hcju/proofwikips huggingface dataset:
Definition:Calendar/Gregorian
The '''Gregorian calendar''' is the calendar which was introduced by [https://en.wikipedia.org/wiki/Pope_Gregory_XIII Pope Gregory XIII] in $1582 \, \text{CE}$. Over the course of the next few centuries it was gradually adopted by the various subcultures of [https://en.wikipedia.org/wiki/Western_culture Western civilization]. It was derived as a refinement of the Julian calendar to correct for the discrepancy between the Julian calendar and the tropical year. The years themselves are assigned the same numbers as their Julian counterparts. An evolving modern convention is to refer to $\text{A.D.}$ and $\text{B.C.}$ as $\text{CE}$ and $\text{BCE}$, for '''common era''' and '''before common era''' respectively. Like the Julian calendar, it divides the year into either $365$ days or $366$ days, according to the year number. The months, similarly, are kept the same as for the Julian calendar. The only difference is in the determination of which years are classified as leap years. Let $y$ be the year number. Then $y$ is a leap year  :$y$ is divisible by $400$ or: :$y$ is divisible by $4$ and $y$ is not divisible by $100$. Thus, for example: :$2016$ was a leap year, because $2016$ is divisible by $4$ and not divisible by $100$. :$1900$ was '''not''' a leap year, because $1900$ is divisible by $100$ but not $400$. :$2000$ was a leap year, as $2000$ is divisible by $400$.
Definition:Calendar/Gregorian/Year
A '''Gregorian year''', also known as the '''civil year''', is the length of a year as defined using the '''Gregorian calendar'''. :$1$ '''Gregorian year''' $= \begin{cases} 366 \, \text {days} & : 400 \divides y \\ 365 \, \text {days} & : 400 \nmid  y \text { and } 100 \divides y \\ 366 \, \text {days} & : 100 \nmid  y \text { and } 4 \divides y \\ 365 \, \text {days} & : 4 \nmid y \end{cases}$ where: :$y$ denotes the number of the year :$4 \divides y$ denotes that $y$ is divisible by $4$ :$4 \nmid y$ denotes that $y$ is not divisible by $4$.
Definition:Calendar/Muslim
The '''Muslim calendar''' is a calendar whose period is based on $12$ full cycles of phases of the moon. Each Muslim year lasts either $354$ or $355$ days, depending on precisely when the relevant new moons happen The years are numbered from the year when [https://en.wikipedia.org/wiki/Muhammad Muhammad] and his followers migrated from [https://en.wikipedia.org/wiki/Mecca Mecca] to [https://en.wikipedia.org/wiki/Medina Medina] in $622 \, \text{A.D}$ by the Julian calendar. Years are denoted with the suffix $\text{A.H.}$, meaning '''Anno Hegirae''', literally '''in the year of the Hejira'''. The start and end of the Muslim year are $10$ to $12$ days earlier every civil year. The $354$ or $355$ days in the Muslim year are divided into $12$ approximately equal sections called months, each either $29$ or $30$ days, depending on the phases of the moon. They are assigned both names and index numbers: {{begin-eqn}} {{eqn | l = 1:       | o =        | c = Muḥarram }} {{eqn | l = 2:       | o =        | c = Ṣafar }} {{eqn | l = 3:       | o =        | c = Rabī‘ al-awwal }} {{eqn | l = 4:       | o =        | c = Rabī‘ ath-thānī }} {{eqn | l = 5:       | o =        | c = Jumādá al-ūlá }} {{eqn | l = 6:       | o =        | c = Jumādá al-ākhirah }} {{eqn | l = 7:       | o =        | c = Rajab }} {{eqn | l = 8:       | o =        | c = Sha‘bān }} {{eqn | l = 9:       | o =        | c = Ramaḍān }} {{eqn | l = 10:       | o =        | c = Shawwāl }} {{eqn | l = 11:       | o =        | c = Dhū al-Qa‘dah }} {{eqn | l = 12:       | o =        | c = Dhū al-Ḥijjah }} {{end-eqn}}
Definition:Calendar/Muslim/Year
A '''Muslim year''', also known as the '''Islamic year''' or '''Hijra year''', is the length of a year as defined using the '''Muslim calendar'''. :$1$ '''Muslim year''' $= \begin{cases} 355 \, \text {days} & : \text {Leap year} \\ 354 \, \text {days} & : \text {Non-leap year} \end{cases}$
Definition:Calendar/Jewish/Year
A '''Jewish year''', also known as the '''Hebrew year''', is the length of a year as defined using the '''Jewish calendar'''. Its length is either $354$ or $355$ days, or longer, if an extra month has been inserted.
Definition:Calendar/Jewish
The '''Jewish calendar''' is a calendar based on both the phases of the moon and on the orbital period of earth around the sun. The whose period is based on $12$ full cycles of phases of the moon. The basic Jewish year is $354$ or $355$ days of $12$ months, each alternately $29$ and $30$ days long. When the error between the Jewish year and tropical year becomes approximately $30$ days, a $13$th month is inerted into the year.
Definition:Lucas-Carmichael Number
Let $n$ have the property that: :$p \divides n \implies \paren {p + 1} \divides \paren {n + 1}$ where: :$p$ is prime :$p \divides n$ denotes that $p$ is a divisor of $n$. Then $n$ is classified as a '''Lucas-Carmichael number'''.
Definition:Complement of Subgroup
Let $G$ be a group with identity $e$. Let $H$ and $K$ be subgroups. Let $HK$ be their subset product and $H \cap K$ their intersection. === Definition 1 === {{Definition:Complement of Subgroup/Definition 1}} === Definition 2 === {{Definition:Complement of Subgroup/Definition 2}}
Definition:Inner Semidirect Product
Let $G$ be a group. Let $H$ be a subgroup of $G$. Let $N$ be a normal subgroup of $G$. Let $H$ and $N$ be complementary. Then $G$ is the '''inner semidirect product''' of $N$ and $H$. This is denoted $G = N \rtimes H$ or $G = H \ltimes N$.
Definition:Semidirect Product
Let $H$ and $N$ be groups. Let $\Aut N$ denote the automorphism group of $N$. Let $\phi: H \to \Aut N$ be a group homomorphism, that is, let $H$ act on $N$. The '''(outer) semidirect product of $N$ and $H$ with respect to $\phi$''' is the cartesian product $N \times H$ with the group operation defined as: :$\tuple {n_1, h_1} \circ \tuple {n_2, h_2} = \tuple {n_1 \cdot \map \phi {h_1} \paren {n_2}, h_1 \cdot h_2}$ {{explain|What specifically is "$\cdot$" here? This is an instance where it is important (in my belief) to be completely explicit as to what the operations are within the groups $H$ and $N$. It may also be clearer to refer to $G_1$ and $G_2$ because $N$ and $H$ have connotations of "normal subgroup" and "subgroup" respectively, and it is highly desirable to be consistent with other pages.}} It is denoted $N \rtimes_\phi H$.
Definition:Continued Fraction/Expansion of Real Number
=== Irrational Number === Let $x$ be an irrational number. The '''continued fraction expansion of $x$''' is the simple continued fraction $\paren {\floor {\alpha_n} }_{n \ge 0}$ where $\alpha_n$ is recursively defined as: :$\alpha_n = \displaystyle \begin{cases} x & : n = 0 \\ \dfrac 1 {\fractpart {\alpha_{n - 1} } } & : n \ge 1 \end{cases}$ where: :$\floor {\, \cdot \,}$ is the floor function :$\fractpart {\, \cdot \,}$ is the fractional part function. === Rational Number === Let $x$ be a rational number. The '''continued fraction expansion of $x$''' is found using the Euclidean Algorithm. {{explain|how}}
Definition:Derivation on Ring
Let $R$ be a ring. A '''derivation''' on $R$ is a group homomorphism $D$ of the additive group of $R$ which satsfies the Leibniz law: :$\forall a, b \in R: \map D {a b} = \map D a b + a \map D b$
Definition:Differential Ring
A '''differential ring''' is a pair $\struct {R, D}$ where: :$R$ is a ring :$D$ is a derivation on $R$.
Definition:Differential Field
A '''differential field''' is a differential ring which is a field.
Definition:Galois Extension/Finite/Definition 3
$L / K$ is a '''Galois extension'''  the order of the automorphism group $\Aut {L / K}$ equals the degree $\index L K$: :$\order {\Aut {L / K} } = \index L K$
Definition:Field Extension/Degree/Finite
$E / F$ is a '''finite field extension'''  its degree $\index E F$ is finite.
Definition:Field Extension/Degree/Infinite
$E / F$ is an '''infinite field extension'''  its degree $\index E F$ is not finite.
Definition:Field Norm/Definition 1
By Vector Space on Field Extension is Vector Space, $L$ is naturally a finite dimensional vector space over $K$. Let $\theta_\alpha$ be the linear operator: :$\theta_\alpha: L \to L: \beta \mapsto \alpha \beta$ The '''field norm''' $\map {N_{L/K} }\alpha$ of $\alpha$ is the determinant of $\theta_\alpha$.
Definition:Field Norm/Definition 2
Let $L / K$ be Galois. By Finite Field Extension has Finite Galois Group, the Galois group $\operatorname{Gal} \left({L / K}\right)$ is finite. The '''field norm''' $N_{L / K} \left({\alpha}\right)$ of $\alpha$ is $\displaystyle \prod_{\sigma \mathop \in \operatorname{Gal} \left({L / K}\right)} \sigma \left({\alpha}\right)$.
Definition:Nilradical of Ring/Definition 1
The '''nilradical''' of $A$ is the subset consisting of all nilpotent elements of $A$.
Definition:Nilradical of Ring/Definition 2
Let $\Spec A$ denote the prime spectrum of $A$. The '''nilradical''' of $A$ is: :$\displaystyle \Nil A = \bigcap_{\mathfrak p \mathop \in \Spec A} \mathfrak p$ That is, it is the intersection of all prime ideals of $A$.
Definition:Permutation Representation
Let $G$ be a group. Let $X$ be a set. Let $\struct {\map \Gamma X, \circ}$ be the symmetric group on $X$. A '''permutation representation''' of $G$ is a group homomorphism from $G$ to $\struct {\map \Gamma X, \circ}$. === Associated to Group Action === {{:Definition:Permutation Representation/Group Action}}
Definition:Faithful Group Action/Definition 1
$\phi$ is '''faithful'''  $e$ is the only element if $G$ which acts trivially: :$\forall g \in G: \paren {\forall x \in X: g * x = x \implies g = e}$
Definition:Faithful Group Action/Definition 2
$\phi$ is '''faithful'''  its permutation representation is injective.
Definition:Free Group Action
Let $G$ be a group with identity $e$ acting on a set $X$. The group action is '''free''' : :$\forall g \in G: \forall x \in X : g * x = x \implies g = e$
Definition:Kernel of Group Action
Let $G$ be a group with identity $e$. Let $X$ be a set. Let $* : G\times X\to X$ be a group action. === Definition 1 === {{Definition:Kernel of Group Action/Definition 1}} === Definition 2 === {{Definition:Kernel of Group Action/Definition 2}}
Definition:Kernel of Group Action/Definition 1
The '''kernel of the group action''' is the set: :$G_0 = \set {g \in G: \forall x \in X: g * x = x}$
Definition:Kernel of Group Action/Definition 2
The '''kernel of the group action''' is the kernel of its permutation representation.
Definition:Permutation Representation/Group Action
Let $\phi: G \times X \to X$ be a group action. Define for $g \in G$ the mapping $\phi_g : X \to X$ by: :$\map {\phi_g} x = \map \phi {g, x}$ The '''permutation representation of $G$ associated to the group action''' is the group homomorphism $G \to \struct {\map \Gamma X, \circ}$ which sends $g$ to $\phi_g$.
Definition:Group Action/Permutation Representation
Let $\struct {\map \Gamma X, \circ}$ be the symmetric group on $X$. Let $\rho: G \to \struct {\map \Gamma X, \circ}$ be a permutation representation. The '''group action of $G$ associated to the permutation representation $\rho$''' is the group action $\phi: G \times X \to X$ defined by: :$\map \phi {g, x} = \map {\rho_g} x$ where $\rho_g : X \to X$ is the permutation representation associated to $\rho$ for $g \in G$ by $\map {\rho_g} x = \map \phi {g, x}$.
Definition:Field of Quotients/Definition 4
A '''field of quotients''' of $D$ is a pair $\struct {F, \iota}$ which is its total ring of fractions, that is, the localization of $D$ at the nonzero elements $D_{\ne 0}$.
Definition:Fixed Field
Let $F$ be a field. Let $G \le \Aut F$ be a subgroup of the automorphism group of $F$. The '''fixed field''' of $G$ is the set: :$\Fix G = \set {f \in F : \forall \sigma \in G : \map \sigma f = f}$
Definition:Group Action Induced on Subgroup
Let $G$ be a group. Let $X$ be a set. Let $\phi : G \times X \to X$ be a group action. Let $H \le G$ be a subgroup. The '''group action induced on $H$''' is the restriction of $\phi$ to $H \times X$. Equivalently, the '''group action induced on $H$''' is the group action associated to the permutation representation: :$\rho \circ \iota : H \to \struct {\map \Gamma X, \circ}$ where: :$\iota : H \to G$ is the inclusion homomorphism :$\rho$ is the  permutation representation of $\phi$ :$\struct {\map \Gamma X, \circ}$ is the symmetric group on $X$.
Definition:Stable Under Group Action
Let $G$ be a group. Let $X$ be a set. Let $\phi : G \times X \to X$ be a group action. Let $S \subset X$ be a subset of $X$. Then $S$ is '''stable (under $\phi$)''' : :$\map \phi {G \times S} \subset S$
Definition:Group Action Induced on Stable Subset
Let $G$ be a group. Let $X$ be a set. Let $\phi : G \times X \to X$ be a group action. Let $S \subset X$ be a stable subset. The '''group action induced on $S$''' is the restriction of $\phi$ to $G \times S$.
Definition:Complete Quotient
Let $x$ be an irrational number. The '''sequence of complete quotients''' is defined recursively by: :$\alpha_0 = x$ :$\alpha_{n+1} = \displaystyle\frac1{\{\alpha_n\}}$ where $\{\cdot\}$ denotes fractional part.
Definition:Parametric Equation
Let $\map \RR {x_1, x_2, \ldots, x_n}$ be a relation on the variables $x_1, x_2, \ldots, x_n$. Let the truth set of $\RR$ be definable as: :$\forall k \in \N: 1 \le k \le n: x_k = \map {\phi_k} t$ where: :$t$ is a variable whose domain is to be defined :each of $\phi_k$ is a mapping whose domain is the domain of $t$ and whose codomain is the domain of $x_k$. Then each of: :$x_k = \map {\phi_k} t$ is a '''parametric equation''' where $t$ is the '''parameter'''. The set: :$\set {\phi_k: 1 \le k \le n}$ is a '''set of parametric equations specifying $\RR$'''. === $2$ Dimensions === {{:Definition:Parametric Equation/2 Dimensions}}
Definition:Basis Expansion/Recurrence/Period
The '''period of recurrence''' is the number of digits in the recurring part after which it repeats itself.
Definition:Force/Dimension
The dimension of measurement of '''force''' is $\mathsf {M L T}^{-2}$. This arises from Newton's Second Law of Motion and its definition as a mass (of dimension $\mathsf M$) multiplied by an acceleration (of dimension $\mathsf {L T}^{-2}$).
Definition:Force/Unit
The units of measurement of '''force''' are as follows: === Newton === {{:Definition:Newton (Unit)}} === Dyne === {{:Definition:Dyne}} Thus we see: :$1 \ \mathrm N = 10^5 \ \mathrm {dyn}$
Definition:Electromotive Force/Dimension
The dimension of measurement of '''electromotive force''' is $\mathsf {M L}^2 \mathsf T^{-3} \mathsf I^{-1}$. This derives from its definition as: :$\dfrac {\text{Work} } {\text {Charge} }$
Definition:Electromotive Force/Unit
The unit of measurement of '''electromotive force'''  is the volt $\mathrm V$: :$1 \ \mathrm V = 1 \ \mathrm J \ \mathrm C^{-1}$ that is, $1$ joule per coulomb.
Definition:Electromotive Force
'''Electromotive force''' is a quantity that measures the source of potential energy in an electrical circuit. It is defined as the amount of work per unit electric charge.
Definition:Inductance
'''Inductance''' is the property of an electrical conductor by which a change in electric current through it induces an electromotive force in both the conductor itself and also in any nearby conductors.
Definition:Inductance/Dimension
The dimension of measurement of '''inductance''' is: :$\mathsf {M L}^2 \mathsf T^{-2} \mathsf I^{-2}$ This derives from its definition as: :$v = L \dfrac {\mathrm d I} {\mathrm d t}$
Definition:Dynamical System
A '''dynamical system''' is a non-linear system in which a function describes the time dependence of a point in a geometrical space.
Definition:Resistance
'''Resistance''' is a measure of how difficult it is to pass an electric current through a conductor. It is defined as the electromotive force needed to push a unit current through that conductor.
Definition:Resistance/Dimension
The dimension of measurement of '''resistance''' is: :$\mathsf {M L}^2 \mathsf T^{−3} \mathsf I^{−2}$ This derives from its definition as: :$R = \dfrac {\text {EMF} } {\text {current} }$
Definition:Resistance/Unit
The unit of measurement of '''resistance'''  is the ohm $\Omega$: :$1 \ \Omega = 1 \ \mathrm{V} \ \mathrm{A}^{-1}$ that is, $1$ volt per ampere.
Definition:Electric Current/Dimension
'''Electric current''' is one of the fundamental dimensions of physics. In dimensional analysis it is assigned the symbol $\mathsf I$.
Definition:Volume/Dimension
'''Volume''' is a dimension of measurement of physics. The dimension of measurement of '''volume''' is: :$L^3$ length cubed.
Definition:Volume/Unit
The SI unit of '''volume''' is the cubic metre: : $\mathrm m^3$ The CGS unit of '''volume''' is the cubic centimetre: :$\mathrm {cm}^3$ or, less formally: :$\mathrm {cc}$ Thus: :$1 \ \mathrm m^3 = 10^6 \ \mathrm {cm}^3 = 1 \,000 \,000 \ \mathrm {cc}$
Definition:Lucas Prime
A '''Lucas prime''' is a Lucas number which happens to be prime.
Definition:Series (Electronics)
Two or more components in an electrical circuit are connected in '''series'''  the current through them flows along a single path through the circuit. {{finish|Needs to be formalised, with the concept of an electrical circuit being mapped by means of a graph. This will need plenty of work.}}
Definition:Anticlockwise
'''Anticlockwise''' is this direction: :400px
Definition:Clockwise
'''Clockwise''' is this direction: :400px
Definition:Rocket
A '''rocket''' is a device whose means of propulsion works by throwing material out of one end at high speed in order to take advantage of the Principle of Conservation of Momentum to move in the opposite direction.
Definition:Cuboid
A '''cuboid''' is a parallelepiped whose faces are all rectangular. :400px
Definition:Moon
'''The moon''' is the single natural satellite of Earth.
Definition:Integration With Respect To
Let $f$ be a function which is integrable on a domain $D$. Let $F$ be the primitive of $f$: :$\displaystyle F = \int \map f x \rd x$ The operation of evaluating the primitive of $f$ is referred to as '''integration with respect to $x$.'''
Definition:Spherical Triangle
A '''spherical triangle''' is a geometric figure described on the surface of a sphere which is bounded by the arcs of three great circles which intersect in $3$ vertices: :500px
Definition:Spherical Triangle/Side
The '''sides''' of a '''spherical triangle''' $T$ are the arcs of the $3$ great circles which form the boundaries of $T$ between the points at which they intersect.
Definition:Linear Momentum/Dimension
The dimension of measurement of '''linear momentum''' is $\mathsf {M L T}^{-1}$.
Definition:Frequency (Physics)
Let $\map f t$ be a periodic function of time $t$. The '''frequency''' of $f$ is the number of periods of $f$ that occur during a unit time interval.
Definition:Zero Operator
Let $H$ be a Hilbert space. Let $A$ be a linear operator on $H$. Suppose that: :$\forall h \in H: A \left({h}\right) = 0$ Then $A$ is called a '''zero operator''', and is denoted as $\mathbf 0$.
Definition:Space Diagonal
Consider a parallelepiped $P$. The '''space diagonals''' of $P$ are the line segments that can be drawn from each vertex $X$ of $P$ to the vertex $Y$ of $P$ which is not contained in any of the faces of $P$ which contain $X$. :400px In the above diagram, the '''space diagonals''' of the parallelepiped $ABCDEFGH$ are $AF$, $BE$, $CH$ and $DG$.
Definition:Fermat-Torricelli Point
Let $ABC$ be a triangle. The '''Fermat Point''' is defined to be the point $P$ which minimizes the following quantity: :$AP + BP + CP$ which is the total distance from the three vertices to point $P$.
Definition:Weakly Abnormal Subgroup
Let $G$ be a group. Let $H$ be a subgroup of $G$. === Definition 1 === {{:Definition:Weakly Abnormal Subgroup/Definition 1}} === Definition 2 === {{:Definition:Weakly Abnormal Subgroup/Definition 2}} === Definition 3 === {{:Definition:Weakly Abnormal Subgroup/Definition 3}}
Definition:Weakly Abnormal Subgroup/Definition 1
$H$ is '''weakly abnormal in $G$''' : :$\forall g \in G: g \in H^{\gen g}$ where $H^{\gen g}$ denotes the smallest subgroup of $G$ containing $H$, generated by the conjugacy action by the cyclic subgroup of $G$ generated by $g$.
Definition:Weakly Pronormal Subgroup
Let $G$ be a group. Let $H$ be a subgroup of $G$. === Definition 1 === {{:Definition:Weakly Pronormal Subgroup/Definition 1}} === Definition 2 === {{:Definition:Weakly Pronormal Subgroup/Definition 2}}
Definition:Trivial Vector Space
Let $V$ be a vector space. Then $V$ is said to be '''trivial''' : :$V = \left\{ {\mathbf 0}\right\}$ where $\mathbf 0$ denotes the zero vector.
Definition:Weakly Pronormal Subgroup/Definition 1
$H$ is '''weakly pronormal in $G$''' : :$\forall g \in G: \exists x \in H^{\gen g}: H^x = H^g$ where: :$H^{\gen g}$ denotes the smallest subgroup of $G$ containing $H$, generated by the conjugacy action by the cyclic subgroup of $G$ generated by $g$ :$H^x$ denotes the conjugate of $H$ by $x$.
Definition:Left Zero Divisor
A '''left zero divisor (in $R$)''' is an element $x \in R$ such that: : $\exists y \in R^*: x \circ y = 0_R$
Definition:Right Zero Divisor
A '''right zero divisor (in $R$)''' is an element $x \in R$ such that: : $\exists y \in R^*: y \circ x = 0_R$
Definition:Exhaust Velocity
Let $R$ be a rocket. The '''exhaust velocity''' of $R$ is the speed at which the propellant of $R$ is emitted out of the back end, to make $R$ travel forwards.
Definition:Usual Ordering
Let $X$ be one of the sets of numbers: $\N$, $\Z$, $\Q$, $\R$. The '''usual ordering''' on $X$ is the conventional counting and measuring order on $X$ that is learned when one is initially introduced to numbers. === Natural Numbers === {{Definition:Ordering on Natural Numbers}} === Integers === {{Definition:Ordering on Integers}} {{expand|This needs to be formalised. The page is included as it is one of the most wanted redline links.}}
Definition:Bid
Let $A$ be an auction. A '''bid''' is a move by a player in $A$ which consists of the submission of a non-negative number. Such a '''bid''' is associated with a payment which is to be made by the winning player in exchange for the prize.
Definition:Fractal
A '''fractal''' is an object that has the property of self-similarity on all scales.
Definition:Universal Cover
A '''universal cover''' is a covering space which is simply connected. {{explain|Context}}
Definition:Conjugacy Action
Let $\struct {G, \circ}$ be a group. The '''(left) conjugacy action''' of $G$ is the left group action $* : G \times G \to G$ defined as: :$\forall g, x \in G: g * x = g \circ x \circ g^{-1}$ The '''right conjugacy action''' of $G$ is the right group action $* : G \times G \to G$ defined as: :$\forall x, g \in G: x * g = g^{-1} \circ x \circ g$
Definition:Subinterval
Let $I$ be a real interval. A '''subinterval''' $J$ of $I$ is a real interval such that $J \subseteq I$. {{Proofread|A purely intuitive definition which may or may not be accurate -- for example, is $\left({a \,.\,.\, b}\right)$ strictly speaking a subinterval of $\left[{a \,.\,.\, b}\right]$?}}
Definition:Subdivision (Real Analysis)/Finite
Let $x_0, x_1, x_2, \ldots, x_{n - 1}, x_n$ be points of $\R$ such that: :$a = x_0 < x_1 < x_2 < \cdots < x_{n - 1} < x_n = b$ Then $\set {x_0, x_1, x_2, \ldots, x_{n - 1}, x_n}$ form a '''finite subdivision of $\closedint a b$'''.
Definition:Subdivision (Real Analysis)/Infinite
Let $x_0, x_1, x_2, \ldots$ be an infinite number of points of $\R$ such that: :$a = x_0 < x_1 < x_2 < \cdots < x_{n - 1} < \ldots \le b$ Then $\set {x_0, x_1, x_2, \ldots}$ forms an '''infinite subdivision of $\closedint a b$'''.
Definition:Orthonormal Basis
Let $\mathbb F$ be a field. Let $V$ be a vector space over $\mathbb F$. Let $S \subset V$ be a subset of $V$. Then $S$ is an '''orthonormal basis''' : :$(1): \quad S$ is a basis. :$(2): \quad S$ is orthonormal.
Definition:Left Module
Let $\struct {R, +_R, \times_R}$ be a ring. Let $\struct {G, +_G}$ be an abelian group. A left module over $R$ is an $R$-algebraic structure $\struct {G, +_G, \circ}_R$ with one operation $\circ$, the '''(left) ring action''', which satisfies the left module axioms: {{:Definition:Left Module Axioms}}