hcju/proofwikips huggingface dataset:
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}}  e3nadl00pqmmtky0opygemoaw128urn
Definition:Nullity
=== Linear Transformation === {{:Definition:Nullity/Linear Transformation}} === Matrix === {{:Definition:Nullity/Matrix}}
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
 :''{{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
 :''{{: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  :''{{: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''' : :$\exists \mathbf B \in \struct {\map {\MM_R} n, +, \times}: \mathbf A \mathbf B = \mathbf I_n = \mathbf B \mathbf A$ 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'''  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  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''' : :$(1): \quad J \subsetneq R$ :$(2): \quad$ There is no ideal $K$ of $R$ such that $J \subsetneq K \subsetneq R$. That is,  $J$ 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}}
Definition:Transcendental (Abstract Algebra)
=== 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}}