === {{:Definition:Characteristic of Ring/Definition 2}} === Definition 3 === {{:Definition:Characteristic of Ring/Definition 3}}


Definition:Prime Field

A '''prime field''' is a field with no proper subfields.


Definition:Prime Subfield

Let $F$ be a field. From Intersection of Subfields is Subfield, the intersection of all the subfields of $F$ is itself a subfield which is unique and minimal. This unique minimal subfield in a field $F$ is called the '''prime subfield of $F$'''.


Definition:Polynomial over Ring

Let $R$ be a commutative ring with unity. === One Variable === {{Definition:Polynomial over Ring/One Variable}} === Multiple Variables === {{Definition:Polynomial over Ring/Multiple Variables}}


=== Field Extension === {{:Definition:Transcendental (Abstract Algebra)/Field Extension}} === Transcendental over Integral Domain === {{:Definition:Transcendental (Abstract Algebra)/Ring}}


Definition:Elementary Function

An '''elementary function''' is one of the following: * The constant function: $\map {f_c} x = c$ where $c \in \R$ * Powers of $x$: $\map f x = x^y$, where $y \in \R$ * Exponentials: $\map f x = e^x$ * Natural logarithms: $\map f x = \ln x$ * Trigonometric functions: $\map f x = \sin x$, $\map f x = \cos x$ * Inverse trigonometric functions: $\map f x = \arcsin x$, $\map f x = \arccos x$ * All functions that are compositions of the above, for example $\map f x = \ln \sin x$, $\map f x = e^{\cos x}$ * All functions obtained by adding, subtracting, multiplying and dividing any of the above types any finite number of times.


Definition:Derivative

Informally, a '''derivative''' is the rate of change of one variable with respect to another. === Real Function === {{:Definition:Derivative/Real Function/Derivative at Point}} === Complex Function === {{:Definition:Derivative/Complex Function/Point}} === Vector-Valued Function === {{:Definition:Derivative/Vector-Valued Function/Point}}


Definition:Differentiation

The process of obtaining the derivative of a differentiable function $f$ with respect to $x$ is known as: : '''differentiation (of $f$) with respect to $x$''' or : '''differentiation WRT $x$'''


Definition:Continuous Function

=== Continuous Complex Function === {{:Definition:Continuous Complex Function}} === Continuous Real Function === {{:Definition:Continuous Real Function}}


Definition:Subdivision (Real Analysis)

Let $\closedint a b$ be a closed interval of the set $\R$ of real numbers. === Finite === {{:Definition:Subdivision (Real Analysis)/Finite}} === Infinite === {{:Definition:Subdivision (Real Analysis)/Infinite}}


Definition:Lower Sum

Let $\closedint a b$ be a closed real interval. Let $f: \closedint a b \to \R$ be a bounded real function. Let $P = \set {x_0, x_1, x_2, \ldots, x_n}$ be a finite subdivision of $\closedint a b$. For all $\nu \in \set {1, 2, \ldots, n}$, let $m_\nu^{\paren f}$ be the infimum of $f$ on the interval $\closedint {x_{\nu - 1} } {x_\nu}$. Then: :$\displaystyle \map {L^{\paren f} } P = \sum_{\nu \mathop = 1}^n m_\nu^{\paren f} \paren {x_\nu - x_{\nu - 1} }$ is called the '''lower sum of $f$ on $\closedint a b$ belonging''' (or '''with respect''') '''to (the subdivision) $P$'''. If there is no ambiguity as to what function is under discussion, $m_\nu$ and $\map L P$ are usually used.


Definition:Upper Sum

Let $\closedint a b$ be a closed real interval. Let $f: \closedint a b \to \R$ be a bounded real function. Let $P = \set {x_0, x_1, x_2, \ldots, x_n}$ be a finite subdivision of $\closedint a b$. For all $\nu \in \set {1, 2, \ldots, n}$, let $M_\nu^{\paren f}$ be the supremum of $f$ on the interval $\closedint {x_{\nu - 1} } {x_\nu}$. Then: :$\displaystyle \map {U^{\paren f} } P = \sum_{\nu \mathop = 1}^n M_\nu^{\paren f} \paren {x_\nu - x_{\nu - 1} }$ is called the '''upper sum of $f$ on $\closedint a b$ belonging''' (or '''with respect''') '''to (the subdivision) $P$'''. If there is no ambiguity as to what function is under discussion, $M_\nu$ and $\map U P$ are often seen.


Definition:Definite Integral

Let $\closedint a b$ be a closed real interval. Let $f: \closedint a b \to \R$ be a real function. === Riemann Integral === {{:Definition:Definite Integral/Riemann|Riemann Integral}} === Darboux Integral === {{:Definition:Definite Integral/Darboux|Darboux Integral}}


Definition:Dedekind Cut

Let $\struct {S, \preceq}$ be a totally ordered set. === Definition 1 === {{:Definition:Dedekind Cut/Definition 1}} === Definition 2 === {{:Definition:Dedekind Cut/Definition 2}}


Definition:Complex Number

{{:Definition:Complex Number/Definition 1}}


Definition:Determinant/Matrix

Let $\mathbf A = \sqbrk a_n$ be a square matrix of order $n$. That is, let: :$\mathbf A = \begin {bmatrix} a_{1 1} & a_{1 2} & \cdots & a_{1 n} \\ a_{2 1} & a_{2 2} & \cdots & a_{2 n} \\  \vdots &  \vdots & \ddots &  \vdots \\ a_{n 1} & a_{n 2} & \cdots & a_{n n} \\ \end {bmatrix}$ === Definition 1 === {{:Definition:Determinant/Matrix/Definition 1}} === Definition 2 === {{:Definition:Determinant/Matrix/Definition 2}}


Definition:Cofactor

Let $R$ be a commutative ring with unity. Let $\mathbf A \in R^{n \times n}$ be a square matrix of order $n$.  Let: : $D = \begin{vmatrix}  a_{11} & a_{12} & \cdots & a_{1n} \\   a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\   a_{n1} & a_{n2} & \cdots & a_{nn}\end{vmatrix}$ be a determinant of order $n$. === Cofactor of an Element === {{:Definition:Cofactor/Element}} === Cofactor of a Minor === {{:Definition:Cofactor/Minor}}


Definition:Discriminant of Polynomial

Let $k$ be a field. Let $\map f X \in k \sqbrk X$ be a polynomial of degree $n$. Let $\overline k$ be an algebraic closure of $k$. Let the roots of $f$ in $\overline k$ be $\alpha_1, \alpha_2, \ldots, \alpha_n$. Then the '''discriminant''' $\map \Delta f$ of $f$ is defined as: :$\displaystyle \map \Delta f := \prod_{1 \mathop \le i \mathop < j \mathop \le n} \paren {\alpha_i - \alpha_j}^2$ === Quadratic Equation === The concept is usually encountered in the context of a quadratic equation $a x^2 + b x + c$: {{:Definition:Discriminant of Polynomial/Quadratic Equation}} === Cubic Equation === In the context of a cubic equation $a x^3 + b x^2 + c x + d$: {{Definition:Discriminant of Polynomial/Cubic Equation}}


Definition:Minor of Determinant

Let $\mathbf A = \sqbrk a_n$ be a square matrix of order $n$. Consider the order $k$ square submatrix $\mathbf B$ obtained by deleting $n - k$ rows and $n - k$ columns from $\mathbf A$. Let $\map \det {\mathbf B}$ denote the determinant of $\mathbf B$. Then $\map \det {\mathbf B}$ is an '''order-$k$ minor''' of $\map \det {\mathbf A}$. Thus a '''minor''' is a determinant formed from the elements (in the same relative order) of $k$ specified rows and columns.


Definition:Diagonal Matrix

Let $\mathbf A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} \\ \end{bmatrix}$ be a square matrix of order $n$. Then $\mathbf A$ is a '''diagonal matrix''' {{iff}} all elements of $\mathbf A$ are zero except for possibly its diagonal elements.


Definition:Triangular Matrix

Let $\mathbf T = \begin {bmatrix} a_{1 1} & a_{1 2} & \cdots & a_{1 n} \\ a_{2 1} & a_{2 2} & \cdots & a_{2 n} \\  \vdots &  \vdots & \ddots &  \vdots \\ a_{m 1} & a_{m 2} & \cdots & a_{m n} \\ \end {bmatrix}$ be a matrix of order $m \times n$. Then $\mathbf T$ is a '''triangular matrix''' {{iff}} all the elements either above or below the diagonal are zero. === Upper Triangular Matrix === {{:Definition:Triangular Matrix/Upper Triangular Matrix}} === Lower Triangular Matrix === {{:Definition:Triangular Matrix/Lower Triangular Matrix}}


Definition:Leading Coefficient of Matrix

Let $\mathbf A = \sqbrk a_{m n}$ be an $m \times n$ matrix. The '''leading coefficient''' of each row of $\mathbf A$ is the leftmost non-zero element of that row. A zero row has no '''leading coefficient'''.


Definition:Echelon Matrix

Let $\mathbf A = \sqbrk a_{m n}$ be an $m \times n$ matrix. === Echelon Form === {{:Definition:Echelon Matrix/Echelon Form}} === Reduced Echelon Form === {{:Definition:Echelon Matrix/Reduced Echelon Form}}


Definition:Complex Conjugate

Let $z = a + i b$ be a complex number. Then the '''(complex) conjugate''' of $z$ is denoted $\overline z$ and is defined as: :$\overline z := a - i b$


Definition:Root (Analysis)

Let $x, y \in \R_{\ge 0}$ be positive real numbers. Let $n \in \Z$ be an integer such that $n \ne 0$. Then $y$ is the '''positive $n$th root of $x$''' {{iff}}: :$y^n = x$ and we write: :$y = \sqrt[n] x$ Using the power notation, this can also be written: :$y = x^{1/n}$


Definition:Power (Algebra)

=== Natural Numbers === {{:Definition:Power (Algebra)/Natural Number}} === Integers === {{:Definition:Power (Algebra)/Integer}} === Rational Numbers === {{:Definition:Power (Algebra)/Rational Number}} === Real Numbers === {{:Definition:Power (Algebra)/Real Number/Definition 1}} === Complex Numbers === {{Definition:Power (Algebra)/Complex Number}}


Definition:Exponential Function

The '''exponential function''' is denoted $\exp$ and can be defined in several ways, as described below.


Definition:Square Root

A '''square root''' of a number $n$ is a number $z$ such that $z$ squared equals $n$.


Definition:Arithmetic Mean

Let $x_1, x_2, \ldots, x_n \in \R$ be real numbers. The '''arithmetic mean''' of $x_1, x_2, \ldots, x_n$ is defined as: :$\displaystyle A_n := \dfrac 1 n \sum_{k \mathop = 1}^n x_k$ That is, to find out the '''arithmetic mean''' of a set of numbers, add them all up and divide by how many there are.


Definition:Geometric Mean

Let $x_1, x_2, \ldots, x_n \in \R_{>0}$ be (strictly) positive real numbers. The '''geometric mean''' of $x_1, x_2, \ldots, x_n$ is defined as: :$\displaystyle G_n := \paren {\prod_{k \mathop = 1}^n x_k}^{1/n}$


Definition:Harmonic Mean

Let $x_1, x_2, \ldots, x_n \in \R$ be real numbers which are all positive. The '''harmonic mean''' of $x_1, x_2, \ldots, x_n$ is defined as: :$\displaystyle H_n := \paren {\frac 1 n \paren {\sum_{k \mathop = 1}^n \frac 1 {x_k} } }^{-1}$ That is, to find the '''harmonic mean''' of a set of $n$ numbers, take the reciprocal of the arithmetic mean of their reciprocals.


Definition:Reciprocal

Let $x \in \R$ be a real number such that $x \ne 0$. Then $\dfrac 1 x$ is called the '''reciprocal of $x$'''.


Definition:Euler's Number

=== As the Limit of a Sequence === {{:Definition:Euler's Number/Limit of Sequence}} === As the Limit of a Series === {{:Definition:Euler's Number/Limit of Series}} === As the Base of the Natural Logarithm === As the base of the Natural Logarithm: {{:Definition:Euler's Number/Base of Logarithm}} === In Terms of the Exponential Function === In terms of the exponential function: {{:Definition:Euler's Number/Exponential Function}} === As the Base of the Exponential with Derivative One at Zero === {{:Definition:Euler's Number/Base of Exponential}}


Definition:Field Extension

Let $F$ be a field. A '''field extension over $F$''' is a field $E$ where $F \subseteq E$. That is, such that $F$ is a subfield of $E$.


Definition:Unbounded Divergent Sequence

=== Real Sequence === {{:Definition:Unbounded Divergent Sequence/Real Sequence}} === Complex Sequence === As the Complex Numbers cannot be Ordered Compatibly with Ring Structure, there is no concept of $-\infty$ in discussions relating to $\C$. So we can use only the following definition: {{:Definition:Unbounded Divergent Sequence/Complex Sequence}}


Definition:Oscillating Sequence

Let $S$ be one of the standard number fields $\Q$, $\R$ or $\C$. Let $\sequence {x_n}$ be a sequence in $S$. Let $\sequence {x_n}$ be divergent. Suppose $\sequence {x_n}$ is ''not'' unbounded. That is, let: :$\neg x_n \to \infty$ as $n \to \infty$ Then $\sequence {x_n}$ is said to '''oscillate'''.


Definition:Field Adjoined Element

{{delete|think it fits more naturally here}} Let $E/F$ be a field extension, $\alpha \in E$. Then: :$F[\alpha] $ denotes the smallest subring of $E$ containing $F \cup \alpha$. :$F(\alpha) $ denotes the smallest subfield of $E$ containing $F \cup \alpha$.  We say this as '''$F$ adjoined with $\alpha$'''. Category:Definitions/Field Theory 53di6f01mi92rlorjbozg4cb00lpybf


Definition:Simple Field Extension

Let $E / F$ be a field extension. Then $E$ is a '''simple extension over $F$''' {{iff}}: : $\exists \alpha \in E: E = F \sqbrk \alpha$ where $F \sqbrk \alpha$ is the field extension generated by $\alpha$.


Definition:Subsequence

Let $\sequence {x_n}$ be a sequence in a set $S$. Let $\sequence {n_r}$ be a strictly increasing sequence in $\N$. Then the composition $\sequence {x_{n_r} }$ is called a '''subsequence of $\sequence {x_n}$'''.


Definition:Limit Superior

Let $\sequence {x_n}$ be a bounded sequence in $\R$. === Definition 1 === {{:Definition:Limit Superior/Definition 1}} === Definition 2 === {{:Definition:Limit Superior/Definition 2}}


Definition:Limit Inferior

Let $\sequence {x_n}$ be a bounded sequence in $\R$. === Definition 1 === {{:Definition:Limit Inferior/Definition 1}} === Definition 2 === {{:Definition:Limit Inferior/Definition 2}}


Definition:Series

=== General Definition === {{:Definition:Series/General}} === Series in a Standard Number Field === The usual context for the definition of a series occurs when $S$ is one of the standard number fields $\Q, \R, \C$. {{:Definition:Series/Number Field}} === Finite === {{:Definition:Series/Finite}}


Definition:Cauchy Criterion

The '''Cauchy criterion''' is the necessary and sufficient condition for an infinite sequence in a complete metric space to converge that the absolute difference between successive terms with sufficiently large indices tends to zero.


Definition:Absolutely Convergent Series

=== General Definition === {{:Definition:Absolutely Convergent Series/General}} === Real Numbers === {{:Definition:Absolutely Convergent Series/Real Numbers}} === Complex Numbers === {{:Definition:Absolutely Convergent Series/Complex Numbers}}


Definition:Rational Function

Let $F$ be a field. Let $P: F \to F$ and $Q: F \to F$ be polynomial functions on $F$. Let $S$ be the set $F$ from which all the roots of $Q$ have been removed. That is: :$S = F \setminus \set {x \in F: \map Q x = 0}$ Then the equation $y = \dfrac {\map P x} {\map Q x}$ defines a mapping from $S$ to $F$. Such a mapping is called a '''rational function'''.


Definition:Semiperimeter

The '''semiperimeter''' of a plane figure is defined to be one half of the perimeter.


Definition:Circumradius

=== Polygon === {{:Definition:Circumradius/Polygon}} === Polyhedron === {{:Definition:Circumradius/Polyhedron}} Category:Definitions/Geometry mx2rosmsqqf5hj1aog5ceifz94uowrp


Definition:Incircle of Triangle

The '''incircle''' of a triangle is the circle tangent to each side of the triangle. This circle is said to be inscribed in the triangle. :410px === Incenter === {{:Definition:Incircle of Triangle/Incenter}} === Inradius === {{:Definition:Incircle of Triangle/Inradius}}


Definition:Excircle of Triangle

Given a triangle, extend two sides in the direction opposite their common vertex. The circle tangent to both of these lines and to the third side of the triangle is called an '''excircle'''. :500px There are three '''excircles''' for every triangle. === Excenter === {{:Definition:Excircle of Triangle/Excenter}} === Exradius === {{:Definition:Excircle of Triangle/Exradius}}


Definition:Circumcircle of Triangle

The '''circumcircle of a triangle''' is the circle that passes through all three of the triangle's vertices. This circle is said to circumscribe the triangle. :320px === Circumcenter === {{:Definition:Circumcircle of Triangle/Circumcenter}} === Circumradius === {{:Definition:Circumcircle of Triangle/Circumradius}}


Definition:Triangle (Geometry)

:300px A '''triangle''' is a polygon with exactly three sides.


Definition:Altitude of Triangle

Let $\triangle ABC$ be a triangle. Let a perpendicular be dropped from $\angle A$ to its opposite side $a$: :400px The line $h_a$ so constructed is called the '''altitude''' of $\angle A$. === Foot of Altitude === {{:Definition:Altitude of Triangle/Foot}}


Definition:Polygon

A '''polygon''' is a closed plane figure made up of an unspecified number of non-crossing straight line segments that join in pairs at their endpoints. For example: :400px


Definition:Quadrilateral

A '''quadrilateral''' is a polygon with exactly four sides.


Definition:Perimeter

The '''perimeter''' of a plane figure is the length of its boundary.


Definition:Bisection

To '''bisect''' a finite geometrical object is to cut it into two equal parts.


Definition:Differentiable Mapping

=== Real Function === {{:Definition:Differentiable Mapping/Real Function}} === Complex Function === {{:Definition:Differentiable Mapping/Complex Function}} === Real-Valued Function === {{:Definition:Differentiable Mapping/Real-Valued Function}} === Vector-Valued Function === {{:Definition:Differentiable Mapping/Vector-Valued Function}} === Between Differentiable Manifolds === {{Definition:Differentiable Mapping between Manifolds}}


Definition:Compact Space

=== Euclidean Space === {{:Definition:Compact Space/Euclidean Space}} === Topology === {{:Definition:Compact Space/Topology}} === Metric Space === {{:Definition:Compact Space/Metric Space}} === Normed Vector Space === {{:Definition:Compact Space/Normed Vector Space}}


Definition:Open Cover

Let $T = \struct {S, \tau}$ be a topological space. Let $\CC$ be a cover for $S$. Then $\CC$ is an '''open cover (of $T$)''' {{iff}}: :$\CC \subseteq \tau$ That is, {{iff}} all the elements of $\CC$ are open sets. === Open Cover of Subset === {{:Definition:Open Cover/Subset}}


Definition:Closed Set

=== Topology === Let $T = \struct {S, \tau}$ be a topological space. Let $H \subseteq S$. {{:Definition:Closed Set/Topology}} === Metric Space === In the context of metric spaces, the same definition applies: {{:Definition:Closed Set/Metric Space}} === Normed Vector Space === {{:Definition:Closed Set/Normed Vector Space}} === Complex Analysis === {{:Definition:Closed Set/Complex Analysis}} === Real Analysis === {{:Definition:Closed Set/Real Analysis/Real Numbers}}


Definition:Submanifold

A '''submanifold''' is a manifold which is completely contained in some other manifold.


Definition:Codimension

Let $X$ be a manifold. Let $Z$ be a submanifold of $X$. The '''codimension of $Z$ in $X$''' is the dimension of $X$ minus the dimension of $Z$. Category:Definitions/Topology 28d2r6r1yrbtw22lu96vpui26vgff5r


Definition:Critical Point (Topology)

Let $f: X \to Y$ be a smooth map of manifolds. A point $x \in X$ is called a '''critical point''' of $f$ {{iff}} $\d f_x: T_x \sqbrk X \to T_y \sqbrk Y$ is not surjective at $x$. {{explain|What are $\d f_x$, $T_x$ and $T_y$ in this context?}} Category:Definitions/Topology e5niy2x3798s0kyysfe4v2x207hvzt6


Definition:Homeomorphism

=== Topological Spaces === Let $T_\alpha = \struct {S_\alpha, \tau_\alpha}$ and $T_\beta = \struct {S_\beta, \tau_\beta}$ be topological spaces. Let $f: T_\alpha \to T_\beta$ be a bijection. {{:Definition:Homeomorphism/Topological Spaces/Definition 1}} === Metric Spaces === The same definition applies to metric spaces: {{:Definition:Homeomorphism/Metric Spaces/Definition 1}} === Manifolds === The same definition applies to manifolds: {{:Definition:Homeomorphism/Manifolds}}


Definition:Hausdorff Space

{{:Definition:Hausdorff Space/Definition 1}}


Definition:Tychonoff Separation Axioms

The '''Tychonoff separation axioms''' are a classification system for topological spaces. They are not axiomatic as such, but conditions that may or may not apply to general or specific topological spaces.


Definition:Differential Form

Let $X$ be a smooth manifold. Let $\map {T_x} X$ denotes the tangent space of $X$. A '''$p$-form''' on $X$ is a function $\omega: \map {T_x} X^p \to \R$ defined at each point of $X$ which takes $p$ vectors as inputs, and outputs a real number.


Definition:Euclidean Space

Let $S$ be one of the standard number fields $\Q$, $\R$, $\C$. Let $S^n$ be a cartesian space for $n \in \N_{\ge 1}$. Let $d: S^n \times S^n \to \R$ be the usual (Euclidean) metric on $S^n$. Then $\tuple {S^n, d}$ is a '''Euclidean space'''.


Definition:Tangent Vector

Let $M$ be a smooth manifold. Let $m \in M$ be a point.  Let $V$ be an open neighborhood of $m$.  Let $C^\infty \left({V, \R}\right)$ be defined as the set of all smooth mappings $f: V \to \R$. === Definition 1 === {{:Definition:Tangent Vector/Definition 1}} === Definition 2 === {{:Definition:Tangent Vector/Definition 2}}


Definition:Tangent Bundle

Let $X$ be a differentiable manifold. The '''tangent bundle''' of $X$ is the disjoint union of all the tangent spaces of $X$ at $x$ as the base point $x$ ranges over the entire manifold: :$\displaystyle \map T X = \coprod_{x \mathop \in X} \map {T_x} X$ {{explain|probably also need to mention the natural differential structure on the tangent bundle.}} Category:Definitions/Topology nvuraom6bujfdc60gq0lg9ek5cpxewn


Definition:P-adic Norm

=== Definition 1 === {{:Definition:P-adic Norm/Definition 1}} === Definition 2 === {{:Definition:P-adic Norm/Definition 2}} === $p$-adic Metric === {{:Definition:P-adic Norm/P-adic Metric}} === $p$-adic Numbers === Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. {{:Definition:P-adic Norm/P-adic Numbers}}


Definition:Homotopy

=== Free Homotopy === {{:Definition:Homotopy/Free}} === Relative Homotopy === {{:Definition:Homotopy/Relative}} === Path-Homotopy === {{:Definition:Homotopy/Path}}


Definition:Homotopy Group

Let $X$ be a topological space, and $x_0 \in X$. Let $\mathbb S^n \subseteq \R^{n+1}$ be the $n$-sphere, and $a \in \mathbb S^n$. Let $\pi_n \left({X, x_0}\right)$ be the set of homotopy classes relative to $a$ of continuous mappings $c: \mathbb S^n \to X$ such that $c(a) = x_0$. Let $* : \pi_n \left({X, x_0}\right) \times \pi_n \left({X, x_0}\right) \to \pi_n \left({X, x_0}\right)$ denote the concatenation of homotopy classes of paths. That is, if $\overline{c_1}, \overline{c_2}$ are two elements of $\pi_n \left({X, x_0}\right)$, then: :$\overline{c_1} * \overline{c_2} = \overline{c_1 \cdot c_2}$ where $\cdot$ denotes the usual concatenation of paths. Then $\left({\pi_n \left({X, x_0}\right), *}\right)$ is the '''$n$th fundamental group''' of $X$. The first homotopy group is usually called the '''fundamental group''' when higher homotopy groups are not in sight. For a path-connected manifold, by Fundamental Group is Independent of Base Point for Path-Connected Space, the isomorphism class of $\pi_1 \left({X, x_0}\right)$ does not depend on $x_0$ and we just write $\pi_1 \left({X}\right)$.


Definition:Concatenation (Topology)

Let $T = \left({S, \tau}\right)$ be a topological space. Let $c_1, c_2: \left[{0  \,.\,.\, 1}\right]^n \to S$ be maps. Let $c_1$ and $c_2$ both satisfy the concatenation criterion $c \left({\partial \left[{0 \,.\,.\, 1}\right]^n}\right) = x_0$. <!-- For two maps $c_1, c_2: \left[{0 \,.\,.\, 1}\right]^n \to X$ which satisfy the concatenation criterion $c \left({\partial \left[{0 \,.\,.\, 1}\right]^n}\right) = x_0$--> Then the '''concatenation''' $c_1 * c_2$ is defined as: :$\left({c_1 * c_2}\right) \left({t_1, t_2, \ldots, t_n}\right) = \begin{cases} c_1 \left({2t_1, t_2, \ldots, t_n}\right) & : t_1 \in \left[{0 \,.\,.\, 1/2}\right] \\  c_2 \left({2t_1-1, t_2, \ldots, t_n}\right) & : t_1 \in \left[{1/2 \,.\,.\, 1}\right] \end{cases} $  where $\left({t_1, \ldots, t_n}\right)$ are coordinates in the $n$-cube. By Continuous Mapping on Finite Union of Closed Sets, $c_1*c_2$ is continuous. This resulting map is continuous, since: * $2 \left({\dfrac 1 2}\right) = 1$ and $2 \left({\dfrac 1 2}\right) - 1 = 0$; * anywhere any coordinate of $\hat t$ is either $1$ or $0$, $\left({c_1*c_2}\right) \left({\hat t}\right) = x_0$. The resulting map also clearly satisfies the concatenation criteria itself.


Definition:Fundamental Group

Let $(X, x_0)$ be a pointed topological space with base point $x_0$. The '''fundamental group''' $\pi_1 \left({X, x_0}\right)$ of $X$ at the base point $x_0$ is the set of homotopy classes of loops with base point $x_0$ with multiplication of homotopy classes of paths.


Definition:Maximum Value of Real Function/Local

Let $f$ be a real function defined on an open interval $\openint a b$. Let $\xi \in \openint a b$. Then $f$ has a '''local maximum at $\xi$''' {{iff}}: :$\exists \openint c d \subseteq \openint a b: \forall x \in \openint c d: \map f x \le \map f \xi$ That is, {{iff}} there is some subinterval on which $f$ attains a maximum within that interval.


Definition:Minimum Value of Real Function/Local

Let $f$ be a real function defined on an open interval $\openint a b$. Let $\xi \in \openint a b$. Then $f$ has a '''local minimum at $\xi$''' {{iff}}: :$\exists \openint c d \subseteq \openint a b: \forall x \in \openint c d: \map f x \ge \map f \xi$ That is, {{iff}} there is some subinterval on which $f$ attains a minimum within that interval.


Definition:Stationary Point

Let $f$ be a real function which is differentiable on the open interval $\openint a b$. Let $\exists \xi \in \openint a b: \map {f'} \xi = 0$, where $\map {f'} \xi$ is the derivative of $f$ at $\xi$. Then $\xi$ is known as a '''stationary point of $f$'''.


Definition:Cauchy Equivalent Metrics

Let $X$ be a set upon which there are two metrics $d_1$ and $d_2$. That is, $\struct {X, d_1}$ and $\struct {X, d_2}$ are two different metric spaces on the same underlying set $X$. Then $d_1$ and $d_2$ are said to be '''Cauchy equivalent''' {{iff}} for every sequence $\sequence {x_n}$ of points in $X$: :$\sequence {x_n}$ is a Cauchy sequence in $\struct {X, d_1} \iff \sequence {x_n}$ is a  is a Cauchy sequence in $\struct {X, d_2}$ {{NamedforDef|Augustin Louis Cauchy|cat = Cauchy}}


Definition:Homology Group

Let $X$ be a topological space. Let the standard n-simplex be denoted: :$\displaystyle \Delta^n := \left\{ {\left({x_0, \ldots, x_n}\right) \in {\R_{\ge 0} }^{n + 1}: \sum_{i \mathop = 1}^{n + 1} x_i = 1}\right\}$ Let $\mathcal C \left({\Delta^n, X}\right)$ be the set of continuous mappings from $\Delta^n$ to $X$.  For $n \ge 0$, let $C_n \left({X}\right)$ be the free abelian group generated by $\mathcal C \left({\Delta^n, X}\right)$.  Then there is a boundary map $\partial_n: C_n \left({X}\right) \to C_{n - 1} \left({X}\right)$ defined as follows. First, there are maps $s^i_n: \Delta^{n - 1} \to \Delta^n$, where $n > 0$ and $0 \le i \le n$, defined by: :$s^i_n \left({x_0, \ldots, x_{n - 1} }\right) = \left({x_0, \ldots, x_{i - 1}, 0, x_i, \ldots, x_{n-1} }\right)$ These can be considered as the inclusion of $\Delta^{n-1}$ as a 'face' of $\Delta^n$. For a continuous function $\phi: \Delta^n \to X$ we define: :$\displaystyle \partial_n \left({\phi}\right) = \sum_{i \mathop = 0}^n \left({-1}\right)^i \phi \circ s^i_n$ This definition, along with the requirement that $\partial_n$ be a group homomorphism, uniquely specifies $\partial_n$. In addition, one has $\partial_{n-1} \partial_n = 0$, meaning that the sequence of groups and morphisms: :$0 \gets C_0 \left({X}\right) \stackrel {\partial_1} {\longleftarrow} C_1 \left({X}\right) \stackrel {\partial_2} {\longleftarrow} C_2 \left({X}\right) \stackrel {\partial_3} {\longleftarrow} \cdots$ are a chain complex. This is demonstrated in Singular Chains form Chain Complex. Let $\partial_0$ denote the map $C_0 \left({X}\right) \to 0$. Thus the $n$th singular homology group of $X$ is defined as the $n$th homology group of this chain complex. Explicitly: Let $B_n \left({X}\right) \subset C_n \left({X}\right)$ denote the image of $\partial_{n+1}$. Let $Z_n \left({X}\right)$ denote the kernel of $\partial_n$. Since $\partial_n \partial_{n + 1} = 0$: :$B_n \left({X}\right) \subseteq Z_n \left({X}\right)$ Then define: :$H_n \left({X}\right) = \dfrac {Z_n \left({X}\right)} {B_n \left({X}\right)}$


Definition:Convex Real Function

Let $f$ be a real function which is defined on a real interval $I$. === Definition 1 === {{:Definition:Convex Real Function/Definition 1}} === Definition 2 === {{:Definition:Convex Real Function/Definition 2}} === Definition 3 === {{:Definition:Convex Real Function/Definition 3}}


Definition:Concave Real Function

Let $f$ be a real function which is defined on a real interval $I$. === Definition 1 === {{:Definition:Concave Real Function/Definition 1}} === Definition 2 === {{:Definition:Concave Real Function/Definition 2}} === Definition 3 === {{:Definition:Concave Real Function/Definition 3}}


Definition:Simply Connected

A path-connected topological space $T = \left({S, \tau}\right)$ is said to be '''simply connected''' if the fundamental group $\pi_1 \left({T}\right)$ is trivial. {{stub|Expand to include the context of regions in the plane.}}


Definition:Primitive (Calculus)

=== Primitive of Real Function === {{:Definition:Primitive (Calculus)/Real}}


Definition:Transcendental Number

A number (either real or complex) is '''transcendental''' {{iff}} it is not algebraic.


Definition:Improper Integral

An '''improper integral''' is a definite integral over an interval which is not closed, that is, open or half open, and whose limits of integration are the end points of that interval. When the end point is not actually in the interval, the conventional definition of the definite integral is not valid.


Definition:Lebesgue Measure

Let $\JJ_{ho}^n$ be the set of half-open $n$-rectangles. Let $\map \BB {\R^n}$ be the Borel $\sigma$-algebra on $\R^n$. Let $\lambda^n$ be the $n$-dimensional Lebesgue pre-measure on $\JJ_{ho}^n$. Any measure $\mu$ extending $\lambda^n$ to $\map \BB {\R^n}$ is called '''$n$-dimensional Lebesgue measure'''.


Definition:Measurable Set

Let $\struct {X, \Sigma}$ be a measurable space. A subset $S \subseteq X$ is said to be '''($\Sigma$-)measurable''' {{iff}} $S \in \Sigma$. === Measurable Set of an Arbitrary Outer Measure === {{:Definition:Measurable Set/Arbitrary Outer Measure}} === Measurable Subset of the Reals === {{:Definition:Measurable Set/Subset of Reals}} === Measurable Subset of $\R^n$ === {{:Definition:Measurable Set/Subsets of Real Space}}


Definition:Strictly Negative

Let $\struct {R, +, \circ, \le}$ be an ordered ring whose zero is $0_R$. Then $x \in R$ is '''strictly negative''' {{iff}}: :$x \le 0_R$ and $x \ne 0_R$ That $x$ is strictly negative may be (more conveniently) denoted $0_R < x$ or $x > 0_R$. Thus, the set of all strictly negative elements of $R$ is denoted: :$R_{< 0_R} := \set {x \in R: x < 0_R}$


Definition:Algebra of Sets

{{:Definition:Algebra of Sets/Definition 1}}


Definition:Quotient Topology/Quotient Space

The '''quotient space of $S$ by $\RR$''' is the topological space whose points are elements of the quotient set of $\RR$ and whose topology is $\tau_\RR$: :$T_\RR := \struct {S / \RR, \tau_\RR}$


Definition:Torus (Topology)

The '''$n$-dimensional torus''' (or '''$n$-torus)''' $\Bbb T^n$ is defined as the space whose points are those of the cross product of $n$ circles: :$\Bbb T^n = \underbrace{\Bbb S^1 \times \Bbb S^1 \times \ldots \times \Bbb S^1}_{n \text{ times}}$ and whose topology $\tau_{\Bbb T^n}$ is defined as: :$U \in \tau_{\Bbb T^n} \iff \exists U_1, U_2, \ldots, U_n \in \tau_{\Bbb S^1} : U = U_1 \times U_2 \times \ldots \times U_n$ where $\tau_{\Bbb S^1}$ is the topology of the circle. {{explain|In this context, what is the "topology of the circle"?}} Category:Definitions/Algebraic Topology 2hjek1yjp7hakl3p0qgvhxiw3md3yve


Definition:Extended Real Number Line

=== Definition 1 === {{:Definition:Extended Real Number Line/Definition 1}} === Definition 2 === {{:Definition:Extended Real Number Line/Definition 2}}


=== Set === {{:Definition:Characteristic Function (Set Theory)/Set}} === Relation === {{:Definition:Characteristic Function (Set Theory)/Relation}}


Definition:Simple Function

Let $\left({X, \Sigma}\right)$ be a measurable space. A real-valued function $f: X \to \R$ is said to be a '''simple function''' {{iff}} it is a finite linear combination of characteristic functions: :$\displaystyle f = \sum_{k \mathop = 1}^n a_k \chi_{S_k}$ where $a_1, a_2, \ldots, a_n$ are real numbers and each of the sets $S_k$ is $\Sigma$-measurable.


Definition:Lebesgue Integral

Let $\lambda^n$ be a Lebesgue measure on $\R^n$. Let $f: \R^n \to \overline \R$ be a Lebesgue integrable function. Then the $\lambda^n$-integral of $f$: :$\displaystyle \int f \rd \lambda^n$ is called the '''Lebesgue integral of $f$'''.


Definition:Skolem Sequence

A '''Skolem sequence of order $n$''' is a (finite) sequence $S = \left\{{s_1, s_2, \ldots, s_{2 n}}\right\}$ of $2 n$ integers for which: :$(1): \quad$ For every $k \in \left\{{1, 2, \ldots, n}\right\}$ there exist exactly two elements $s_i, s_j \in S$ so that $s_i = s_j = k$ and: :$(2): \quad$ If $s_i = s_j = k$ and $i < j$ then $j - i = k$. {{NamedforDef|Thoralf Albert Skolem|cat = Skolem}} Category:Definitions/Number Theory f7hnlsqeuoxpb7zqpn3nkg6eyl7o8i8


Definition:Counting Measure

Let $\struct {X, \Sigma}$ be a measurable space. The '''counting measure (on $X$)''', denoted $\size {\, \cdot \,}$, is the measure defined by: :$\size {\, \cdot \,}: \Sigma \to \overline \R, \ \size E := \begin {cases} \map \# E & : \text {$E$ is finite} \\ +\infty & : \text {$E$ is infinite} \end{cases}$ where $\overline \R$ denotes the extended real numbers, and $\#$ denotes cardinality. That $\size {\, \cdot \,}$ is actually a measure is shown on Counting Measure is Measure.


Definition:Standard Discrete Metric

The '''standard discrete metric''' on a set $S$ is the metric satisfying: :$\map d {x, y} = \begin{cases} 0 & : x = y  \\  1 & : x \ne y \end{cases}$ This can be expressed using the Kronecker delta notation as: :$\map d {x, y} = 1 - \delta_{x y}$ The resulting metric space $M = \struct {S, d}$ is '''the standard discrete metric space on $S$'''.


Definition:Euclidean Metric/Real Vector Space

The '''Euclidean metric''' on $\R^n$ is defined as: :$\displaystyle d_2 \left({x, y}\right) := \left({\sum_{i \mathop = 1}^n \left({x_i - y_i}\right)^2}\right)^{1 / 2}$ where $x = \left({x_1, x_2, \ldots, x_n}\right), y = \left({y_1, y_2, \ldots, y_n}\right) \in \R^n$.


Definition:Connected Sum

The '''connected sum''' of two manifolds $A^n, B^n$ of dimension $n$ is defined as follows: Let $\Bbb D^n$ be a closed n-disk. Let $\alpha: \Bbb D^n \to A^n$ be a continuous (or, in the case of smooth manifolds, a smooth) injection. Let $\beta: \Bbb D^n \to B^n$ be a similar function.   Define the set: :$S = \paren {A^n \setminus \map \alpha {\paren {\Bbb D^n}^\circ} } \cup \paren {B^n \setminus \map \beta {\paren {\Bbb D^n}^\circ} }$ where: :$\setminus$ denotes set difference :$\paren {\Bbb D^n}^\circ$ denotes the interior of $B^n$. Define an equivalence relation $\sim$ on $S$ as: :$x \sim y \iff \paren {\paren {x = y} \lor \paren {\map {\alpha^{-1} } x = \map {\beta^{-1} } y} }$ Since the interiors of the disks were removed from the manifolds, it necessarily follows that: :$\map {\alpha^{-1} } x, \map {\beta^{-1} } y \in \partial \Bbb D^n$ The '''connected sum''' $A^n \# B^n$ is defined as the quotient space of $S$ under $\sim$. Category:Definitions/Topology mqyj5hjf7j2aem5z5zo7otzc6peyd2b


Definition:Sphere

=== Geometry === {{:Definition:Sphere/Geometry}} === Topology === {{:Definition:Sphere/Topology}} === Metric Space === {{:Definition:Sphere/Metric Space}} === Normed Division Ring === {{:Definition:Sphere/Normed Division Ring}} === Normed Vector Space === {{:Definition:Sphere/Normed Vector Space}} === P-adic Numbers === The definition of an '''sphere''' in the context of the $p$-adic numbers is a direct application of the definition of an sphere in a normed division ring: {{:Definition:Sphere/P-adic Numbers}} Category:Definitions/Topology Category:Definitions/Metric Spaces Category:Definitions/Normed Division Rings tntunx5oet71ckpg8veon8tihn7moi3


Definition:Argument of Complex Number

Let $z = x + i y$ be a complex number. An '''argument of $z$''', or $\arg z$, is formally defined as a solution to the pair of equations: :$(1): \quad \dfrac x {\cmod z} = \map \cos {\arg z}$ :$(2): \quad \dfrac y {\cmod z} = \map \sin {\arg z}$ where $\cmod z$ is the modulus of $z$. From Sine and Cosine are Periodic on Reals, it follows that if $\theta$ is an '''argument''' of $z$, then so is $\theta + 2 k \pi$ where $k \in \Z$ is ''any'' integer.


Definition:Cantor Set

=== As a Limit of Intersections === {{:Definition:Cantor Set/Limit of Intersections}} === From Ternary Representation === {{:Definition:Cantor Set/Ternary Representation}} === As a Limit of a Decreasing Sequence === {{:Definition:Cantor Set/Limit of Decreasing Sequence}} These definitions are all equivalent, as shown on Equivalence of Definitions of Cantor Set.


Definition:Dihedral Group

The '''dihedral group''' $D_n$ of order $2 n$ is the group of symmetries of the regular $n$-gon.


Definition:Euler-Mascheroni Constant

The '''Euler-Mascheroni constant''' $\gamma$ is the real number that is defined as: {{begin-eqn}} {{eqn | l = \gamma       | o = :=       | r = \lim_{n \mathop \to +\infty} \paren {\sum_{k \mathop = 1}^n \frac 1 k - \int_1^n \frac 1 x \rd x} }} {{eqn | r = \lim_{n \mathop \to +\infty} \paren {H_n - \ln n} }} {{end-eqn}} where $H_n$ is the harmonic series and $\ln$ is the natural logarithm.


Definition:Alternating Group

Let $S_n$ denote the symmetric group on $n$ letters. For any $\pi \in S_n$, let $\map \sgn \pi$ be the sign of $\pi$. The kernel of the mapping $\sgn: S_n \to C_2$ is called the '''alternating group on $n$ letters''' and denoted $A_n$.


Definition:Dicyclic Group

For even $n$, the '''dicyclic group''' $\Dic n$ of order $4 n$ is the group having the presentation: :$\Dic n = \gen {x, y: x^{2 n} = e, y^2 = x^n, y^{-1} x y = x^{-1} }$


Definition:Quaternion Group

The dicyclic group $\Dic 2$ is known as the '''quaternion group'''. The elements of $\Dic 2$ are: :$\Dic 2 = \set {e, a, a^2, a^3, b, a b, a^2 b, a^3 b}$


Definition:Power Series

=== Real Domain === {{:Definition:Power Series/Real Domain}} === Complex Domain === {{:Definition:Power Series/Complex Domain}}


Definition:Interval of Convergence

Let $\xi \in \R$ be a real number. Let $\displaystyle \map S x = \sum_{n \mathop = 0}^\infty a_n \paren {x - \xi}^n$ be a power series in $x$ about $\xi$. Then the set of values of $x$ for which $\map S x$ converges is called the '''interval of convergence''' of $S$.


Definition:Radius of Convergence

=== Real Domain === {{:Definition:Radius of Convergence/Real Domain}} === Complex Domain === {{:Definition:Radius of Convergence/Complex Domain}}


Definition:Conditionally Convergent Series

A series which is convergent, but not absolutely convergent, is said to be '''conditionally convergent'''.


Definition:Divergent (Analysis)

=== Divergent Sequence === {{:Definition:Divergent Sequence}} === Divergent Series === {{:Definition:Divergent Series}} === Divergent Function === {{:Definition:Divergent Function}} === Divergent Improper Integral === {{:Definition:Divergent Improper Integral}}


Definition:Taylor Series

Let $f$ be a real function which is smooth on the open interval $\openint a b$. Let $\xi \in \openint a b$. Then the '''Taylor series expansion of $f$''' about the point $\xi$ is: :$\displaystyle \sum_{n \mathop = 0}^\infty \frac {\paren {x - \xi}^n} {n!} \map {f^{\paren n} } \xi$ It is ''not'' necessarily the case that this power series is convergent with sum $\map f x$.


Definition:Analytic Function

=== Real Analytic Function === {{:Definition:Analytic Function/Real Numbers}} === Complex Analytic Function === {{:Definition:Analytic Function/Complex Plane}}


Definition:Sine

=== Definition from Triangle === {{:Definition:Sine/Definition from Triangle}} === Definition from Circle === The sine of an angle in a right triangle can be extended to the full circle as follows: {{:Definition:Sine/Definition from Circle}} === Real Numbers === {{:Definition:Sine/Real Function}} === Complex Numbers === {{:Definition:Sine/Complex Function}}


Definition:Cosine

=== Definition from Triangle === {{:Definition:Cosine/Definition from Triangle}} === Definition from Circle === {{:Definition:Cosine/Definition from Circle}} === Real Numbers === {{:Definition:Cosine/Real Function}} === Complex Numbers === {{:Definition:Cosine/Complex Function}}


Definition:Trigonometric Function

There are six basic '''trigonometric functions''': sine, cosine, tangent, cotangent, secant, and cosecant.


Definition:Secant Function

=== Definition from Triangle === {{:Definition:Secant Function/Definition from Triangle}} === Definition from Circle === {{:Definition:Secant Function/Definition from Circle}} === Real Function === {{:Definition:Secant Function/Real}} === Complex Function === {{:Definition:Secant Function/Complex}}


Definition:Cosecant

=== Definition from Triangle === {{:Definition:Cosecant/Definition from Triangle}} === Definition from Circle === {{:Definition:Cosecant/Definition from Circle}} === Real Function === {{:Definition:Cosecant/Real Function}} === Complex Function === {{:Definition:Cosecant/Complex Function}}


Definition:Cotangent

=== Definition from Triangle === {{:Definition:Cotangent/Definition from Triangle}} === Definition from Circle === {{:Definition:Cotangent/Definition from Circle}} === Real Function === {{:Definition:Cotangent/Real Function}} === Complex Function === {{:Definition:Cotangent/Complex Function}}


Definition:Holomorphic Function

=== Complex Function === {{:Definition:Holomorphic Function/Complex Plane}} === Vector-Valued Function === {{definition wanted}}


Definition:Complex Function

A '''complex function''' is a function whose domain and codomain are subsets of the set of complex numbers $\C$.


Definition:Periodic Function

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


Definition:Pi

The real number $\pi$ (pronounced '''pie''') is an irrational number (see proof here) whose value is approximately $3.14159\ 26535\ 89793\ 23846\ 2643 \ldots$


Definition:Decomposable Set

A set $S \subset \R^n$ is '''decomposable in $m$ sets $A_1, \ldots, A_m \subset \R^n$''' if there exist isometries $\phi_1, \ldots, \phi_m: \R^n \to \R^n$ such that: :$(1):\quad \displaystyle S = \bigcup_{k \mathop = 1}^m \phi_k \left({A_k}\right)$  :$(2):\quad \forall i \ne j: \phi_i \left({A_i}\right) \cap \phi_j \left({A_j}\right) = \varnothing$ Such a union is known as a '''decomposition'''. {{explain|The definition of Definition:Irreducible Space suggests that the definition may apply to a general topological space, not only $\R^n$. This needs to be clarified.}}


Definition:Isometry (Metric Spaces)

=== Definition 1 === {{:Definition:Isometry (Metric Spaces)/Definition 1}} === Definition 2 === {{:Definition:Isometry (Metric Spaces)/Definition 2}} Such metric spaces $M_1$ and $M_2$ are defined as being '''isometric'''. === Isometry Into === {{:Definition:Isometry (Metric Spaces)/Into}}


Definition:Equidecomposable

Two sets $S, T \subset \R^n$ are said to be '''equidecomposable''' if there exists a set: :$X = \set {A_1, \ldots, A_m} \subset \powerset {\R^n}$ where $\powerset {\R^n}$ is the power set of $\R^n$, such that both $S$ and $T$ are decomposable into the elements of $X$.


Definition:Asymptotically Equal

=== Sequences === {{:Definition:Asymptotically Equal/Sequences/Definition 1}} === Functions === {{:Definition:Asymptotically Equal/Functions}} === General Definition === {{:Definition:Asymptotically Equal/General Definition/Point}}


Definition:Inverse Sine/Arcsine

{{:Definition:Real Arcsine}}


Definition:Inverse Cosine/Real/Arccosine

{{:Graph of Arccosine Function|Graph}} From Shape of Cosine Function, we have that $\cos x$ is continuous and strictly decreasing on the interval $\closedint 0 \pi$. From Cosine of Multiple of Pi, $\cos \pi = -1$ and $\cos 0 = 1$. Therefore, let $g: \closedint 0 \pi \to \closedint {-1} 1$ be the restriction of $\cos x$ to $\closedint 0 \pi$. Thus from Inverse of Strictly Monotone Function, $\map g x$ admits an inverse function, which will be continuous and strictly decreasing on $\closedint {-1} 1$. This function is called '''arccosine of $x$''' and is written $\arccos x$. Thus: :The domain of $\arccos x$ is $\closedint {-1} 1$ :The image of $\arccos x$ is $\closedint 0 \pi$.


Definition:Inverse Tangent/Real/Arctangent

{{:Graph of Arctangent Function}} From Shape of Tangent Function, we have that $\tan x$ is continuous and strictly increasing on the interval $\openint {-\dfrac \pi 2} {\dfrac \pi 2}$. From the same source, we also have that: :$\tan x \to + \infty$ as $x \to \dfrac \pi 2 ^-$ :$\tan x \to - \infty$ as $x \to -\dfrac \pi 2 ^+$ Let $g: \openint {-\dfrac \pi 2} {\dfrac \pi 2} \to \R$ be the restriction of $\tan x$ to $\openint {-\dfrac \pi 2} {\dfrac \pi 2}$. Thus from Inverse of Strictly Monotone Function, $\map g x$ admits an inverse function, which will be continuous and strictly increasing on $\R$. This function is called '''arctangent''' of $x$ and is written $\arctan x$. Thus: :The domain of $\arctan x$ is $\R$ :The image of $\arctan x$ is $\openint {-\dfrac \pi 2} {\dfrac \pi 2}$.


Definition:Limit Point/Topology

Let $T = \left({S, \tau}\right)$ be a topological space. === Limit Point of Set === {{:Definition:Limit Point/Topology/Set}} === Limit Point of Point === The concept of a '''limit point''' can be sharpened to apply to individual points, as follows: {{:Definition:Limit Point/Topology/Point}} === Limit Point of Sequence === {{:Definition:Limit Point/Topology/Sequence}}


Definition:Vacuous Truth

Let $P \implies Q$ be a conditional statement. Suppose that $P$ is false. Then the statement $P \implies Q$ is a '''vacuous truth''', or '''is vacuously true'''. It is frequently encountered in the form: :$\forall x: \map P x \implies \map Q x$ when the propositional function $\map P x$ is false for all $x$. Such a statement is also a '''vacuous truth'''. For example, the statement: :'''All cats who are expert chess-players are also fluent in ancient Sanskrit''' is '''(vacuously) true''', because (as far as the author knows) there ''are'' no cats who are expert chess-players.


Definition:Metric Subspace

Let $\struct {A, d}$ be a metric space. Let $H \subseteq A$. Let $d_H: H \times H \to \R$ be the restriction $d \restriction_{H \times H}$ of $d$ to $H$. That is, let $\forall x, y \in H: \map {d_H} {x, y} = \map d {x, y}$. Then $d_H$ is the '''metric induced on $H$ by $d$''' or the '''subspace metric of $d$ (with respect to $H$)'''. The metric space $\struct {H, d_H}$ is called a '''metric subspace of $\struct {A, d}$'''.


Definition:Equivalent Metrics

Let $X$ be a set upon which there are two metrics $d_1$ and $d_2$. That is, $\struct {X, d_1}$ and $\struct {X, d_2}$ are two different metric spaces on the same set $X$. Let $\sequence {x_n}$ be a sequence in $X$. Let $n \to \infty$. Suppose that $x_n \to x$ in $\struct {X, d_1}$ {{iff}} $x_n \to x$ in $\struct {X, d_2}$. Then $d_1$ and $d_2$ are '''equivalent metrics'''.


Definition:Taxicab Metric

The '''taxicab metric''' on $A_{1'} \times A_{2'}$ is defined as: : $d_1 \left({x, y}\right) := d_{1'} \left({x_1, y_1}\right) + d_{2'} \left({x_2, y_2}\right)$ where $x = \left({x_1, x_2}\right), y = \left({y_1, y_2}\right) \in A_{1'} \times A_{2'}$.


Let $M_1 = \struct {A, d_1}$ and $M_2 = \struct {A, d_2}$ be metric spaces on the same underlying set $A$. === Definition 1 === {{:Definition:Topologically Equivalent Metrics/Definition 1}} === Definition 2 === {{:Definition:Topologically Equivalent Metrics/Definition 2}}


Definition:Lipschitz Equivalence

=== Metric Spaces === {{:Definition:Lipschitz Equivalence/Metric Spaces}} === Metrics === {{:Definition:Lipschitz Equivalence/Metrics/Definition 1}}


Definition:Inverse Cotangent/Real/Arccotangent

{{:Graph of Arccotangent Function}} From Shape of Cotangent Function, we have that $\cot x$ is continuous and strictly decreasing on the interval $\openint 0 \pi$. From the same source, we also have that: :$\cot x \to + \infty$ as $x \to 0^+$ :$\cot x \to - \infty$ as $x \to \pi^-$ Let $g: \openint 0 \pi \to \R$ be the restriction of $\cot x$ to $\openint 0 \pi$. Thus from Inverse of Strictly Monotone Function, $\map g x$ admits an inverse function, which will be continuous and strictly decreasing on $\R$. This function is called '''arccotangent''' of $x$ and is written $\arccot x$. Thus: :The domain of $\arccot x$ is $\R$ :The image of $\arccot x$ is $\openint 0 \pi$.


Definition:Natural Logarithm

=== Positive Real Numbers === {{:Definition:Natural Logarithm/Positive Real}} === Complex Numbers === {{:Definition:Natural Logarithm/Complex}}


Definition:Discrete Topology

Let $S \ne \O$ be a set. Let $\tau = \powerset S$ be the power set of $S$. That is, let $\tau$ be the set of all subsets of $S$: :$\tau := \set {H: H \subseteq S}$ Then $\tau$ is called '''the discrete topology on $S$''' and $\struct {S, \tau} = \struct {S, \powerset S}$ '''the discrete space on $S$''', or just '''a discrete space'''. === Finite Discrete Topology === {{:Definition:Discrete Topology/Finite}} === Infinite Discrete Topology === {{:Definition:Discrete Topology/Infinite}}


Definition:Indiscrete Topology

Let $S \ne \O$ be a set. Let $\tau = \set {S, \O}$. Then $\tau$ is called the '''indiscrete topology''' on $S$.


Definition:Fourier Transform

The '''Fourier transform''' of a Lebesgue integrable function $f: \R^N \to \C$ is the function $\map \FF f: \R^N \to \C$ given by: :$\displaystyle \map \FF {\map f \xi} := \int_{\R^N} \map f {\mathbf x} e^{-2 \pi i \mathbf x \cdot \xi} \rd \mathbf x$ for $\xi \in \R^N$. Here, the product $\mathbf x \cdot \xi$ in the exponential is the dot product of the vectors $\mathbf x$ and $\mathbf \xi$.


Definition:Metrizable Topology

Let $\struct {S, d}$ be a metric space. Let $\struct {S, \tau}$ be the topological space induced by $d$. Then for any topological space which is homeomorphic to such a $\struct {S, \tau}$, it and its topology are defined as '''metrizable'''.


Definition:Finer Topology

Let $S$ be a set. Let $\tau_1$ and $\tau_2$ be topologies on $S$. === Definition 1 === {{Definition:Finer Topology/Definition 1}} === Definition 2 === {{Definition:Finer Topology/Definition 2}} This can be expressed as: :$\tau_1 \ge \tau_2 := \tau_1 \supseteq \tau_2$ === Strictly Finer === {{:Definition:Finer Topology/Strictly Finer}}


Definition:Coarser Topology

Let $S$ be a set. Let $\tau_1$ and $\tau_2$ be topologies on $S$. Let $\tau_1 \subseteq \tau_2$. Then $\tau_1$ is said to be '''coarser''' than $\tau_2$. This can be expressed as: :$\tau_1 \le \tau_2 := \tau_1 \subseteq \tau_2$ === Strictly Coarser === {{:Definition:Coarser Topology/Strictly Coarser}}


Definition:Convolution (Measure Theory)

Let $\BB^n$ be the Borel $\sigma$-algebra on $\R^n$, and let $\lambda^n$ be Lebesgue measure on $\R^n$. === Convolution of Measurable Functions === {{:Definition:Convolution of Measurable Functions}} === Convolution of Measurable Function and Measure === {{:Definition:Convolution of Measurable Function and Measure}} === Convolution of Measures === {{:Definition:Convolution of Measures}}


Definition:Meromorphic Function

Let $U \subset \C$ be an open set. A '''meromorphic function on $U$''' is a holomorphic function on the complement of a discrete subset $P \subset U$ such that: :for all $p\in P$, $f$ has either a pole or removable singularity at $p$.


Definition:Uniform Convergence

=== Metric Space === {{:Definition:Uniform Convergence/Metric Space}} === Real Numbers === The above definition can be applied directly to the real numbers treated as a metric space: {{:Definition:Uniform Convergence/Real Numbers}} === Infinite Series === {{:Definition:Uniform Convergence/Infinite Series}}


Definition:Pointwise Convergence

Let $\sequence {f_n}$ be a sequence of real functions defined on $D \subseteq \R$. Suppose that: :$\displaystyle \forall x \in D: \lim_{n \mathop \to \infty} \map {f_n} x = \map f x$ That is: :$\forall x \in D: \forall \epsilon \in \R_{>0}: \exists N \in \R: \forall n > N: \size {\map {f_n} x - \map f x} < \epsilon$ Then '''$\sequence {f_n}$ converges to $f$ pointwise on $D$ as $n \to \infty$'''. (See the definition of convergence of a sequence).


Definition:Sub-Basis

=== Analytic Sub-Basis === {{:Definition:Sub-Basis/Analytic Sub-Basis}} === Synthetic Sub-Basis === {{:Definition:Sub-Basis/Synthetic Sub-Basis}}


Definition:Entire Function

A holomorphic self-map $f$ of the complex plane is called an '''entire function'''.


Definition:Topological Subspace

Let $T = \struct {S, \tau}$ be a topological space. Let $H \subseteq S$ be a non-empty subset of $S$. Define: :$\tau_H := \set {U \cap H: U \in \tau} \subseteq \powerset H$ where $\powerset H$ denotes the power set of $H$. Then the topological space $T_H = \struct {H, \tau_H}$ is called a '''(topological) subspace''' of $T$.


Definition:Closure (Topology)

Let $T = \struct {S, \tau}$ be a topological space. Let $H \subseteq S$. === Definition 1 === {{:Definition:Closure (Topology)/Definition 1}} === Definition 2 === {{:Definition:Closure (Topology)/Definition 2}} === Definition 3 === {{:Definition:Closure (Topology)/Definition 3}} === Definition 4 === {{:Definition:Closure (Topology)/Definition 4}} === Definition 5 === {{:Definition:Closure (Topology)/Definition 5}} === Definition 6 === {{:Definition:Closure (Topology)/Definition 6}}


Definition:Sigma-Algebra

{{:Definition:Sigma-Algebra/Definition 1}}


Definition:Borel Sigma-Algebra

=== Topological Spaces === Let $\struct {S, \tau}$ be a topological space The '''Borel sigma-algebra''' $\map {\mathcal B} {S, \tau}$ of $\struct {S, \tau}$ is the $\sigma$-algebra generated by $\tau$. That is, it is the $\sigma$-algebra generated by the set of open sets in $S$. === Metric Spaces === Let $\struct {X, \norm {\,\cdot\,} }$ be a metric space. The '''Borel sigma-algebra''' (or '''$\sigma$-algebra''') on $\struct {X, \norm {\,\cdot\,} }$ is the $\sigma$-algebra generated by the open sets in $\powerset X$. By the definition of a topology induced by a metric, this definition is a particular instance of the definition of a Borel $\sigma$-algebra on topological spaces. === Borel Set === {{:Definition:Borel Sigma-Algebra/Borel Set}}


Definition:Lebesgue Sigma-Algebra

Let $\tau$ be the Euclidean topology on $\R^n$. Let $\mathcal N$ be the collection of null sets in $\R^n$. The '''Lebesgue sigma-algebra''' on $\R^n$ is the sigma-algebra generated by the set $\tau \cup \mathcal N$. {{namedfor|Henri Léon Lebesgue}} {{SUBPAGENAME}} {{SUBPAGENAME}} 74sbdzjrem7wc3tly1t0hqfqiuwfr5y


Definition:Null Set

Let $\left({X, \Sigma, \mu}\right)$ be a measure space. A set $N \in \Sigma$ is called a '''($\mu$-)null set''' {{iff}} $\mu \left({N}\right) = 0$. === Family of Null Sets === The family of '''$\mu$-null sets''', $\left\{{N \in \Sigma: \mu \left({N}\right) = 0}\right\}$, is denoted $\mathcal N_{\mu}$.


Definition:Complex Modulus

Let $z = a + i b$ be a complex number, where $a, b \in \R$. Then the '''(complex) modulus of $z$''' is written $\cmod z$ and is defined as the square root of the sum of the squares of the real and imaginary parts: :$\cmod z := \sqrt {a^2 + b^2}$


Definition:Real-Valued Function

Let $f: S \to T$ be a function. Let $S_1 \subseteq S$ such that $\map f {S_1} \subseteq \R$. Then $f$ is said to be '''real-valued on $S_1$'''. That is, $f$ is defined as '''real-valued on $S_1$''' {{iff}} the image of $S_1$ under $f$ lies entirely within the set of real numbers $\R$. A '''real-valued function''' is a function $f: S \to \R$ whose codomain is the set of real numbers $\R$. That is, $f$ is '''real-valued''' {{iff}} it is '''real-valued''' over its entire domain.


Definition:Complex-Valued Function

Let $f: S \to T$ be a function. Let $S_1 \subseteq S$ such that $f \left({S_1}\right) \subseteq \C$. Then $f$ is defined as '''complex-valued on $S_1$'''. That is, $f$ is defined as '''complex-valued on $S_1$''' if the image of $S_1$ under $f$ lies entirely within the set of complex numbers $\C$. A '''complex-valued function''' is a function $f: S \to \C$ whose codomain is the set of complex numbers $\C$. That is $f$ is complex-valued iff it is complex-valued over its entire domain.


Definition:Usual Metric

Let $X$ be one of the standard number fields $\Q$, $\R$, $\C$. Let $X^n$ be a cartesian space on $X$. The '''usual metric''' on $X^n$ is the Euclidean metric on $X^n$: === Real Numbers === {{:Definition:Euclidean Metric/Real Vector Space}} === Rational Numbers === {{:Definition:Euclidean Metric/Rational Space}} === Complex Plane === {{:Definition:Euclidean Metric/Complex Plane}} Category:Definitions/Examples of Metric Spaces Category:Definitions/Euclidean Metric qy3nckivsilneywbe9yipzdhkmtyufq


Definition:Absolute Continuity

Let $I \subseteq \R$ be a real interval. A real function $f: I \to \R$ is said to be '''absolutely continuous''' if it satisfies the following property: :For every $\epsilon > 0$ there exists $\delta > 0$ such that the following property holds: ::For every finite set of disjoint closed real intervals $\closedint {a_1} {b_1}, \dotsc, \closedint {a_n} {b_n} \subseteq I$ such that: :::$\displaystyle \sum_{i \mathop = 1}^n \size {b_i - a_i} < \delta$ ::it holds that: :::$\displaystyle \sum_{i \mathop = 1}^n \size {\map f {b_i} - \map f {a_i} } < \epsilon$


Definition:Uniform Continuity

=== Metric Spaces === {{:Definition:Uniform Continuity/Metric Space}} === Real Numbers === {{:Definition:Uniform Continuity/Real Numbers}}


Definition:Inner Product

Let $\C$ be the field of complex numbers. Let $\F$ be a subfield of $\C$. Let $V$ be a vector space over $\F$. An '''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= Bilinearity         | q = \forall x, y \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} }} {{axiom | n = 4         | lc= Positiveness         | q = \forall x \in V         | m = \quad \innerprod x x = 0 \implies x = \mathbf 0_V }} {{end-axiom}} That is, an inner product is a semi-inner product with the additional condition $(4)$. 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'         | lc= Symmetry         | q = \forall x, y \in V         | m = \innerprod x y = \innerprod y x }} {{end-axiom}} === Inner Product Space === {{:Definition:Inner Product Space}}


Definition:Inner Product Space

An '''inner product space''' is a vector space together with an associated inner product.


Definition:Isolated Point (Topology)

Let $T = \struct {S, \tau}$ be a topological space. === Isolated Point of Subset === {{:Definition:Isolated Point (Topology)/Subset}} === Isolated Point of Space === When $H = S$ the definition applies to the entire topological space $T = \struct {S, \tau}$: {{:Definition:Isolated Point (Topology)/Space}}


Definition:Everywhere Dense

Let $T = \struct {S, \tau}$ be a topological space. Let $H \subseteq S$ be a subset. === Definition 1 === {{:Definition:Everywhere Dense/Definition 1}} === Definition 2 === {{:Definition:Everywhere Dense/Definition 2}}


Definition:Sequentially Compact Space

Let $T = \left({S, \tau}\right)$ be a topological space. Let $H \subseteq S$. Then $H$ is '''sequentially compact in $T$''' {{iff}} every infinite sequence in $H$ has a subsequence which converges to a point in $S$.


Definition:Totally Bounded Metric Space

=== Definition 1 === {{:Definition:Totally Bounded Metric Space/Definition 1}} === Definition 2 === {{:Definition:Totally Bounded Metric Space/Definition 2}}


Definition:Interior (Topology)

Let $T = \struct {S, \tau}$ be a topological space. Let $H \subseteq S$. === Definition 1 === {{:Definition:Interior (Topology)/Definition 1}} === Definition 2 === {{:Definition:Interior (Topology)/Definition 2}} === Definition 3 === {{:Definition:Interior (Topology)/Definition 3}}


Definition:Nowhere Dense

Let $T = \struct {S, \tau}$ be a topological space. Let $H \subseteq S$. === Definition 1 === {{:Definition:Nowhere Dense/Definition 1}} === Definition 2 === {{:Definition:Nowhere Dense/Definition 2}}


Definition:Quotient Topology

Let $T = \left({S, \tau}\right)$ be a topological space. Let $\mathcal R \subseteq S \times S$ be an equivalence relation on $S$. Let $q_\mathcal R: S \to S / \mathcal R$ be the quotient mapping induced by $\mathcal R$. === Definition 1 === {{:Definition:Quotient Topology/Definition 1}} === Definition 2 === {{:Definition:Quotient Topology/Definition 2}}


Definition:Topological Property

Let $P$ be a property whose domain is the set of all topological spaces. Suppose that whenever $\map P T$ holds, then so does $\map P {T'}$, where $T$ and $T'$ are topological spaces which are homeomorphic. Then $P$ is known as a '''topological property'''. Loosely, a topological property is one which is preserved under homeomorphism.


Definition:Cover of Set

Let $S$ be a set. A '''cover for $S$''' is a set of sets $\CC$ such that: :$\displaystyle S \subseteq \bigcup \CC$ where $\bigcup \CC$ denotes the union of $\CC$.


Definition:Subcover

Let $S$ be a set. Let $\mathcal U$ be a cover for $S$. A '''subcover of $\mathcal U$ for $S$''' is a set $\mathcal V \subseteq \mathcal U$ such that $\mathcal V$ is also a cover for $S$. === Finite Subcover === {{:Definition:Subcover/Finite}} === Countable Subcover === {{:Definition:Subcover/Countable}}


Definition:Relatively Compact Subspace

Let $T = \left({S, \tau}\right)$ be a topological space. Let $T_H = \left({H, \tau_H}\right)$ be a subspace of $T$. Let $\operatorname{cl} \left({H}\right)$ be the closure of $H$ in $T$. Then $T_H$ is '''relatively compact in $T$''' {{iff}} $\operatorname{cl} \left({H}\right)$ is compact.


Definition:Inscribe

Let a geometric figure $A$ be constructed in the interior of another geometric figure $B$ such that: :$(1): \quad$ $A$ and $B$ have points in common :$(2): \quad$ No part of $A$ is outside $B$. Then $A$ is '''inscribed''' inside $B$. === Circle in Polygon === {{:Definition:Inscribe/Circle in Polygon}} === Polygon in Circle === {{:Definition:Inscribe/Polygon in Circle}} === Polygon in Polygon === {{:Definition:Inscribe/Polygon in Polygon}}


Definition:Circumscribe

=== Circle around Polygon === {{:Definition:Circumscribe/Circle around Polygon}} === Polygon around Circle === {{:Definition:Circumscribe/Polygon around Circle}} === Polygon around Polygon === {{:Definition:Circumscribe/Polygon around Polygon}}


Definition:Fixed Point

Let $f: S \to T$ be a mapping. Then a '''fixed point''' (or '''fixed element''') '''of $S$ under $f$''' is an $x \in S$ such that $\map f x = x$.


Definition:Coefficient

A '''coefficient''' is a constant which is used in a particular context to be multiplied by a variable that is under consideration. === Binomial Coefficient === {{:Definition:Binomial Coefficient/Integers/Definition 3}} === Polynomial Coefficient === For the usage of this term in the context of polynomial theory: {{:Definition:Polynomial Coefficient}}


Definition:Lemma

A '''lemma''' is a statement which is proven during the course of reaching the proof of a theorem. Logically there is no qualitative difference between a '''lemma''' and a theorem. They are both statements whose value is either true or false. However, a '''lemma''' is seen more as a stepping-stone than a theorem in itself (and frequently takes a lot more work to prove than the theorem to which it leads). Some lemmas are famous enough to be named after the mathematician who proved them (for example: Abel's Lemma and Urysohn's Lemma), but they are still categorised as second-class citizens in the aristocracy of mathematics. :''Always the lemma, never the theorem.''


Definition:Euclidean Geometry

Euclidean geometry is the branch of geometry in which the parallel postulate applies. An assumption which is currently under question is whether or not ordinary space is itself Euclidean. '''Euclidean geometry''' adheres to Euclid's postulates.


Definition:Uniform Equivalence

=== Metric Spaces === {{:Definition:Uniform Equivalence/Metric Spaces}} === Metrics === {{:Definition:Uniform Equivalence/Metrics}} Category:Definitions/Metric Spaces rbfy2keax03l4o8uavh8o5ccqlogn3b


Definition:A Priori

'''A priori''' knowledge is the sort of knowledge which comes from reason alone. That is, it does not require the exercise of experience to know it. For example: :''If Fred Bloggs has committed a crime, then he is guilty'' as opposed to: :''Fred Bloggs has committed the crime of usury.''


Definition:Clopen Set

Let $T = \struct {S, \tau}$ be a topological space. Let $X \subseteq S$ such that $X$ is both open in $T$ and closed in $T$. Then $X$ is described as '''clopen'''.


Definition:Connected (Topology)

=== Topological Space === {{Definition:Connected (Topology)/Topological Space}} === Set of Topological Space === {{Definition:Connected (Topology)/Set}} === Points in Topological Space === {{:Definition:Connected (Topology)/Points}}


Definition:Position

'''Position''' is an abstract geometrical concept that defines '''where something is'''. It is only possible to define the '''position''' of an object in relation to the '''position''' of another object. Even when one tries to consider an '''''absolute'' position''', one has to describe it in relation to a frame of reference, which itself needs to have a '''position''' (which ultimately is itself arbitrarily defined).


Definition:Angle Bisector

:300px Let $\angle ABC$ be an angle. The '''angle bisector''' of $\angle ABC$ is the straight line which bisects $\angle ABC$. In the above diagram, $BD$ is the '''angle bisector''' of $\angle ABC$. Thus $\angle ABD \cong \angle DBC$ and $\angle ABD + \angle DBC = \angle ABC$.


Definition:Darboux Function

Let $S \subseteq \R$. Let $f: S \to \R$ be a real function. Then $f$ is '''Darboux''' {{iff}}, given any $a, b \in S$ and $y \in \R$ such that $a < b$ and $y$ is between $\map f a$ and $\map f b$, there exists a $c \in S$ such that $a \le c \le b$ and $\map f c = y$. That is, for every '''intermediate value''' between $\map f a$ and $\map f b$, that value is the image of some '''intermediate value''' between $a$ and $b$.


Definition:Path (Topology)

Let $T = \struct {S, \tau}$ be a topological space. Let $I \subset \R$ be the closed real interval $\closedint a b$. A '''path in $T$''' is a continuous mapping $\gamma: I \to S$. The mapping $\gamma$ can be referred as: :a '''path (in $T$) joining $\map \gamma a$ and $\map \gamma b$''' or: :a '''path (in $T$) from $\map \gamma a$ to $\map \gamma b$'''. It is common to refer to a point $z = \map \gamma t$ as a '''point on the path $\gamma$''', even though $z$ is in fact on the image of $\gamma$. === Initial Point === {{:Definition:Path (Topology)/Initial Point}} === Final Point === {{:Definition:Path (Topology)/Final Point}} === Endpoint === {{:Definition:Path (Topology)/Endpoint}}


Definition:Congruence (Geometry)

In the field of Euclidean geometry, two geometric figures are '''congruent''' if they are, informally speaking, both "the same size and shape". That is, one figure can be overlaid on the other figure with a series of rotations, translations, and reflections. Specifically: : all corresponding angles of the '''congruent''' figures must have the same measurement : all corresponding sides of the '''congruent''' figures must be be the same length.


Definition:WLOG

Suppose there are several cases which need to be investigated. If the same argument can be used to dispose of two or more of these cases, then it is acceptable in a proof to pick just one of these cases, and announce this fact with the words: '''Without loss of generality, ...''', or just '''WLOG'''.


Definition:Sector

A '''sector''' of a circle is the area bounded by two radii and an arc. {{EuclidSaid}} :''{{:Definition:Euclid's Definitions - Book III/10 - Sector of Circle}}'' {{EuclidDefRefNocat|III|10|Sector of Circle}} In the diagram below, $BAC$ is a '''sector'''.  :300px In fact there are two '''sectors''', together making up the whole of the circle. When describing a sector by means of its three defining points (as in $BAC$ above), the convention is to report them in the following order: :$(1):\quad$ The point on the circumference at the end of the clockwise radius :$(2):\quad$ The point at the center of the circle :$(3):\quad$ The point on the circumference at the end of the anticlockwise radius Thus in the '''sector''' above, $BAC$ describes the '''sector''' indicated by $\theta$, while the '''sector''' comprising the rest of the circle is described by $CAB$. === Angle of Sector === {{:Definition:Sector/Angle}}


Definition:Component (Topology)

Let $T = \left({S, \tau}\right)$ be a topological space. Let the relation $\sim $ be defined on $T$ as follows: :$x \sim y$ {{iff}} $x$ and $y$ are connected in $T$. That is, {{iff}} there exists a connected set of $T$ that contains both $x$ and $y$. === Definition 1 === {{:Definition:Component (Topology)/Definition 1}} === Definition 2 === {{:Definition:Component (Topology)/Definition 2}} === Definition 3 === {{:Definition:Component (Topology)/Definition 3}}


Definition:Totally Disconnected Space

A topological space $T = \struct {S, \tau}$ is a '''totally disconnected space''' {{iff}} all components of $T$ are singletons.


Definition:Extremal Length

Let $\Gamma$ be a set of rectifiable curves in the complex plane $\C$. We consider conformal metrics of the form $\map \rho z \cmod {\d z}$, where: :$\rho: \C \to \hointr 0 \to$ is Borel measurable and: :the area $\displaystyle \map A \rho := \iint \rho^2 \paren {x + i y} \rd x \rd y$ is finite and positive. Every $\gamma \in \Gamma$ has a distance function with respect to such a metric, defined by: :$\displaystyle \map L {\gamma, \rho} := \int_\gamma \map \rho z \size {\d z}$ We define: :$\displaystyle \map L {\Gamma, \rho} := \inf_{\gamma \mathop \in \Gamma} \map L {\gamma, \rho}$ and: :$\displaystyle \map \lambda \Gamma := \sup_\rho \frac {\map L {\Gamma, \rho}^2} {\map A \rho}$ The quantity $\map \lambda \Gamma$ is called the '''extremal length''' of the curve family $\Gamma$. Its reciprocal: :$\mod \Gamma := \dfrac 1 {\map \lambda \Gamma}$ is called the '''modulus''' of $\Gamma$.