hcju/proofwikips huggingface dataset:
Definition:R-Algebraic Structure Homomorphism
Let $R$ be a ring. Let $\struct {S, \ast_1, \ast_2, \ldots, \ast_n, \circ}_R$ and $\struct {T, \odot_1, \odot_2, \ldots, \odot_n, \otimes}_R$ be $R$-algebraic structures. Let $\phi: S \to T$ be a mapping. Then $\phi$ is an '''$R$-algebraic structure homomorphism''' : :$(1): \quad \forall k \in \closedint 1 n: \forall x, y \in S: \map \phi {x \ast_k y} = \map \phi x \odot_k \map \phi y$ :$(2): \quad \forall x \in S: \forall \lambda \in R: \map \phi {\lambda \circ x} = \lambda \otimes \map \phi x$ where $\closedint 1 n = \set {1, 2, \ldots, n}$ denotes an integer interval. Note that this definition also applies to modules and vector spaces.
Definition:Ring Monomorphism
Let $\struct {R, +, \circ}$ and $\struct {S, \oplus, *}$ be rings. Let $\phi: R \to S$ be a (ring) homomorphism. Then $\phi$ is a ring monomorphism  $\phi$ is an injection.
Definition:Group Monomorphism
Let $\struct {G, \circ}$ and $\struct {H, *}$ be groups. Let $\phi: G \to H$ be a (group) homomorphism. Then $\phi$ is a group monomorphism  $\phi$ is an injection.
Definition:Ring Epimorphism
Let $\struct {R, +, \circ}$ and $\struct {S, \oplus, *}$ be rings. Let $\phi: R \to S$ be a (ring) homomorphism. Then $\phi$ is a ring epimorphism  $\phi$ is a surjection.
Definition:Totally Ordered Field
Let $\struct {F, +, \circ, \preceq}$ be an ordered ring. Let $\struct {F, +, \circ}$ be a field. Let the ordering $\preceq$ be a total ordering. Then $\struct {F, +, \circ, \preceq}$ is a '''totally ordered field'''.
Definition:Group Epimorphism
Let $\struct {G, \circ}$ and $\struct {H, *}$ be groups. Let $\phi: G \to H$ be a (group) homomorphism. Then $\phi$ is a group epimorphism  $\phi$ is a surjection.
Definition:F-Homomorphism
Let $R, S$ be rings with unity. Let $F$ be a subfield of both $R$ and $S$. Then a ring homomorphism $\varphi: R \to S$ is called an '''$F$-homomorphism''' if: :$\forall a \in F: \map \phi a = a$ That is, $\phi \restriction_F = I_F$ where: :$\phi \restriction_F$ is the restriction of $\phi$ to $F$ :$I_F$ is the identity mapping on $F$.
Definition:Homomorphism (Graph Theory)
Let $G = \struct {\map V G, \map E G}$ and $H = \struct {\map V H, \map E H}$ be graphs. Let there exist a mapping $F: \map V G \to \map V H$ such that: :for each edge $\set {u, v} \in \map E G$ :there exists an edge $\set {\map F u, \map F v} \in \map E H$. Then $G$ and $H$ are '''homomorphic'''. The mapping $F$ is called a '''homomorphism''' from $G$ to $H$.
Definition:Spanning Subgraph
Let $G$ be a graph. A '''spanning subgraph for $G$''' is a subgraph of $G$ which contains every vertex of $G$. This is also called a '''factor of $G$'''. {{questionable|Should it not also be specified that $G$ should be connected, and so should such a subgraph?}}
Definition:Induced Subgraph
Let $G = \struct {\map V G, \map E G}$ be a graph. Let $V' \subseteq V$ be a subset of vertices of $G$. The '''subgraph of $G$ induced by $V'$''' is the subgraph $G' = \struct {\map {V'} {G'}, \map {E'} {G'} }$ of $G$ such that: :$G'$ has the vertex set $V'$ :For all $u, v \in V'$, $e = u v \in \map E G$  $e \in \map {E'} {G'}$. That is, it contains all the edges of $G$ that connect elements of the given subset of the vertex set of $G$, and only those edges.
Definition:Universal Graph
Let $\Bbb G$ be a set of graphs. A '''universal graph in $\Bbb G$''' is a graph $K$ in which every element of $\Bbb G$ can be embedded in $K$. Note that $K$ need not itself be an element of $\Bbb G$.  psjyq6jsu0vkx9s97sgljejl6sxgyxr
Definition:Undirected Graph
An '''undirected graph''' is a graph whose edge set considered as a relation is symmetric. That is, if $G$ is an '''undirected graph''' and $e = u v$ is an edge in $G$, then $v u$ is also an edge in $G$. In other words, if you can "go from $u$ to $v$", you can also "go from $v$ to $u$".
Definition:Acyclic Graph
An '''acyclic graph''' is a graph or digraph with no cycles. An acyclic connected undirected graph is a tree. An acyclic disconnected undirected graph is a forest.
Definition:Girth
Let $G$ be a graph. The '''girth''' of $G$ is the smallest length of any cycle in $G$. An acyclic graph is defined as having a girth of infinity.
Definition:Circumference (Graph Theory)
Let $G$ be a graph. The '''circumference''' of $G$ is the longest length of any cycle in $G$. An acyclic graph is defined as having a circumference of infinity.
Definition:Strict Ordering
=== Definition 1 === {{:Definition:Strict Ordering/Asymmetric and Transitive}} === Definition 2 === {{:Definition:Strict Ordering/Antireflexive and Transitive}}
Definition:Strict Partial Ordering
Let $\left({S, \prec}\right)$ be a relational structure. Let $\prec$ be a strict ordering. Then $\prec$ is a '''strict partial ordering''' on $S$ iff $\prec$ is not connected. That is, iff $\left({S, \prec}\right)$ has at least one pair which is non-comparable: :$\exists x, y \in S: x \not \prec y \land y \not \prec x$
Definition:Strict Total Ordering
Let $\struct {S, \prec}$ be a relational structure. Let $\prec$ be a strict ordering. Then $\prec$ is a '''strict total ordering''' on $S$  $\struct {S, \prec}$ has no non-comparable pairs: :$\forall x, y \in S: x \ne y \implies x \prec y \lor y \prec x$ That is,  $\prec$ is connected.
Definition:Strict Weak Ordering
A '''strict weak ordering''' on a set $S$ is a relation $\mathcal R$ such that: :$(1): \quad \mathcal R$ is a strict partial ordering :$(2): \quad$ The incomparability relation $\mathcal R'$ defined as: ::::$a \mathrel {\mathcal R'} b  := \neg \left({a \mathrel {\mathcal R} b}\right) \land \neg \left({b \mathrel {\mathcal R} a}\right)$ :::is transitive.
Definition:Heronian Mean
The '''Heronian mean''' of two numbers $x$ and $y$ is defined as: :$\displaystyle H = \frac {x + \sqrt{xy} + y} 3$ It can also be defined as: :$\displaystyle H = \frac 2 3 \left({\frac {x + y} 2}\right) + \frac 1 3 \sqrt {x y}$ Thus it is seen to be a weighted mean of their arithmetic mean and geometric mean. {{NamedforDef|Heron of Alexandria|cat=Heron}}
Definition:Weighted Mean
Let $S = \sequence {x_1, x_2, \ldots, x_n}$ be a sequence of real numbers. Let $\map W x$ be a weight function to be applied to the terms of $S$. The '''weighted mean''' of $S$ is defined as: :$\bar x := \dfrac {\displaystyle \sum_{i \mathop = 1}^n \map W {x_i} x_i} {\displaystyle \sum_{i \mathop = 1}^n \map W {x_i} }$ This means that elements of $S$ with a larger weight contribute more to the '''weighted mean''' than those with a smaller weight. If we write: :$\forall i: 1 \le i \le n: w_i = \map W {x_i}$ we can write this '''weighted mean''' as: :$\bar x := \dfrac {w_1 x_1 + w_2 x_2 + \cdots + w_n x_n} {w_1 + w_2 + \cdots + w_n}$ From the definition of the weight function, none of the weights can be negative. While some of the weights may be zero, not ''all'' of them can, otherwise we would be dividing by zero. === Normalized Weighted Mean === {{:Definition:Weighted Mean/Normalized}}
Definition:Reflexive Closure
=== Definition 1 === {{:Definition:Reflexive Closure/Union with Diagonal}} === Definition 2 === {{:Definition:Reflexive Closure/Smallest Reflexive Superset}} === Definition 3 === {{:Definition:Reflexive Closure/Intersection of Reflexive Supersets}}
Definition:Reflexive Reduction
Let $\RR$ be a relation on a set $S$. The '''reflexive reduction''' of $\RR$ is denoted $\RR^\ne$, and is defined as: :$\RR^\ne := \RR \setminus \set {\tuple {x, x}: x \in S}$
Definition:Transitive Closure (Relation Theory)
=== Definition 1 === {{:Definition:Transitive Closure (Relation Theory)/Smallest Transitive Superset}} === Definition 2 === {{:Definition:Transitive Closure (Relation Theory)/Intersection of Transitive Supersets}} === Definition 3 === {{:Definition:Transitive Closure (Relation Theory)/Finite Chain}} === Definition 4 === {{:Definition:Transitive Closure (Relation Theory)/Union of Compositions}}
Definition:Symmetric Closure
Let $\mathcal R$ be a relation on a set $S$. === Definition 1 === {{:Definition:Symmetric Closure/Definition 1}} === Definition 2 === {{:Definition:Symmetric Closure/Definition 2}}
Definition:Reflexive Transitive Closure
Let $\mathcal R$ be a relation on a set $S$. === Smallest Reflexive Transitive Superset === {{:Definition:Reflexive Transitive Closure/Smallest Reflexive Transitive Superset}} === Reflexive Closure of Transitive Closure === {{:Definition:Reflexive Transitive Closure/Reflexive Closure of Transitive Closure}} === Transitive Closure of Reflexive Closure === {{:Definition:Reflexive Transitive Closure/Transitive Closure of Reflexive Closure}}
Definition:Continuous Real Function
=== Continuity at a Point === {{Definition:Continuous Real Function/Point}} === Continuous Everywhere === {{Definition:Continuous Real Function/Everywhere}} === Continuity on a Subset of Domain === {{Definition:Continuous Real Function/Subset}}
Definition:Continuous Mapping (Topology)
Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be topological spaces. Let $f: S_1 \to S_2$ be a mapping from $S_1$ to $S_2$. == Continuous at a Point == {{Definition:Continuous Mapping (Topology)/Point}} == Continuous on a Set == {{Definition:Continuous Mapping (Topology)/Set}} == Continuous Everywhere == {{Definition:Continuous Mapping (Topology)/Everywhere}}
Definition:Discontinuous Mapping
=== Discontinuous Real Function === {{:Definition:Discontinuous Mapping/Real Function}} === Discontinuous Topological Space === {{:Definition:Discontinuous Mapping/Topological Space}}
Definition:Continuous Mapping (Metric Space)
Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be metric spaces. Let $f: A_1 \to A_2$ be a mapping from $A_1$ to $A_2$. Let $a \in A_1$ be a point in $A_1$. === Continuous at a Point === {{Definition:Continuous Mapping (Metric Space)/Point}} === Continuous on a Space === {{Definition:Continuous Mapping (Metric Space)/Space}}
Definition:Universe (Set Theory)
Sets are considered to be subsets of some large '''universal set''', also called the '''universe'''. Exactly what this '''universe''' is will vary depending on the subject and context. When discussing particular sets, it should be made clear just what that '''universe''' is. However, note that from There Exists No Universal Set, this '''universe''' cannot be ''everything that there is''. The traditional symbol used to signify the '''universe''' is $\mathfrak A$. However, this is old-fashioned and inconvenient, so some newer texts have taken to using $\mathbb U$ or just $U$ instead. With this notation, this definition can be put into symbols as: :$\forall S: S \subseteq \mathbb U$ The use of $\mathbb U$ or a variant is not universal: some sources use $X$.
Definition:Ring of Sets
{{:Definition:Ring of Sets/Definition 1}}
Definition:Ring (Abstract Algebra)/Ring Axioms
A ring is an algebraic structure $\struct {R, *, \circ}$, on which are defined two binary operations $\circ$ and $*$, which satisfy the following conditions: {{begin-axiom}} {{axiom | n = \text A 0         | q = \forall a, b \in R         | m = a * b \in R         | lc= Closure under addition }} {{axiom | n = \text A 1         | q = \forall a, b, c \in R         | m = \paren {a * b} * c = a * \paren {b * c}         | lc= Associativity of addition }} {{axiom | n = \text A 2         | q = \forall a, b \in R         | m = a * b = b * a         | lc= Commutativity of addition }} {{axiom | n = \text A 3         | q = \exists 0_R \in R: \forall a \in R         | m = a * 0_R = a = 0_R * a         | lc= Identity element for addition: the zero }} {{axiom | n = \text A 4         | q = \forall a \in R: \exists a' \in R         | m = a * a' = 0_R = a' * a         | lc= Inverse elements for addition: negative elements }} {{axiom | n = \text M 0         | q = \forall a, b \in R         | m = a \circ b \in R         | lc= Closure under product }} {{axiom | n = \text M 1         | q = \forall a, b, c \in R         | m = \paren {a \circ b} \circ c = a \circ \paren {b \circ c}         | lc= Associativity of product }} {{axiom | n = \text D         | q = \forall a, b, c \in R         | m = a \circ \paren {b * c} = \paren {a \circ b} * \paren {a \circ c}         | lc= Product is distributive over addition }} {{axiom | m = \paren {a * b} \circ c = \paren {a \circ c} * \paren {b \circ c} }} {{end-axiom}} These criteria are called the '''ring axioms'''.
Definition:Group Axioms
A group is an algebraic structure $\struct {G, \circ}$ which satisfies the following four conditions: {{begin-axiom}} {{axiom | n = \text G 0         | lc= Closure         | q = \forall a, b \in G         | m = a \circ b \in G }} {{axiom | n = \text G 1         | lc= Associativity         | q = \forall a, b, c \in G         | m = a \circ \paren {b \circ c} = \paren {a \circ b} \circ c }} {{axiom | n = \text G 2         | lc= Identity         | q = \exists e \in G: \forall a \in G         | m = e \circ a = a = a \circ e }} {{axiom | n = \text G 3         | lc= Inverse         | q = \forall a \in G: \exists b \in G         | m = a \circ b = e = b \circ a }} {{end-axiom}} These four stipulations are called the '''group axioms'''.
Definition:Field Axioms
The properties of a field are as follows. For a given field $\struct {F, +, \circ}$, these statements hold true: {{begin-axiom}} {{axiom | n = \text A 0         | lc= Closure under addition         | q = \forall x, y \in F         | m = x + y \in F }} {{axiom | n = \text A 1         | lc= Associativity of addition         | q = \forall x, y, z \in F         | m = \paren {x + y} + z = x + \paren {y + z} }} {{axiom | n = \text A 2         | lc= Commutativity of addition         | q = \forall x, y \in F         | m = x + y = y + x }} {{axiom | n = \text A 3         | lc= Identity element for addition         | q = \exists 0_F \in F: \forall x \in F         | m = x + 0_F = x = 0_F + x         | rc= $0_F$ is called the zero }} {{axiom | n = \text A 4         | lc= Inverse elements for addition         | q = \forall x: \exists x' \in F         | m = x + x' = 0_F = x' + x         | rc= $x'$ is called a negative element }} {{axiom | n = \text M 0         | lc= Closure under product         | q = \forall x, y \in F         | m = x \circ y \in F }} {{axiom | n = \text M 1         | lc= Associativity of product         | q = \forall x, y, z \in F         | m = \paren {x \circ y} \circ z = x \circ \paren {y \circ z} }} {{axiom | n = \text M 2         | lc= Commutativity of product         | q = \forall x, y \in F         | m = x \circ y = y \circ x }} {{axiom | n = \text M 3         | lc= Identity element for product         | q = \exists 1_F \in F, 1_F \ne 0_F: \forall x \in F         | m = x \circ 1_F = x = 1_F \circ x         | rc= $1_F$ is called the unity }} {{axiom | n = \text M 4         | lc= Inverse elements for product         | q = \forall x \in F^*: \exists x^{-1} \in F^*         | m = x \circ x^{-1} = 1_F = x^{-1} \circ x }} {{axiom | n = \text D         | lc= Product is distributive over addition         | q = \forall x, y, z \in F         | m = x \circ \paren {y + z} = \paren {x \circ y} + \paren {x \circ z} }} {{end-axiom}} These are called the '''field axioms'''.
Definition:Unit of System of Sets
Let $\mathcal S$ be a system of sets. Let $U \in \mathcal S$ such that: :$\forall A \in \mathcal S: A \cap U = A$ Then $U$ is '''the unit of $\mathcal S$'''. Note that, for a given system of sets, if $U$ exists then it is unique.  72s55wkujcn21d1czz8k1faomsc1q94
Definition:Semiring of Sets
A '''semiring of sets''' or '''semi-ring of sets''' is a system of sets $\SS$, subject to: :$(1):\quad \O \in \SS$ :$(2):\quad A, B \in \SS \implies A \cap B \in \SS$; that is, $\SS$ is $\cap$-stable :$(3):\quad$ If $A, A_1 \in \SS$ such that $A_1 \subseteq A$, then there exists a finite sequence $A_2, A_3, \ldots, A_n \in \SS$ such that: ::$(3a):\quad \displaystyle A = \bigcup_{k \mathop = 1}^n A_k$ ::$(3b):\quad$ The $A_k$ are pairwise disjoint Alternatively, criterion $(3)$ can be replaced by: :$(3'):\quad$ If $A, B \in \SS$, then there exists a finite sequence of pairwise disjoint sets $A_1, A_2, \ldots, A_n \in \SS$ such that $\displaystyle A \setminus B = \bigcup_{k \mathop = 1}^n A_k$.
Definition:Continuous Complex Function
As the complex plane is a metric space, the same definition of continuity applies to complex functions as to metric spaces.
Definition:Triangle (Geometry)/Isosceles
An '''isosceles triangle''' is a triangle in which two sides are the same length. :300px
Definition:Collinear
Three (or more) points are '''collinear'''  there exists a straight line that passes through all the points.
Definition:Triangle (Geometry)/Right-Angled
A '''right-angled triangle''' is a triangle in which one of the vertices is a right angle. :450px Note that in order to emphasise the nature of the right angle in such a triangle, a small square is usually drawn inside it.
Definition:Chord
=== Chord of a Circle === :300px {{:Definition:Chord of Circle}} === Chord of a Polygon === {{:Definition:Chord of Polygon}} === Chord of a Parabola === {{:Definition:Chord of Parabola}} === Chord of a general Curve === {{:Definition:Chord of Curve}} There are other sorts of chords, still to be documented.
Definition:Row Matrix
A '''row matrix''' is a $1 \times n$ matrix: :$\mathbf R = \begin{bmatrix} r_{1 1} & r_{1 2} & \cdots & r_{1 n} \end{bmatrix}$ That is, it is a matrix with only one row.
Definition:Complements of Parallelograms
Let $ABDC$ and $EFHG$ be two parallelograms with the same angles, which share a diagonal, such that $ABDC \cap EFHG \ne \O$. :400px Then the two parallelograms $CIGK$ and $BJHL$ are known as the '''complements''' of the parallelograms $ABDC$ and $EFHG$.
Definition:Diameter of Parallelogram
Let $ABCD$ be a parallelogram: :400px The '''diameters''' of $ABCD$ are the lines $AC$ and $BD$ joining their opposite vertices.
Definition:Conjugate Angles
The '''conjugate''' of an angle $\theta$ is the angle $\phi$ such that: :$\theta + \phi = 2 \pi$ where $\theta$ and $\pi$ are expressed in radians. That is, it is the angle that makes the given angle equal to a full angle. Equivalently, the '''conjugate''' of an angle $\theta$ is the angle $\phi$ such that: :$\theta + \phi = 360 \degrees$ where $\theta$ and $\pi$ are expressed in degrees. Thus, '''conjugate angles''' are two angles whose measures add up to the measure of $4$ right angles. That is, their measurements add up to $360$ degrees or $2 \pi$ radians.
Definition:Porism
The '''porism''' is a term whose precise definition is difficult to pin down. It arises from the works of {{AuthorRef|Euclid}}, and appears to have two different meanings: # A result which follows directly, or in passing, during the course of proving something else; that is, a corollary. # A proposition that asserts that there may be conditions under which a particular problem has either no solutions or an indeterminate number of them.
Definition:Perpendicular Bisector
Let $AB$ be a line segment. The '''perpendicular bisector''' of $AB$ is the straight line which: :is perpendicular to $AB$ :passes through the point which bisects $AB$. :300px
Definition:Concentric Circles
Circles are said to be '''concentric''' if they both have the same point as the center. :300px
Definition:Baire Space (Topology)
Let $T = \left({S, \tau}\right)$ be a topological space. === Definition 1 === {{:Definition:Baire Space (Topology)/Definition 1}} === Definition 2 === {{:Definition:Baire Space (Topology)/Definition 2}} === Definition 3 === {{:Definition:Baire Space (Topology)/Definition 3}} === Definition 4 === {{:Definition:Baire Space (Topology)/Definition 4}}
Definition:Sequence/Infinite Sequence
An '''infinite sequence''' is a sequence whose domain is infinite.
Definition:Baire Space (Set Theory)
The '''Baire space''' $\mathbf B$ is defined as the set of all infinite sequences of natural numbers. It can also be defined as the Cartesian product of a countably infinite number of copies of the set of natural numbers.
Definition:Lindelöf Space
A topological space $T = \struct {S, \tau}$ is a '''Lindelöf space'''  every open cover of $S$ has a countable subcover.
Definition:Concyclic Points
A set of $4$ or more points $S = \set {P_1, P_2, \ldots, P_n}$ is '''concyclic''' if they all lie on the circumference of a circle. :320px In the above diagram, points $A, B, C, D$ are '''concyclic''', as they all lie on the circumference of the circle centered at $O$.
Definition:Ramsey Number
Ramsey's Theorem states that in any coloring of the edges of a sufficiently large complete graph, one will find monochromatic complete subgraphs. More precisely, for any given number of colors $c$, and any given integers $n_1, \ldots, n_c$, there is a number $\map R {n_1, \ldots, n_c}$ such that: :if the edges of a complete graph of order $\map R {n_1, \ldots, n_c}$ are colored with $c$ different colours, then for some $i$ between $1$ and $c$, it must contain a complete subgraph of order $n_i$ whose edges are all color $i$. This number $\map R {n_1, \ldots, n_c}$ is called the '''Ramsey number''' for $n_1, \ldots, n_c$. {{NamedforDef|Frank Plumpton Ramsey|cat = Ramsey}}
Definition:Triangle (Graph Theory)
The complete graph $K_3$ of order $3$ is called a '''triangle'''. :250px
Definition:Claw
The complete bipartite graph $K_{1, 3}$ is known as a '''claw'''. :250px
Definition:Group Product
=== Group Law === {{:Definition:Group Product/Group Law}} === Product Element === {{:Definition:Group Product/Product Element}}
Definition:Coloring
=== Vertex Coloring === {{:Definition:Coloring/Vertex Coloring}} === Edge Coloring === {{:Definition:Coloring/Edge Coloring}}
Definition:Incident (Graph Theory)/Planar Graph
Let $G = \left({V, E}\right)$ be a planar graph: Then a face of $G$ is '''incident to''' an edge $e$ of $G$ if $e$ is one of those which surrounds the face. Similarly, a face of $G$ is '''incident to''' a vertex $v$ of $G$ if $v$ is at the end of one of those incident edges. In the above graph, for example, the face $ABHC$ is incident to: : the edges $AB, BH, HC, CA$ : the vertices $A, B, H, C$.
Definition:Monochromatic Graph
A '''monochromatic graph''' is a colored graph (either vertex-colored or edge-colored, depending on the context) in which each of the vertices or edges is assigned the same color.
Definition:Indeterminate Equation
An '''indeterminate equation''' is an equation for which there is more than one unknown. Hence, unless other restrictions are imposed, there may be an infinite number of solutions.
Definition:Equation
An '''equation''' is a mathematical statement that states that two expressions are equal. For expressions $A$ and $B$, this would usually be portrayed: :$A = B$ where $A$ is known as the  and $B$ the .
Definition:Expression
An '''expression''' is a well-formed word of mathematical symbols which together can be considered as a single object. {{expand|Match to natural language use of the word}}
Definition:Algebraic Number
An algebraic number is an algebraic element of the field extension $\C / \Q$. That is, it is a complex number that is a root of a polynomial with rational coefficients.
Definition:Cyclic Quadrilateral
A '''cyclic quadrilateral''' is a quadrilateral which can be circumscribed: :300px
Definition:Recursive Sequence
A '''recursive sequence''' is a sequence where each term is defined from earlier terms in the sequence. A famous example of a recursive sequence is the Fibonacci sequence: :$F_n = F_{n-1} + F_{n-2}$ The equation which defines this sequence is called a '''recurrence relation''' or '''difference equation'''. === Initial Terms === {{:Definition:Recursive Sequence/Initial Terms}}
Definition:Lebesgue Space
Let $\struct {X, \Sigma, \mu}$ be a measure space, and let $p \in \R$, $p \ge 1$. The '''(real) Lebesgue $p$-space of $\mu$''' is defined as: :$\map {\mathcal L^p} \mu := \set {f: X \to \R: f \in \map {\mathcal M} \Sigma, \displaystyle \int \size f^p \rd \mu < \infty}$ where $\map {\mathcal M} \Sigma$ denotes the space of $\Sigma$-measurable functions. On $\map {\mathcal L^p} \mu$, we can introduce the $p$-seminorm $\norm {\, \cdot \,}_p$ by: :$\forall f \in \mathcal L^p: \norm f_p := \paren {\displaystyle \int \size f^p \rd \mu}^{1 / p}$ Next, define the equivalence $\sim$ by: :$f \sim g \iff \norm {f - g}_p = 0$ The resulting quotient space: :$\map {L^p} \mu := \map {\mathcal L^p} \mu / \sim$ is also called '''(real) Lebesgue $p$-space'''.