hcju/proofwikips huggingface dataset:
Definition:That which produces Medial Whole with Medial Area
Let $a, b \in \R_{>0}$ be in the forms: :$a = \dfrac {\rho \lambda^{1/4} } {\sqrt 2} \sqrt {1 + \dfrac k {\sqrt {1 + k^2} } }$ :$b = \dfrac {\rho \lambda^{1/4} } {\sqrt 2} \sqrt {1 - \dfrac k {\sqrt {1 + k^2} } }$ where: : $\rho$ is a rational number : $k$ is a rational number whose square root is irrational : $\lambda$ is a rational number whose square root is irrational. Then $a - b$ is '''that which produces a medial whole with a medial area'''. {{:Euclid:Proposition/X/78}}
Definition:Apotome/Whole
The real number $a$ is called the '''whole''' of the apotome.
Definition:Apotome/Annex
The real number $b$ is called the '''annex''' of the apotome.
Definition:Apotome/Terms
The '''terms''' of $a - b$ are the elements $a$ and $b$. === Whole === {{Definition:Apotome/Whole}} === Annex === {{Definition:Apotome/Annex}}
Definition:Common Section
Let $A$ and $B$ be planes. The '''common section''' of $A$ and $B$ is the intersection of $A$ and $B$.
Definition:Similar Planes
'''Similar planes''' are plane figures which are similar.
Definition:Containment of Solid Figure
A solid geometric figure is said to be '''contained''' by the surfaces that form its boundary.
Definition:Similar Solid Figures
{{EuclidDefinition|book=XI|def=9|name=Similar Solid Figures}}
Definition:Similar Equal Solid Figures
{{EuclidDefinition|book=XI|def=10|name=Similar Equal Solid Figures}}
Definition:Solid Angle
{{EuclidDefinition|book=XI|def=11|name=Solid Angle}} === Containment of Solid Angle === {{:Definition:Solid Angle/Containment}} === Vertex of Solid Angle === {{:Definition:Solid Angle/Vertex}}
Definition:Plane of Reference
The '''plane of reference''' is a conceptual plane in a spatial context which is arbitrarily distinguished from the others.
Definition:Plane Surface/Side
From the definition of surface, it follows that a plane locally separates space into two '''sides'''. Thus the '''sides''' of a plane are the parts of that space into which the plane separates it.
Definition:Solid Angle/Containment
The three plane angles which together form a solid angle are said to '''contain''' that solid angle.
Definition:Parallelepiped
A '''parallelepiped''' is a polyhedron formed by three pairs of parallel planes: :500px In the above example, the pairs of parallel planes are: :$ABCD$ and $HGFE$ :$ADEH$ and $BCFG$ :$ABGH$ and $DCFE$
Definition:Geometric Figure/Three-Dimensional Figure/Face
The '''faces''' of a three-dimensional figure are the surfaces which form its extremities.
Definition:Parallelepiped/Base
One of the faces of the parallelepiped is chosen arbitrarily, distinguished from the others and called the '''base of the parallelepiped'''. It is usual to choose the '''base''' to be the one which is conceptually on the bottom. In the above, $ABCD$ would conventionally be identified as being the '''base'''
Definition:Octahedron
An '''octahedron''' is a polyhedron which has $8$ faces.
Definition:Tetrahedron
A '''tetrahedron''' is a polyhedron which has $4$ (triangular) faces.
Definition:Dodecahedron
A '''dodecahedron''' is a polyhedron which has $12$ faces.
Definition:Hexahedron
A '''hexahedron''' is a polyhedron which has $6$ faces. === Cube === {{:Definition:Hexahedron/Regular}}
Definition:Icosahedron
An '''icosahedron''' is a polyhedron which has $20$ faces.
Definition:Polyhedron/Edge
The '''edges''' of a polyhedron are the sides of the polygons which constitute its faces.
Definition:Polyhedron/Vertex
The '''vertices''' of a polyhedron are the vertices of the polygons which constitute its faces.
Definition:Elevated
=== Elevated Point === {{:Definition:Elevated/Point}} === Elevated Line === {{:Definition:Elevated/Line}}
Definition:Angle/Vertex
The point at which the lines containing an angle meet is known as the '''vertex''' of that angle.
Definition:Solid Angle/Vertex
The common vertex of the angles containing a solid angle is known as the '''vertex''' of that solid angle.
Definition:Prism/Base
The '''bases''' of a prism are the two parallel polygons which form the faces at either end of the prism. In the above diagram, the faces $ABCDE$ and $FGHIJ$ are the '''bases''' of the prism.
Definition:Pyramid/Base
The polygon of a pyramid to whose vertices the apex is joined is called the '''base''' of the pyramid. In the above diagram, $ABCDE$ is the '''base''' of the pyramid $ABCDEQ$.
Definition:Pyramid/Apex
The vertex of a pyramid which is the common vertex of its triangular faces is called the '''apex''' of the pyramid. In the above diagram, $Q$ is the '''apex'''.
Definition:Square Pyramid
:300px A '''square pyramid''' is a pyramid whose base is a square.
Definition:Pyramid/Height
The '''height''' of a pyramid is the length of the perpendicular from the plane of the base to its apex. In the above diagram, $h$ is the height.
Definition:Cylinder/Height
The '''height''' of a cylinder is the length of a line segment drawn perpendicular to the base and its opposite plane. In the above diagram, $h$ is the '''height''' of the cylinder $ACBDFE$.
Definition:Height of Solid Figure
The '''height''' of a solid figure is the length of a perpendicular from the base to the point or points most distant from it. === Height of Parallelepiped === {{:Definition:Height of Parallelepiped}} === Height of Prism === :500px {{:Definition:Height of Prism}} === Height of Pyramid === {{:Definition:Height of Pyramid}} === Height of Cone === {{:Definition:Height of Cone}} === Height of Cylinder === {{:Definition:Height of Cylinder}}
Definition:Base of Solid Figure
The '''base''' of a solid figure is one of its faces which has been distinguished from the others in some way. The solid figure is usually oriented so that the '''base''' is situated at the bottom. === Base of Parallelepiped === :500px {{:Definition:Base of Parallelepiped}} === Base of Pyramid === :400px {{:Definition:Base of Pyramid}} === Base of Prism === :500px {{:Definition:Base of Prism}} === Base of Cone === :300px {{:Definition:Base of Cone}} ==== Base of Right Circular Cone ==== :300px {{:Definition:Base of Right Circular Cone}} === Base of Cylinder === :300px {{:Definition:Base of Cylinder}}
Definition:Decagon
A '''decagon''' is a polygon with exactly $10$ sides. :300px
Definition:Geometric Sequence/Term
The elements: :$x_n$ for $n = 0, 1, 2, 3, \ldots$ are the '''terms''' of $\sequence {x_n}$.
Definition:Proportion/Inverse
Two real variables $x$ and $y$ are '''inversely proportional'''  their product is a constant: :$\forall x, y \in \R: x \propto \dfrac 1 y \iff \exists k \in \R, k \ne 0: x y = k$
Definition:Proportion/Joint
Two real variables $x$ and $y$ are '''jointly proportional''' to a third real variable $z$  the product of $x$ and $y$ is a constant multiple of $z$: :$\forall x, y \in \R: x y \propto z \iff \exists k \in \R, k \ne 0: x y = k z$
Definition:Proportion/Continued
Four magnitudes $a, b, c, d$ are '''in continued proportion'''  $a : b = b : c = c : d$.
Definition:Linear Measure/Distance
The '''distance''' between two points $A$ and $B$ in space is defined as the length of a straight line that would be drawn from $A$ to $B$.
Definition:Physical Constant
A '''physical constant''' is a physical quantity that is believed to be unchanging in both space and time.
Definition:Recursive Sequence/Initial Terms
Let $S$ be a recursive sequence. In order for $S$ to be defined, it is necessary to define the '''initial term''' (or terms) explicitly. For example, in the Fibonacci sequence, the '''initial terms''' are defined as: :$F_0 = 0, F_1 = 1$
Definition:Proper Subset/Improper
$S$ is an '''improper subset''' of $T$  $S$ is a subset of $T$ but specifically ''not'' a proper subset of $T$. That is, either: :$S = T$ or: :$S = \O$
Definition:Set Union/Family of Sets/Two Sets
Let $I = \set {\alpha, \beta}$ be an indexing set containing exactly two elements. Let $\family {S_i}_{i \mathop \in I}$ be a family of sets indexed by $I$. From the definition of the union of $S_i$: :$\displaystyle \bigcup_{i \mathop \in I} S_i := \set {x: \exists i \in I: x \in S_i}$ it follows that: :$\displaystyle \bigcup \set {S_\alpha, S_\beta} := S_\alpha \cup S_\beta$
Definition:Set Intersection/Family of Sets/Two Sets
Let $I = \set {\alpha, \beta}$ be an indexing set containing exactly two elements. Let $\family {S_i}_{i \mathop \in I}$ be a family of sets indexed by $I$. From the definition of the intersection of $S_i$: :$\displaystyle \bigcap_{i \mathop \in I} S_i := \set {x: \forall i \in I: x \in S_i}$ it follows that: :$\displaystyle \bigcap \set {S_\alpha, S_\beta} := S_\alpha \cap S_\beta$
Definition:Cartesian Product/Countable
Let $\sequence {S_n}_{n \mathop \in \N}$ be an infinite sequence of sets.  The '''cartesian product''' of $\sequence {S_n}$ is defined as: :$\displaystyle \prod_{k \mathop = 1}^\infty S_k = \set {\tuple {x_1, x_2, \ldots, x_n, \ldots}: \forall k \in \N: x_k \in S_k}$ It defines the concept: :$S_1 \times S_2 \times \cdots \times S_n \times \cdots$ Thus $\displaystyle \prod_{k \mathop = 1}^\infty S_k$ is the set of all infinite sequences $\tuple {x_1, x_2, \ldots, x_n, \ldots}$ with $x_k \in S_k$.
Definition:Composition of Mappings/General Definition
Let $f_1: S_1 \to S_2, f_2: S_2 \to S_3, \ldots, f_n: S_n \to S_{n + 1}$ be mappings such that the domain of $f_k$ is the same set as the codomain of $f_{k - 1}$. Then the '''composite of $f_1, f_2, \ldots, f_n$''' is defined and denoted as: {{begin-eqn}} {{eqn | l = \forall x \in S_1: \map {\paren {f_n \circ \cdots \circ f_2 \circ f_1} } x       | o = :=       | r = \begin {cases} \map {f_1} x & : n = 1 \\ \map {f_n} {\map {\paren {f_{n - 1} \circ \cdots \circ f_2 \circ f_1} } x} : & n > 1 \end {cases}       | c =  }} {{eqn | r = \map {f_n} {\dotsm \map {f_2} {\map {f_1} x} \dotsm}       | c =  }} {{end-eqn}}
Definition:Inverse Mapping/Definition 1
Let $f: S \to T$ be a mapping. Let $f^{-1} \subseteq T \times S$ be the inverse of $f$: :$f^{-1} := \set {\tuple {t, s}: \map f s = t}$ Let $f^{-1}$ itself be a mapping: :$\forall y \in T: \tuple {y, x_1} \in f^{-1} \land \tuple {y, x_2} \in f^{-1} \implies x_1 = x_2$ and :$\forall y \in T: \exists x \in S: \tuple {y, x} \in f$ Then $f^{-1}$ is called the '''inverse mapping of $f$'''.
Definition:Chebyshev Distance
The '''Chebyshev distance''' on $A_1 \times A_2$ is defined as: :$\map {d_\infty} {x, y} := \max \set {\map {d_1} {x_1, y_1}, \map {d_2} {x_2, y_2} }$ where $x = \tuple {x_1, x_2}, y = \tuple {y_1, y_2} \in A_1 \times A_2$.
Definition:Taxicab Metric/General Definition
The '''taxicab metric''' on $\displaystyle \mathcal A = \prod_{i \mathop = 1}^n A_{i'}$ is defined as: : $\displaystyle d_1 \left({x, y}\right) := \sum_{i \mathop = 1}^n d_{i'} \left({x_i, y_i}\right)$ where $x = \left({x_1, x_2, \ldots, x_n}\right), y = \left({y_1, y_2, \ldots, y_n}\right) \in \mathcal A$.
Definition:Chebyshev Distance/General Definition
The '''Chebyshev distance''' on $\displaystyle \mathcal A = \prod_{i \mathop = 1}^n A_i$ is defined as: :$\displaystyle \map {d_\infty} {x, y} = \max_{i \mathop = 1}^n \set {\map {d_i} {x_i, y_i} }$ where $x = \tuple {x_1, x_2, \ldots, x_n}, y = \tuple {y_1, y_2, \ldots, y_n} \in \mathcal A$.
Definition:Euclidean Metric
The '''Euclidean metric''' on $A_{1'} \times A_{2'}$ is defined as: :$\map {d_2} {x, y} := \paren {\paren {\map {d_{1'} } {x_1, y_1} }^2 + \paren {\map {d_{2'} } {x_2, y_2} }^2}^{1/2}$ where $x = \tuple {x_1, x_2}, y = \tuple {y_1, y_2} \in A_{1'} \times A_{2'}$.
Definition:Euclidean Metric/General Definition
The '''Euclidean metric''' on $\displaystyle \mathcal A = \prod_{i \mathop = 1}^n A_{i'}$ is defined as: : $\displaystyle d_2 \left({x, y}\right) := \left({\sum_{i \mathop = 1}^n \left({d_{i'} \left({x_i, y_i}\right)}\right)^2}\right)^{\frac 1 2}$ where $x = \left({x_1, x_2, \ldots, x_n}\right), y = \left({y_1, y_2, \ldots, y_n}\right) \in \mathcal A$.
Definition:Taxicab Metric/Real Vector Space
The '''taxicab metric''' on $\R^n$ is defined as: : $\displaystyle d_1 \left({x, y}\right) := \sum_{i \mathop = 1}^n \left\vert {x_i - y_i}\right\vert$ 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:Chebyshev Distance/Real Vector Space
The '''Chebyshev distance''' on $\R^n$ is defined as: :$\displaystyle \map {d_\infty} {x, y}:= \max_{i \mathop = 1}^n \set {\size {x_i - y_i} }$ where $x = \tuple {x_1, x_2, \ldots, x_n}, y = \tuple {y_1, y_2, \ldots, y_n} \in \R^n$.
Definition:Standard Discrete Metric/Real Number Plane
The '''discrete metric''' on $\R^2$ is defined as: : $\displaystyle d_0 \left({x, y}\right) := \begin{cases} 0 & : x = y \\ 1 & : \exists i \in \left\{{1, 2}\right\}: x_i \ne y_i \end{cases}$ where $x = \left({x_1, x_2}\right), y = \left({y_1, y_2}\right) \in \R^2$.
Definition:Euclidean Metric/Real Number Plane
The '''Euclidean metric''' on $\R^2$ is defined as: :$\displaystyle \map {d_2} {x, y} := \sqrt {\paren {x_1 - y_1}^2 + \paren {x_2 - y_2}^2}$ where $x = \tuple {x_1, x_2}, y = \tuple {y_1, y_2} \in \R^2$.
Definition:Taxicab Metric/Graphical Example
This diagram shows the open $\epsilon$-ball of point $A$ in the $\left({\R^2, d_1}\right)$ metric space where $d_1$ is the taxicab metric. :300px Note that $\epsilon = \epsilon_1 + \epsilon_2$. Neither the boundary lines nor the extreme points are actually part of the open $\epsilon$-ball.
Definition:P-Product Metric/Real Vector Space
The '''$p$-product metric''' on $\R^n$ is defined as: :$\displaystyle d_p \left({x, y}\right) := \left({\sum_{i \mathop = 1}^n \left\vert{x_i - y_i}\right\vert^p}\right)^{\frac 1 p}$ 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:Taxicab Metric/Real Number Plane
The '''taxicab metric''' on $\R^2$ is defined as: :$\map {d_1} {x, y} := \size  {x_1 - y_1} + \size {x_2 - y_2}$ where $x = \tuple {x_1, x_2}$, $y = \tuple {y_1, y_2} \in \R^2$.
Definition:Chebyshev Distance/Real Number Plane
The '''Chebyshev distance''' on $\R^2$ is defined as: :$\map {d_\infty} {x, y}:= \max \set {\size {x_1 - y_1}, \size {x_2 - y_2} }$ where $x = \tuple {x_1, x_2}, y = \tuple {y_1, y_2} \in \R^2$.
Definition:Euclidean Metric/Ordinary Space
The '''Euclidean metric''' on $\R^3$ is defined as: :$\map {d_2} {x, y} := \sqrt {\paren {x_1 - y_1}^2 + \paren {x_2 - y_2}^2 + \paren {x_3 - y_3}^2}$ where $x = \tuple {x_1, x_2, x_3}, y = \tuple {y_1, y_2, y_3} \in \R^3$.
Definition:Euclidean Metric/Complex Plane
The '''Euclidean metric''' on $\C$ is defined as: :$\displaystyle \forall z_1, z_2 \in \C: \map d {z_1, z_2} := \size {z_1 - z_2}$ where $\size {z_1 - z_2}$ denotes the modulus of $z_1 - z_2$.
Definition:Euclidean Metric/Rational Number Plane
The '''Euclidean metric''' on $\Q^2$ is defined as: :$\displaystyle d_2 \left({x, y}\right) := \sqrt{\left({x_1 - y_1}\right)^2 + \left({x_2 - y_2}\right)^2}$ where $x = \left({x_1, x_2}\right), y = \left({y_1, y_2}\right) \in \Q^2$.
Definition:Continuous Real Function/Open Interval
Let $f$ be a real function defined on an open interval $\openint a b$. Then $f$ is '''continuous on $\openint a b$'''  it is continuous at every point of $\openint a b$.
Definition:Continuous Real Function/Closed Interval/Definition 1
The function $f$ is '''continuous on $\left[{a \,.\,.\, b}\right]$'''  it is: :$(1): \quad$ continuous at every point of $\left({a \,.\,.\, b}\right)$ :$(2): \quad$ continuous on the right at $a$ :$(3): \quad$ continuous on the left at $b$. That is, if $f$ is to be continuous over the ''whole'' of a closed interval, it needs to be continuous at the end points. Because we only have "access" to the function on one side of each end point, all we can do is insist on continuity on the side of the end points on which the function is defined.
Definition:Continuous Real Function/Half Open Interval
Let $f$ be a real function defined on a half open interval $\hointl a b$. Then $f$ is '''continuous on $\hointl a b$'''  it is: :$(1): \quad$ continuous at every point of $\openint a b$ :$(2): \quad$ continuous on the left at $b$. Let $f$ be a real function defined on a half open interval $\hointr a b$. Then $f$ is '''continuous on $\hointr a b$'''  it is: :$(1): \quad$ continuous at every point of $\openint a b$ :$(2): \quad$ continuous on the right at $a$.
Definition:Supremum Metric/Bounded Real Functions on Interval
Let $\left[{a \,.\,.\, b}\right] \subseteq \R$ be a closed real interval. Let $A$ be the set of all bounded real functions $f: \left[{a \,.\,.\, b}\right] \to \R$. Let $d: A \times A \to \R$ be the function defined as: :$\displaystyle \forall f, g \in A: d \left({f, g}\right) := \sup_{x \mathop \in \left[{a \,.\,.\, b}\right]} \left\vert{f \left({x}\right) - g \left({x}\right)}\right\vert$ where $\sup$ denotes the supremum. $d$ is known as the '''supremum metric''' on $A$.
Definition:Supremum Metric
Let $S$ be a set. Let $M = \left({A', d'}\right)$ be a metric space. Let $A$ be the set of all bounded mappings $f: S \to M$. Let $d: A \times A \to \R$ be the function defined as: :$\displaystyle \forall f, g \in A: d \left({f, g}\right) := \sup_{x \mathop \in S} d' \left({f \left({x}\right), g \left({x}\right)}\right)$ where $\sup$ denotes the supremum. $d$ is known as the '''supremum metric''' on $A$.
Definition:Supremum Metric/Continuous Real Functions
Let $\left[{a \,.\,.\, b}\right] \subseteq \R$ be a closed real interval. Let $A$ be the set of all continuous functions $f: \left[{a \,.\,.\, b}\right] \to \R$. Let $d: A \times A \to \R$ be the function defined as: :$\displaystyle \forall f, g \in A: d \left({f, g}\right) := \sup_{x \mathop \in \left[{a \,.\,.\, b}\right]} \left\vert{f \left({x}\right) - g \left({x}\right)}\right\vert$ where $\sup$ denotes the supremum. $d$ is known as the '''supremum metric''' on $A$.
Definition:L1 Metric
=== Closed Real Interval === {{:Definition:L1 Metric/Closed Real Interval}}
Definition:L1 Metric/Closed Real Interval
Let $S$ be the set of all real functions which are continuous on the closed interval $\left[{a \,.\,.\, b}\right]$. Let the real-valued function $d: S \times S \to \R$ be defined as: :$\displaystyle \forall f, g \in S: d_1 \left({f, g}\right) := \int_a^b \left\vert{f \left({t}\right) - g \left({t}\right)}\right\vert \ \mathrm d t$ Then $d_1$ is the '''$L^1$ metric''' on $\left[{a \,.\,.\, b}\right]$.
Definition:L2 Metric/Closed Real Interval
Let $S$ be the set of all real functions which are continuous on the closed interval $\closedint a b$. Let the real-valued function $d: S \times S \to \R$ be defined as: :$\displaystyle \forall f, g \in S: \map d {f, g} := \paren {\int_a^b \paren {\map f t - \map g t}^2 \rd t}^{\frac 1 2}$ Then $d$ is the '''$L^2$ metric''' on $\closedint a b$.
Definition:L2 Metric
=== Closed Real Interval === {{:Definition:L2 Metric/Closed Real Interval}}
Definition:Bounded Metric Space/Definition 1
'''$M'$ is bounded (in $M$)''' : :$\exists a \in A, K \in \R: \forall x \in B: \map {d_B} {x, a} \le K$ That is, there exists an element of $A$ within a finite distance of all elements of $B$.
Definition:Bounded Metric Space/Definition 2
$M'$ is '''bounded''' : :$\exists K \in \R: \forall x, y \in M': \map {d_B} {x, y} \le K$ That is, there exists a finite distance such that all pairs of elements of $B$ are within that distance.
Definition:Hilbert Sequence Space
Let $d_2: A \times A: \to \R$ be the real-valued function defined as: :$\displaystyle \forall x = \sequence {x_i}, y = \sequence {y_i} \in A: \map {d_2} {x, y} := \paren {\sum_{k \mathop \ge 0} \paren {x_k - y_k}^2}^{\frac 1 2}$ The metric space $\struct {A, d_2}$ is the '''Hilbert sequence space on $\R$''' and is denoted $\ell^2$.
Definition:Supremum Metric/Bounded Real-Valued Functions
Let $S$ be a set. Let $A$ be the set of all bounded real-valued functions $f: S \to \R$. Let $d: A \times A \to \R$ be the function defined as: :$\displaystyle \forall f, g \in A: d \left({f, g}\right) := \sup_{x \mathop \in S} \left\vert{f \left({x}\right) - g \left({x}\right)}\right\vert$ where $\sup$ denotes the supremum. $d$ is known as the '''supremum metric''' on $A$.
Definition:Supremum of Real Sequence
Let $\sequence {x_n}$ be a real sequence. Let $\set {x_n: n \in \N}$ admit a supremum. Then the '''supremum''' of $\sequence {x_n}$) is defined as: :$\displaystyle \map \sup {\sequence {x_n} } = \map \sup {\set {x_n: n \in \N} }$
Definition:Infimum of Real Sequence
Let $\sequence {x_n}$ be a real sequence. Let $\set {x_n: n \in \N}$ admit an infimum. Then the '''infimum''' of $\sequence {x_n}$) is defined as: :$\map \inf {\sequence {x_n} } = \map \inf {\set {x_n: n \in \N} }$
Definition:Supremum Metric/Bounded Real Sequences
Let $A$ be the set of all bounded real sequences. Let $d: A \times A \to \R$ be the function defined as: :$\displaystyle \forall \left\langle{x_i}\right\rangle, \left\langle{y_i}\right\rangle \in A: d \left({\left\langle{x_i}\right\rangle, \left\langle{y_i}\right\rangle}\right) := \sup_{n \mathop \in \N} \left\vert{x_n - y_n}\right\vert$ where $\sup$ denotes the supremum. $d$ is known as the '''supremum metric''' on $A$.
Definition:Supremum Metric/Bounded Continuous Mappings
Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ be metric spaces. Let $A$ be the set of all continuous mappings $f: M_1 \to M_2$ which are also bounded. Let $d: A \times A \to \R$ be the function defined as: :$\displaystyle \forall f, g \in A: d \left({f, g}\right) := \sup_{x \mathop \in A_1} d_2 \left({f \left({x}\right), g \left({x}\right)}\right)$ where $\sup$ denotes the supremum. $d$ is known as the '''supremum metric''' on $A$.
Definition:Supremum Metric/Differentiability Class
Let $\left[{a \,.\,.\, b}\right] \subseteq \R$ be a closed real interval. Let $r \in \N$ be a natural number. Let $\mathscr D^r \left[{a \,.\,.\, b}\right]$ be the set of all continuous functions $f: \left[{a \,.\,.\, b}\right] \to \R$ which are of differentiability class $r$. Let $d: \mathscr D^r \left[{a \,.\,.\, b}\right] \times \mathscr D^r \left[{a \,.\,.\, b}\right] \to \R$ be the function defined as: :$\displaystyle \forall f, g \in \mathscr D^r \left[{a \,.\,.\, b}\right]: d \left({f, g}\right) := \sup_{\substack {x \mathop \in \left[{a \,.\,.\, b}\right] \\ i \in \left\{ {0, 1, 2, \ldots, r}\right\} } } \left\vert{f^{\left({i}\right)} \left({x}\right) - g^{\left({i}\right)} \left({x}\right)}\right\vert$ where: : $f^{\left({i}\right)}$ denotes the $i$th derivative of $f$ : $f^{\left({0}\right)}$ denotes $f$ : $\sup$ denotes the supremum. $d$ is known as the '''supremum metric''' on $\mathscr D^r \left[{a \,.\,.\, b}\right]$.
Definition:Lp Metric
=== Closed Real Interval === {{:Definition:Lp Metric/Closed Real Interval}}
Definition:Lp Metric/Closed Real Interval
Let $S$ be the set of all real functions which are continuous on the closed interval $\left[{a \,.\,.\, b}\right]$. Let $p \in \R_{\ge 1}$. Let the real-valued function $d: S \times S \to \R$ be defined as: :$\displaystyle \forall f, g \in S: d \left({f, g}\right) := \left({\int_a^b \left\vert{f \left({t}\right) - g \left({t}\right)}\right\vert^p \ \mathrm d t}\right)^{\frac 1 p}$ Then $d$ is the '''$L^p$ metric''' on $\left[{a \,.\,.\, b}\right]$.
Definition:P-Sequence Metric/Real Sequences
Let $d_p: A \times A: \to \R$ be the real-valued function defined as: :$\displaystyle \forall x = \left\langle{x_i}\right\rangle, y = \left\langle{y_i}\right\rangle \in A: d_p \left({x, y}\right) := \left({\sum_{k \mathop \ge 0} \left\vert{x_k - y_k}\right\vert^p}\right)^{\frac 1 p}$ The metric space $\left({A, d_p}\right)$ is the '''$p$-sequence space on $\R$''' and is denoted $\ell^p$.
Definition:P-Sequence Metric
=== Real Sequences === {{:Definition:P-Sequence Metric/Real Sequences}}
Definition:P-adic Valuation/Integers
The '''$p$-adic valuation (on $\Z$)''' is the mapping $\nu_p^\Z: \Z \to \N \cup \set {+\infty}$ defined as: :$\map {\nu_p^\Z} n := \begin {cases} +\infty & : n = 0 \\ \sup \set {v \in \N: p^v \divides n} & : n \ne 0 \end{cases}$ where: :$\sup$ denotes supremum :$p^v \divides n$ expresses that $p^v$ divides $n$.