hcju/proofwikips huggingface dataset:
Definition:Power (Algebra)/Real Number/Definition 3
First let $x > 1$. Let $r$ be expressed by its decimal expansion: :$r = n \cdotp d_1 d_2 d_3 \ldots$ For $k \in \Z_{> 0}$, let $\psi_1, \psi_2 \in \Q$ be rational numbers defined as: {{begin-eqn}} {{eqn | l = \psi_1       | r = n + \sum_{j \mathop = 1}^k \frac {d_1} {10^k} = n + \frac {d_1} {10} + \cdots + \frac {d_k} {10^k} }} {{eqn | l = \psi_2       | r = \psi_1 + \dfrac 1 {10^k} }} {{end-eqn}} Then $x^r$ is defined as the (strictly) positive real number $\xi$ defined as: :$\displaystyle \lim_{k \mathop \to \infty} x^{\psi_1} \le \xi \le x^{\psi_2}$ In this context, $x^{\psi_1}, x^{\psi_2}$ denote $x$ to the rational powers $\psi_1$ and $\psi_2$. Next let $x < 1$. Then $x^r$ is defined as: :$x^r := \left({\dfrac 1 x}\right)^{-r}$ Finally, when $x = 1$: :$x^r = 1$
Definition:Power (Algebra)/Real Number/Definition 3/Binary Expansion
First let $x > 1$. Let $r$ be expressed in binary notation: :$r = n \cdotp d_1 d_2 d_3 \ldots$ where $d_1, d_2, d_3 \ldots$ are in $\left\{ {0, 1}\right\}$. For $k \in \Z_{> 0}$, let $\psi_1, \psi_2 \in \Q$ be rational numbers defined as: {{begin-eqn}} {{eqn | l = \psi_1       | r = n + \sum_{j \mathop = 1}^k \frac {d_1} {2^k} = n + \frac {d_1} 2 + \cdots + \frac {d_k} {2^k} }} {{eqn | l = \psi_2       | r = \psi_1 + \dfrac 1 {2^k} }} {{end-eqn}} Then $x^r$ is defined as the (strictly) positive real number $\xi$ defined as: :$\displaystyle \lim_{k \mathop \to \infty} x^{\psi_1} \le \xi \le x^{\psi_2}$ In this context, $x^{\psi_1}, x^{\psi_2}$ denote $x$ to the rational powers $\psi_1$ and $\psi_2$. Next let $x < 1$. Then $x^r$ is defined as: :$x^r := \left({\dfrac 1 x}\right)^{-r}$ Finally, when $x = 1$: :$x^r = 1$
Definition:General Logarithm/Common/Historical Note
'''Common logarithms''' were developed by {{AuthorRef|Henry Briggs}}, as a direct offshoot of the work of {{AuthorRef|John Napier}}. After seeing the tables that {{AuthorRef|John Napier|Napier}} published, {{AuthorRef|Henry Briggs|Briggs}} consulted {{AuthorRef|John Napier|Napier}}, and suggested defining them differently, using base $10$. In $1617$, {{AuthorRef|Henry Briggs|Briggs}} published a set of tables of logarithms of the first $1000$ positive integers. In $1624$, he published tables of logarithms which included $30 \, 000$ logarithms going up to $14$ decimal places. Before the advent of cheap means of electronic calculation, '''common logarithms''' were widely used as a technique for performing multiplication.
Definition:Bit
A '''bit''' is a '''binary digit'''.
Definition:Product Notation (Algebra)/Propositional Function/Iverson's Convention
Let $\displaystyle \prod_{R \left({j}\right)} a_j$ be the product over all $a_j$ such that $j$ satisfies $R$. This can also be expressed: :$\displaystyle \prod_{j \mathop \in \Z} a_j^{\left[{R \left({j}\right)}\right]}$ where $\left[{R \left({j}\right)}\right]$ is Iverson's convention.
Definition:Uncountable Sum
Let $X$ be an uncountable set. Let $f: X \to \left[{0 \,.\,.\, +\infty}\right]$ be an extended real-valued function. The '''uncountable sum''' of $f$ over $X$ is defined to be the supremum of the finite sums: :$\displaystyle \sum_{x \mathop \in X} f \left({x}\right) := \sup \left\{ {\sum_{x \mathop \in F} f \left({x}\right): F \subseteq X, F \text{ finite} }\right\}$
Definition:Supremum of Set/Real Numbers/Propositional Function
Let $\family {a_j}_{j \mathop \in I}$ be a family of elements of the real numbers $\R$ indexed by $I$. Let $\map R j$ be a propositional function of $j \in I$. Then we can define the '''supremum of $\family {a_j}_{j \mathop \in I}$''' as: :$\displaystyle \sup_{\map R j} a_j := \text{ the supremum of all $a_j$ such that $\map R j$ holds}$ If more than one propositional function is written under the supremum sign, they must ''all'' hold.
Definition:Supremum of Set/Real Numbers/Propositional Function/Finite Range
Let the fiber of truth of $\map R j$ be finite. Then the '''supremum of $\family {a_j}_{j \mathop \in I}$''' can be expressed as: :$\displaystyle \max_{\map R j} a_j = \text{ the maxmum of all $a_j$ such that $\map R j$ holds}$ and can be referred to as the '''maximum''' of $\family {a_j}_{j \mathop \in I}$.
Definition:Supremum of Set/Real Numbers/Propositional Function/Vacuous Supremum
Take the indexed supremum: :$\displaystyle \sup _{\Phi \left({j}\right)} a_j$ where $\Phi \left({j}\right)$ is a propositional function of $j$. Suppose that there are no values of $j$ for which $\Phi \left({j}\right)$ is true. Then $\displaystyle \sup_{\Phi \left({j}\right)} a_j$ is defined as being $-\infty$. This supremum is called a '''vacuous supremum'''. This is because: :$\forall a \in \R: \sup \left\{ {a, -\infty}\right\} = a$ Hence for all $j$ for which $\Phi \left({j}\right)$ is false, the supremum is unaffected. In this context $-\infty$ is considered as '''minus infinity''', the hypothetical quantity that has the property: : $\forall n \in \Z: -\infty< n$
Definition:Rounding/Integer
Let $y \in \R$ such that: :$y = \floor {x + \dfrac 1 2}$ Then $y$ is defined as '''$x$ rounded to the nearest integer'''.
Definition:Rounding/Treatment of Half
Consider the situation when $\dfrac x {10^n} + \dfrac 1 2$ is an integer. That is, $\dfrac x {10^n}$ is exactly midway between the two integers $\dfrac x {10^n} - \dfrac 1 2$ and $\dfrac x {10^n} + \dfrac 1 2$. Recall that the general philosophy of the process of rounding is to find the closest approximation to $x$ to a given power of $10$. Thus there are two equally valid such approximations: :$\dfrac x {10^n} - \dfrac 1 2$ and $\dfrac x {10^n} + \dfrac 1 2$ between which $\dfrac x {10^n}$ is exactly midway. There are a number of conventions which determine which is to be used.
Definition:General Dirichlet Series
Let $a_n$ be a sequence in $\C$. Let $\sequence {\lambda_n}$ be a strictly increasing sequence of non-negative real numbers whose limit is infinity. A '''general Dirichlet series of type $\lambda_n$''' is a complex function $f: \C \to \C$ defined by the series: :$\displaystyle \map f s = \sum_{n \mathop = 1}^\infty a_n e^{-\lambda_n s}$
Definition:Sufficiently Small
Let $P$ be a property of real numbers. Then: : '''$P \left({x}\right)$ holds for all sufficiently small $x$''' : :$\exists \epsilon \in \R: \forall x \in \R: \left\lvert{x}\right\rvert \le \epsilon: P \left({x}\right)$ That is, : :''There exists a real number $\epsilon$ such that for every (real) number not more than $\epsilon$ in in absolute value, the property $P$ holds.'' It is not necessarily the case, for a given property $P$ about which such a statement is made, that the value of $\epsilon$ actually needs to be known, merely that such a value can be demonstrated to exist.
Definition:Extended Real Sequence
An '''extended real sequence''' is a sequence (usually infinite) whose codomain is the extended real number line.
Definition:Limit Superior of Extended Real Sequence
Let $\left \langle {x_n} \right \rangle$ be an extended real sequence. The '''limit superior''' of $\left \langle {x_n} \right \rangle$ is defined as: :$\displaystyle \limsup x_n : = \inf_{k \mathop \ge 1} \left({\sup_{n \mathop \ge k} x_n}\right)$
Definition:Limit Inferior of Extended Real Sequence
Let $\left \langle {x_n} \right \rangle$ be an extended real sequence. The '''limit inferior''' of $\left \langle {x_n} \right \rangle$ is defined as: :$\displaystyle \liminf x_n : = \sup_{k \mathop \ge 1} \left({\inf_{n \mathop \ge k} x_n}\right)$
Definition:N-Cube (Euclidean Space)
Let $R, c \in \R$ be real numbers with $R > 0$. Let $\struct {\R^n, d}$ be a Euclidean $n$-Space equipped with the usual metric $d$. An '''$n$-cube''' is a subset of $\struct {\R^n, d}$ defined as the cartesian product of closed real intervals of the form: :$\displaystyle \prod_{i \mathop = 1}^n \closedint {c - R} {c + R}_i$ where $\closedint {c - R} {c + R}_i$ is an interval in the $i$th factor of $\R^n$. The $n$-cube can be concisely expressed as: :$\closedint {c - R} {c + R}^n$ in contexts where the indices of the product are unimportant.
Definition:Diameter of Subset of Metric Space
Let $M = \left({A, d}\right)$ be a metric space. Let $S \subseteq A$ be subset of $A$. Then the '''diameter''' of $S$ is the extended real number defined by: : $\operatorname {diam} \left({S}\right) := \begin{cases} \sup \left\{ {d \left({x, y}\right): x, y \in S }\right\}, \ \text{if this quantity is finite} \\ +\infty, \ \text{otherwise} \end{cases}$ In the finite case, this is, by the definition of the supremum, the smallest real number $D$ such that any two points of $S$ are at most a distance $D$ apart.
Definition:Arbitrarily Small
Let $P$ be a property of real numbers. We say that '''$P \left({x}\right)$ holds for arbitrarily small $\epsilon$''' (or '''there exist arbitrarily small $x$ such that $P \left({x}\right)$ holds''') : :$\forall \epsilon \in \R_{> 0}: \exists x \in \R: \left\lvert{x}\right\rvert \le \epsilon: P \left({x}\right)$ That is: :''For any real number $a$, there exists a (real) number not more than $a$ such that the property $P$ holds.'' or, more informally and intuitively: :''However small a number you can think of, there will be an even smaller one for which $P$ still holds.''
Definition:Half-Plane of Convergence
Let $f \left({s}\right)$ be a (general) Dirichlet series. The '''half-plane of convergence of $f$''' is the set: :$\left\{ {s: s \in \C, \operatorname{Re} \left({s}\right) > \sigma_0}\right\}$ where $\sigma_0$ is abscissa of convergence of $f$.
Definition:Stirling Numbers of the First Kind/Unsigned/Complex Numbers
{{AuthorRef|Donald E. Knuth}}, in his {{BookLink|The Art of Computer Programming: Volume 1: Fundamental Algorithms|Donald E. Knuth|ed = 3rd|edpage = Third Edition}} of $1997$, suggests an extension of the '''unsigned Stirling numbers of the first kind''' $\displaystyle {r \brack r - m}$ to the real and complex numbers. However, beyond stating that such a number is a polynomial in $r$ of degree $2 m$, and providing a few examples, he goes no further than that, and the details of this extension are unclear.
Definition:Stirling Numbers of the Second Kind/Complex Numbers
{{AuthorRef|Donald E. Knuth}}, in his {{BookLink|The Art of Computer Programming: Volume 1: Fundamental Algorithms|Donald E. Knuth|ed = 3rd|edpage = Third Edition}} of $1997$, suggests an extension of the '''Stirling numbers of the second kind''' $\displaystyle {r \brace r - m}$ to the real and complex numbers. However, beyond stating that such a number is a polynomial in $r$ of degree $2 m$, and providing a few examples, he goes no further than that, and the details of this extension are unclear.
Definition:Minimal Uncountable Well-Ordered Set
Let $\Omega$ be an uncountable well-ordered set. Then $\Omega$ is the '''minimal uncountable well-ordered set''' if every initial segment in $\Omega$ is countable.
Definition:Combinatorial Number System
The '''combinatorial number system''' is a system for representing a positive integer $m$ by a sequence of digits which are the upper coefficient of a sequence of $n$ binomial coefficients for some $n \in \Z_{>0}$: :$m := k_1 k_2 k_3 \ldots k_n$ where: :$m = \dbinom {k_1} 1 + \dbinom {k_2} 2 + \dbinom {k_3} 3 + \cdots + \dbinom {k_n} n$ :$0 \le k_1 < k_2 < \cdots < k_n$
Definition:Combination with Repetition
Let $S$ be a (finite) set with $n$ elements. A '''$k$ combination of $S$ with repetition''' is a multiset with $k$ elements selected from $S$.
Definition:Bernoulli Polynomial
The '''Bernoulli polynomials''' $B_n \left({x}\right)$ are the terms of the sequence of polynomials defined as:  :$\displaystyle B_n \left({x}\right) = \sum_{k \mathop = 0}^n \binom n k B_{n - k} x^k$ where $B_k$ are the Bernoulli numbers.
Definition:Midline of Triangle
Let $\triangle ABC$ be a triangle. Let $D$ and $E$ be the midpoints of any two of the sides of $\triangle ABC$. The line $DE$ is a '''midline of $\triangle ABC$'''. 400px
Definition:Second Order Fibonacci Number
The '''second order Fibonacci numbers''' are a sequence $\left \langle {\mathcal F_n}\right \rangle$ of integers which is formally defined recursively for all $n \in \Z_{\ge 0}$ as: :$\mathcal F_n = \begin{cases} 0 & : n = 0 \\ 1 & : n = 1 \\ \mathcal F_{n - 1} + \mathcal F_{n - 2} + F_{n - 2} & : \text{otherwise} \end{cases}$ where $F_{n - 2}$ denotes the $n - 2$th Fibonacci number.
Definition:Fibonomial Coefficient
Let $n \in \Z_{\ge 0}$ and $k \in \Z$. Then the '''Fibonomial coefficient''' $\dbinom n k$ is defined as: :$\dbinom n k_\mathcal F = \begin{cases} 0 & : n < 0, n > k \\ 1 & : n \ge 0, k = 0 \\ \dfrac {F_n F_{n - 1} \cdots F_{n - k + 1} } {F_k F_{k - 1} \cdots F_1} = \displaystyle \prod_{j \mathop = 1}^k \dfrac {F_{n - k + j} } {F_j} & : \text{otherwise} \end{cases}$ where $F_n$ denotes the $n$th Fibonacci number.
Definition:Tower in Set
Let $X$ be a set. Let $T$ be any non-empty subset of $X$ Let $c$ be a fixed choice function on the non-empty subsets $T$ of $X$. Let $\preccurlyeq$ be a well-ordering on $T$. The well-ordered set $\left({T,\preccurlyeq}\right)$ is a '''tower in $X$''' , for all $t \in T$: :$t = c \left({X \setminus S_t \left({T}\right)}\right)$ where $S_t \left({T}\right)$ is the initial segment of $T$ determined by $t$.
Definition:Zeckendorf Representation
'''Zeckendorf representation''' is a system for representing a positive integer $m$ by a sequence of digits which are the indices of a sequence of $r$ Fibonacci numbers: :$n := k_1 k_2 k_3 \ldots k_r$ where: :$n = F_{k_1} + F_{k_2} + F_{k_3} + \cdots + F_{k_r}$ :$k_1 \gg k_2 \gg k_3 \gg \cdots \gg k_r \gg 0$ where $n \gg k$ denotes that $n \ge k + 2$.
Definition:Golden Mean Number System
The '''golden mean number system''' is a system for representing a non-negative real number $x$ by a sequence of zeroes and  ones using the golden mean $\phi$ as a number base.
Definition:Proper Subtower in Set
Let $X$ be a set. Let $\left({T, \preccurlyeq}\right)$ be a tower in $X$. Then $\left({T, \preccurlyeq}\right)$ is a '''proper subtower in $X$''' : : $T$ is a proper subset of some set $T' \subseteq X$ and: : $\left({T', \preccurlyeq}\right)$ is a tower in $X$.
Definition:Golden Mean Number System/Simplification
Let $x \in \R_{\ge 0}$ have a representation which includes the string $011$, say: :$x = p011q$ where $p$ and $q$ are strings in $\left\{ {0, 1}\right\}$. From 100 in Golden Mean Number System is Equivalent to 011, $x$ can also be written as: :$x = p100q$ The expression $p100q$ is a '''simplification''' of $p011q$.
Definition:String/Infinite
A '''string''' $S$ in $\mathcal A$ is an '''infinite string'''  the sequence of symbols of which it is composed is infinite.
Definition:String/Finite
A '''string''' $S$ in $\mathcal A$ is a '''finite string'''  the sequence of symbols of which it is composed is finite.
Definition:Fibonacci String
Consider the alphabet $\left\{ {\text {a}, \text {b} }\right\}$. For all $n \in \Z_{>0}$, let $S_n$ be the (finite) string formed as: :$S_n = \begin{cases} \text {a} & : n = 1 \\ \text {b} & : n = 2 \\ S_{n - 1} S_{n - 2} & : n > 2 \end{cases}$ where $S_{n - 1} S_{n - 2}$ denotes that $S_{n - 1}$ and $S_{n - 2}$ are concatenated. The terms of the sequence $\left\langle{S_n}\right\rangle$ are '''Fibonacci strings'''.
Definition:Fibonacci Nim
'''Fibonacci nim''' is a two-person game whose rules are as follows: :$(1): \quad$ The game starts with one pile of $n$ counters. :$(2): \quad$ The first player removes a number of counters $c_1$ such that $1 \le c_1 < n$. :$(3): \quad$ Each player takes it in turns to remove $c_n$ counters such that $1 \le c_n \le 2 c_{n - 1}$ where $c_{n - 1}$ is the number of counters taken in the other player's prevous move. :$(3): \quad$ The person who removes the last counter (or counters) is the winner.
Definition:Generating Function/Parameter/Also denoted as
The symbol used for the '''parameter''' varies. $x$ is often used instead. In the field of probability theory $s$ tends to be the symbol of choice. Some authors use $\zeta$.
Definition:Linearly Recurrent Sequence
A '''linearly recurrent sequence''' is a sequence which can be defined by a recurrence relation of the form: :$a_n = c_1 a_{n - 1} + c_2 a_{n - 2} + \cdots + c_m a_{n - m}$ with appropriate initial values for $a_1, a_2, \ldots, a_{n - 1}$.
Definition:Generating Function/Extraction of Coefficient
Let $\map G z$ be the generating function for the sequence $S = \sequence {a_n}$. The '''coefficient of $z^n$ extracted from $\map G z$''' is the $n$th term of $S$, and can be denoted: :$\sqbrk {z^n} \map G z := a_n$
Definition:Elementary Symmetric Function
Let $a, b \in \Z$ be integers such that $b \ge a$. Let $U$ be a set of $n = b - a + 1$ numbers $\set {x_a, x_{a + 1}, \ldots, x_b}$. Let $m \in \Z_{>0}$ be a (strictly) positive integer. An '''elementary symmetric function of degree $m$''' is a polynomial which can be defined by the formula: {{begin-eqn}} {{eqn | l = \map {e_m} U       | r = \sum_{a \mathop \le j_1 \mathop < j_2 \mathop < \mathop \cdots \mathop < j_m \mathop \le b} \paren {\prod_{i \mathop = 1}^m x_{j_i} }       | c =  }} {{eqn | r = \sum_{a \mathop \le j_1 \mathop < j_2 \mathop < \mathop \cdots \mathop < j_m \mathop \le b} x_{j_1} x_{j_2} \cdots x_{j_m}       | c =  }} {{end-eqn}} That is, it is the sum of all products of $m$ distinct elements of $\set {x_a, x_{a + 1}, \dotsc, x_b}$.
Definition:Power Sum
Let $S$ be a finite set of numbers. Let $p \in \R$ be a real number. Then the '''$p$th power sum''' of $S$ is defined as: :$S_p = \displaystyle \sum_{x \mathop \in S} x^p$ That is, it is the summation of the $p$th powers of all the elements of $S$.
Definition:Sequence/Doubly Subscripted
A '''doubly subscripted sequence''' is a mapping whose domain is a subset of the cartesian product $\N \times \N$ of the set of natural numbers $\N$ with itself. It can be seen that a '''doubly subscripted sequence''' is an instance of a family of elements indexed by $\N^2$. A '''doubly subscripted sequence''' can be denoted $\left\langle{a_{m n} }\right\rangle_{m, \, n \mathop \ge 0}$
Definition:Generating Function/Doubly Subscripted Sequence
Let $A = \left \langle {a_{m, n} }\right \rangle$ be a doubly subscripted sequence in $\R$ for $m, n \in \Z_{\ge 0}$. Then $\displaystyle G_A \left({w, z}\right) = \sum_{m, \, n \mathop \ge 0} a_{m n} w^m z^n$ is called the '''generating function''' for the sequence $A$.
Definition:Complex Arithmetic
'''Complex arithmetic''' is the branch of arithmetic which concerns the manipulation of complex numbers
Definition:Conjugate Pair
Let $z \in \C$ be a complex number. Let $\overline z$ be the complex conjugate of $z$. Then $z$ and $\overline z$ are a '''conjugate pair'''.
Definition:Complex Algebra
'''Complex algebra''' is the branch of algebra which specifically involves complex arithmetic. Thus it is the branch of mathematics which studies the techniques of manipulation of expressions in the domain of complex numbers.
Definition:Half-Plane
Let $P$ denote the plane. Let $L$ denote an infinite straight line in $P$. Then each of the areas of $P$ on either side of $L$ is a '''half-plane'''.
Definition:Circle of Apollonius
Let $A, B$ be distinct points in the plane. Let $\lambda \in \R_{>0}$ be a strictly positive real number. Let $X$ be the locus of points in the plane such that: :$X A = \map \lambda {X B}$ :400px Then $X$ is a '''circle of Apollonius'''.
Definition:Root of Unity/Complex/Order
Let $z \in U_n$. The '''order''' of $z$ is the smallest $p \in \Z_{> 0}$ such that: :$z^p = 1$
Definition:Cauchy Sequence/Normed Vector Space
Let $\struct {V, \norm {\,\cdot\,} }$ be a normed vector space. Let $\sequence {x_n}$ be a sequence in $V$. Then $\sequence {x_n}$ is a '''Cauchy sequence''' : : $\forall \epsilon \in \R_{>0}: \exists N \in \N: \forall m, n \in \N: m, n \ge N: \norm {x_n - x_m} < \epsilon$
Definition:Cauchy Sequence/Normed Division Ring
Let $\struct {R, \norm {\,\cdot\,} } $ be a normed division ring Let $\sequence {x_n}$ be a sequence in $R$. Then $\sequence {x_n} $ is a '''Cauchy sequence in the norm $\norm {\, \cdot \,}$''' : :$\sequence {x_n}$ is a cauchy sequence in the metric induced by the norm $\norm {\, \cdot \,}$ That is: : $\forall \epsilon \in \R_{>0}: \exists N \in \N: \forall m, n \in \N: m, n \ge N: \norm {x_n - x_m} < \epsilon$
Definition:Sextic Equation
Let $f \paren x = a x^6 + b x^5 + c x^4 + d x^3 + e x^2 + f x + g$ be a polynomial function over a field $\mathbb k$ of degree $6$. Then the equation $f \paren x = 0$ is the general '''sextic equation''' over $\mathbb k$.
Definition:Convergent Sequence/Complex Numbers
=== Definition 1 === {{:Definition:Convergent Sequence/Complex Numbers/Definition 1}} === Definition 2 === {{:Definition:Convergent Sequence/Complex Numbers/Definition 2}}
Definition:Convergent Sequence/Complex Numbers/Definition 2
Let $\sequence {z_k} = \sequence {x_k + i y_k}$ be a sequence in $\C$. $\sequence {z_k}$ '''converges to the limit $c = a + i b$'''  both: :$\forall \epsilon \in \R_{>0}: \exists N \in \R: n > N \implies \size {x_n - a} < \epsilon \text { and } \size {y_n - b} < \epsilon$ where $\size {x_n - a}$ denotes the absolute value of $x_n - a$.
Definition:Series/Complex
A '''complex series''' $S_n$ is the limit to infinity of the sequence of partial sums of a complex sequence $\sequence {a_n}$: {{begin-eqn}} {{eqn | l = S_n       | r = \lim_{N \mathop \to \infty} \sum_{n \mathop = 1}^N a_n       | c =  }} {{eqn | r = \sum_{n \mathop = 1}^\infty a_n       | c =  }} {{eqn | r = a_1 + a_2 + a_3 + \cdots       | c =  }} {{end-eqn}}
Definition:Bounded Mapping/Normed Division Ring
Let $\struct {R, \norm {\,\cdot\,}}$ be a normed division ring. Let $f: S \to R$ be a mapping from $S$ into $R$. Then $f$ is '''bounded'''  the real-valued function $\norm {\,\cdot\,} \circ f: S \to \R$ is bounded, where $\norm {\,\cdot\,} \circ f$ is the composite of $\norm {\,\cdot\,}$ and $f$. That is, $f$ is '''bounded''' if there is a constant $K \in \R_{\ge 0}$ such that $\norm{f \paren {s}} \le K$ for all $s \in S$.
Definition:Bounded Sequence/Normed Division Ring
Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring. Let $\sequence {x_n}$ be a sequence in $R$. Then $\sequence {x_n}$ is '''bounded''' : :$\exists K \in \R$ such that $\forall n \in \N: \norm {x_n} \le K$
Definition:Bounded Sequence/Normed Division Ring/Unbounded
$\sequence {x_n}$ is '''unbounded'''  it is not bounded.
Definition:Bounded Sequence/Metric Space
Let $M$ be a metric space. Let $\sequence {x_n}$ be a sequence in $M$. Then $\sequence {x_n}$ is a '''bounded sequence'''  $\sequence {x_n}$ is bounded in $M$. That is: :$\exists K \in \R: \forall n, m \in \N: \map d {x_n, x_m} \le K$
Definition:Bounded Sequence/Metric Space/Unbounded
$\sequence {x_n}$ is '''unbounded'''  it is not bounded.
Definition:Metric Induced by Norm on Division Ring
Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring. Then the '''induced metric''' or the '''metric induced by $\norm {\,\cdot\,}$''' is the map $d: R \times R \to \R_{\ge 0}$ defined as: :$d \paren {x, y} = \norm {x - y}$
Definition:Series/Real
A '''real series''' $S_n$ is the limit to infinity of the sequence of partial sums of a real sequence $\sequence {a_n}$: {{begin-eqn}} {{eqn | l = S_n       | r = \lim_{N \mathop \to \infty} \sum_{n \mathop = 1}^N a_n       | c =  }} {{eqn | r = \sum_{n \mathop = 1}^\infty a_n       | c =  }} {{eqn | r = a_1 + a_2 + a_3 + \cdots       | c =  }} {{end-eqn}}
Definition:Non-Archimedean/Norm (Division Ring)
=== Definition 1 === {{:Definition:Non-Archimedean/Norm (Division Ring)/Definition 1}}
Definition:Non-Archimedean/Norm (Division Ring)/Archimedean
A norm $\norm {\, \cdot \,}$ on a division ring $R$ is '''Archimedean'''  it is not non-Archimedean.
Definition:Bell Number/Sequence
The sequence of Bell numbers begins: :$(1), 1, 2, 5, 15, 52, 203, 877, 4140, 21 \, 147, \ldots$ where the first $(1)$ is the zeroth '''Bell number''' $B_0$.
Definition:Divisor (Algebra)/Also known as
A '''divisor''' can also be referred to as a '''factor'''.