Definition:Membership Relation

Let $S$ be a set. Let $\powerset S$ denote the power set of $S$. Let $\mathcal R \subseteq S \times \powerset S$ be the relation defined as: :$\tuple {x, A} \in \mathcal R \iff x \in A$ Thus $\mathcal R$ is the relation between elements of $S$ and subsets of $S$ expressing '''membership'''.


Definition:Image (Set Theory)/Relation/Element

Let $s \in S$. The '''image of $s$ by''' (or '''under''') '''$\RR$''' is defined as: :$\map \RR s := \set {t \in T: \tuple {s, t} \in \RR}$ That is, $\map \RR s$ is the set of all elements of the codomain of $\RR$ related to $s$ by $\RR$.


Definition:Image (Set Theory)/Relation

=== Image of a Relation === {{:Definition:Image (Set Theory)/Relation/Relation}} === Image of an Element === {{:Definition:Image (Set Theory)/Relation/Element}} === Image of a Subset === {{:Definition:Image (Set Theory)/Relation/Subset}}


Definition:Image (Set Theory)/Mapping

{{transclude:Definition:Image (Set Theory)/Mapping/Mapping   |section = tc   |title = Image of a Mapping   |link = true   |header = 3   |increase = 1 }} === Image of an Element === {{:Definition:Image (Set Theory)/Mapping/Element}} === Image of a Subset === {{:Definition:Image (Set Theory)/Mapping/Subset}}


Definition:Image (Set Theory)/Mapping/Element

Let $s \in S$. The '''image of $s$ (under $f$)''' is defined as: :$\Img s = \map f s = \displaystyle \bigcup \set {t \in T: \tuple {s, t} \in f}$ That is, $\map f s$ is the element of the codomain of $f$ related to $s$ by $f$.


Definition:Image (Set Theory)/Mapping/Subset

Let $f: S \to T$ be a mapping. Let $X \subseteq S$ be a subset of $S$.


Definition:Tree (Set Theory)

Let $\struct {T, \preceq}$ be an ordered set. Let $\struct {T, \preceq}$ be such that for every $t \in T$, the lower closure of $t$: :$t^\preceq := \set {s \in T: s \preceq t}$  is well-ordered by $\preceq$. Then $\struct {T, \preceq}$ is a '''tree'''. === Branch === A '''branch''' of a tree $T$ is a maximal chain in $T$. === Subtree === A '''subtree''' of a tree $\struct {T, \preceq}$ is an ordered subset $\struct {S, \preceq}$ with the property that: :for every $\forall s \in S: \forall t \in T: t \preceq s \implies t \in S$ Category:Definitions/Set Theory djo29gkqm3npxpyyu1pnk44i2estlzc


Definition:Basic Open Set

Let $T = \left({S, \tau}\right)$ be a topological space. Let $\mathcal B \subseteq \tau$ be a basis for $T$. Let $U \in \mathcal B$. Then $U$ is a '''basic open set''' of $T$. That is, a '''basic open set''' of a topology is an open set of that topology which is an element of a basis for that topology. The basis itself needs to be specified for this definition to make sense. Category:Definitions/Topology Category:Definitions/Topological Bases p16ojsxmpoxw5t9h7f3eos4sdcjv7po


Definition:Sine Integral Function

The '''sine integral function''' is the real function $\Si: \R \to \R$ defined as: :$\map \Si x = \begin{cases} \displaystyle \int_{t \mathop \to 0}^{t \mathop = x} \frac {\sin t} t \rd t & : x \ne 0 \\ \vphantom x \\ 0 & : x = 0 \\ \end{cases}$


Definition:Greatest

=== Ordered Set === {{:Definition:Greatest/Ordered Set}} === Greatest Set === {{:Definition:Greatest Set by Set Inclusion}} === Mapping === {{:Definition:Greatest/Mapping}}


Definition:Smallest

=== Ordered Set === {{:Definition:Smallest/Ordered Set}} === Smallest Set === {{:Definition:Smallest Set by Set Inclusion}} === Mapping === {{:Definition:Smallest/Mapping}}


Definition:Arc Length

Let $y = \map f x$ be a real function which is: : continuous on the closed interval $\closedint a b$ and: : continuously differentiable on the open interval $\openint a b$. The '''arc length''' $s$ of $f$ between $a$ and $b$ is defined as:  :$s := \displaystyle \int_a^b \sqrt {1 + \paren {\frac {\d y} {\d x} }^2} \rd x$


Definition:Greatest/Ordered Set

Let $\struct {S, \preceq}$ be an ordered set. An element $x \in S$ is '''the greatest element (of $S$)''' {{iff}}: :$\forall y \in S: y \preceq x$ That is, every element of $S$ '''precedes, or is equal to,''' $x$. The Greatest Element is Unique, so calling it '''''the'' greatest element''' is justified. Thus for an element $x$ to be the '''greatest element''', all $y \in S$ must be comparable to $x$.


Definition:Smallest/Ordered Set

Let $\struct {S, \preceq}$ be an ordered set. An element $x \in S$ is '''the smallest element''' {{iff}}: :$\forall y \in S: x \preceq y$ That is, $x$ '''strictly precedes, or is equal to,''' every element of $S$. The Smallest Element is Unique, so calling it '''''the'' smallest element''' is justified. The '''smallest element''' of $S$ is denoted $\min S$. For an element to be the '''smallest element''', all $y \in S$ must be comparable to $x$.


Definition:Radicand

Let $\sqrt[n] P$ denote the $n$th root of an expression. Then $P$ is the '''radicand''' of $\sqrt[n] P$. That is, it is '''the object whose $n$th root is to be taken'''.


Definition:Finite Ring

A finite ring is a ring $\left({R, +, \circ}\right)$ such that $R$ is a finite set.


Definition:Removable Discontinuity

=== Real Function === {{Definition:Removable Discontinuity of Real Function}} {{definition wanted|discuss functions of $n$ variables, complex functions}}


Definition:Generalized Sum

Let $\left({G, +}\right)$ be a commutative topological semigroup. Let $\left({g_i}\right)_{i \in I}$ be an indexed subset of $G$. Consider the set $\mathcal F$ of finite subsets of $I$. Let $\subseteq$ denote the subset relation on $\mathcal F$. By virtue of Finite Subsets form Directed Set, $\left({\mathcal F, \subseteq}\right)$ is a directed set. Define the net: :$\phi: \mathcal F \to G$ by: : $\displaystyle \phi \left({F}\right) = \sum_{i \mathop \in F} g_i$ Then $\phi$ is denoted: : $\displaystyle \sum \left\{{g_i: i \in I}\right\}$ and referred to as a '''generalized sum'''.


Definition:Dimension (Hilbert Space)

Let $H$ be a Hilbert space, and let $E$ be a basis of $H$. Then the '''dimension''' $\dim H$ of $H$ is defined as $\card E$, the cardinality of $E$.


Definition:Successor Set

Let $S$ be a set. The '''successor (set) of $S$''' is defined and denoted: :$S^+ := S \cup \set S$


Definition:Infinite Successor Set

Let $S$ be a set. Then $S$ is an '''infinite successor set''' {{iff}}: :$\O \in S$ :$x \in S \implies x^+ \in S$ where $\O$ is the empty set, and $x^+$ denotes the successor set of $x$. === Axiomatic Set Theory === The concept of '''infinite successor set''' is axiomatised in the Axiom of Infinity in Zermelo-Fraenkel set theory: :$\exists x: \paren {\paren {\exists y: y \in x \land \forall z: \neg \paren {z \in y} } \land \forall u: u \in x \implies \paren {u \cup \set u \in x} }$


Definition:Undefined Term

For a definition to not be circular, the definer must use already defined terms.  However, this process cannot go on indefinitely. If we were to insist on ''everything'' being defined only using previously defined terms, we would enter an infinite regress. Concepts that are not defined in terms of previously defined concepts are called '''undefined terms'''. An undefined term is frequently explained by using an ostensive definition: that is, a statement that ''shows'' what something is, rather than ''explains''.


Definition:Power (Algebra)/Natural Number

Let $\N$ denote the natural numbers. For each $m \in \N$, recursively define $e_m: \N \to \N$ to be the mapping: :$e_m \left({n}\right) = \begin{cases} 1 & : n = 0 \\ m \times e_m \left({x}\right) & : n = x + 1 \end{cases}$ where: : $+$ denotes natural number addition. : $\times$ denotes natural number multiplication. $e_m \left({n}\right)$ is then expressed as a binary operation in the form: :$m^n := e_m \left({n}\right)$ and is called '''$m$ to the power of $n$'''.


Definition:Topological Semigroup

Let $\left({S, \circ}\right)$ be a semigroup. On that same underlying set $S$, let $\left({S, \tau}\right)$ be a topological space. Then $\left({S, \circ, \tau}\right)$ is said to be a '''topological semigroup''' if: :$\circ: \left({S, \tau}\right) \times \left({S, \tau}\right) \to \left({S, \tau}\right)$ is a continuous mapping where $\left({S, \tau}\right) \times \left({S, \tau}\right)$ is considered as $S \times S$ with the product topology.


Definition:Isomorphism (Hilbert Spaces)

Let $H, K$ be Hilbert spaces. Denote by $\innerprod \cdot \cdot_H$ and $\innerprod \cdot \cdot_K$ their respective inner products. An '''isomorphism''' between $H$ and $K$ is a map $U: H \to K$, such that: :$(1): \quad U$ is a linear map :$(2): \quad U$ is surjective :$(3): \quad \forall g, h \in H: \innerprod g h_H = \innerprod {U g} {U h}_K$ These three requirements may be summarized by stating that $U$ be a surjective isometry. Furthermore, Surjection that Preserves Inner Product is Linear shows that requirement $(1)$ is superfluous. If such an '''isomorphism''' $U$ exists, $H$ and $K$ are said to be '''isomorphic'''. As the name '''isomorphism''' suggests, Hilbert Space Isomorphism is Equivalence Relation.


Definition:Isometry (Hilbert Spaces)

Let $H, K$ be Hilbert spaces, and denote by $\innerprod \cdot \cdot_H$ and $\innerprod \cdot \cdot_K$ their respective inner products. A linear map $U: H \to K$ is called an '''isometry''' {{iff}}: :$\forall g,h \in H: \innerprod g h_H = \innerprod {U g} {U h}_K$ {{explain|Notation $U g, U h$ may need to be amended so as to make it clear that it means the application of the mapping $U$ on both of $g$ and $h$. If this is indeed what it means (as is probable) then suggest $\map U g$ and $\map U h$.}}


Definition:Unitary Operator

A '''unitary operator''' is an isomorphism from a Hilbert space $H$ to itself.


Definition:Riemann Sum

Let $f$ be a real function defined on the closed interval $\mathbb I = \closedint a b$. Let $\Delta$ be a subdivision of $\mathbb I$. For $1 \le i \le n$: :let $\Delta x_i = x_i - x_{i - 1}$ :let $c_i \in \closedint {x_{i - 1} } {x_i}$. The summation: :$\displaystyle \sum_{i \mathop = 1}^n \map f {c_i} \Delta x_i$ is called a '''Riemann sum''' of $f$ for the subdivision $\Delta$. === Geometric Interpretation === {{stub}}


Definition:Total Relation

Let $\mathcal R \subseteq S \times S$ be a relation on a set $S$. Then $\mathcal R$ is defined as '''total''' {{iff}}: :$\forall a, b \in S: \tuple {a, b} \in \mathcal R \lor \tuple {b, a} \in \mathcal R$ That is, {{iff}} every pair of elements is related (either or both ways round).


Definition:Signed Area

Consider the plane region described by a graph in the $xy$-plane. The '''signed area''' of the region is the area such that: : The area of the graph on or above the $x$-axis is defined as positive : The area of the graph below the $x$-axis is defined as negative. {{explain|As this is a definition which requires the use of spatial relationships, might be good to have a diagram of this.}}


Definition:Initial Segment

Let $\struct {S, \preceq}$ be a well-ordered set. Let $a \in S$. The '''initial segment (of $S$) determined by $a$''' is defined as: :$S_a := \set {b \in S: b \preceq a \land b \ne a}$ which can also be rendered as: :$S_a := \set {b \in S: b \prec a}$ That is, $S_a$ is the set of all elements of $S$ that strictly precede $a$. That is, $S_a$ is the strict lower closure of $a$ (in $S$). By extension, $S_a$ is described as '''an initial segment (of $S$)'''.


Definition:Preordering/Preordered Set

Let $S$ be a set. Let $\precsim$ be a preordering on $S$. Then the relational structure $\left({S, \precsim}\right)$ is called a '''preordered set'''.


Definition:Convergent Net

Let $\left({X, \tau}\right)$ be a topological space, and let $\left({I, \leq}\right)$ be a directed set. Let $\left({x_i}\right)_{i \in I}$ be a net. The net $\left({x_i}\right)$ is said to '''converge to''' $x \in X$, denoted $\displaystyle x_i \to x$ or $\lim x_i = x$, iff: :$\forall U \in \tau: x \in U \implies \exists i_0 \in I: \forall i \ge i_0: x_i \in U$ That is, for every open $U$ with $x \in U$, there exists an $i_0 \in I$ such that forall $i \ge i_0$, $x_i \in U$. If $x_i \to x$, then $x$ is called a '''limit (point) of $\left({x_i}\right)$'''. A net $\left({x_i}\right)_{i \in I}$ is called '''convergent''' if there is an $x \in X$ such that $x_i \to x$. If such an $x$ does not exist, the net is said to be '''divergent'''. === Cluster Point === The net $\left({x_i}\right)$ is said to '''cluster at''' $x \in X$, denoted $\displaystyle x_i \mathop{\longrightarrow}_{\text{cl}} x$, iff: :$\forall U \in \tau, i_0 \in I: x \in U \implies \exists i \ge i_0: x_i \in U$ That is, for every open $U$ with $x \in U$, and for every $i_0 \in I$, there is an $i \ge i_0$ such that $x_i \in U$. If $\displaystyle x_i \mathop{\longrightarrow}_{\text{cl}} x$, then $x$ is called a '''cluster point of $\left({x_i}\right)$'''.


Definition:Precede

Let $\left({S, \preceq}\right)$ be an ordered set. Let $a, b \in S$ such that $a \preceq b$. Then '''$a$ precedes $b$'''.


Definition:Strictly Precede

=== Definition 1 === {{:Definition:Strictly Precede/Definition 1}} === Definition 2 === {{:Definition:Strictly Precede/Definition 2}}


Definition:Strictly Succeed

Let $\left({S, \preceq}\right)$ be an ordered set. Let $a \prec b$. That is, let $a$ strictly precede $b$. Then '''$b$ strictly succeeds $a$'''. This can be expressed symbolically as: :$b \succ a$


Definition:Succeed

Let $\left({S, \preceq}\right)$ be an ordered set. Let $a, b \in S$ such that $a \preceq b$. Then '''$b$ succeeds $a$'''.


Definition:Between

Let $\left({S, \preceq}\right)$ be an ordered set. Let $a, b, c \in S$ such that $a \preceq b$ and $b \preceq c$. Then '''$b$ is between $a$ and $c$'''.


Definition:Strictly Between

Let $\left({S, \preceq}\right)$ be an ordered set. Let $a, b, c \in S$ such that $a \prec b$ and $b \prec c$. That is, such that: :$a \preceq b$ and $a \ne b$ :$b \preceq c$ and $b \ne c$ Then '''$b$ is strictly between $a$ and $c$'''.


Definition:Circular Proof

A '''circular proof''' is a proof $A$ in terms of another proof $B$ such that the truth value of $B$ depends (either directly or indirectly) on the truth value of $A$. A '''circular proof''' is valid, as every statement implies itself.  The problem of a '''circular proof''' is that if the truth of a conclusion is [http://en.wiktionary.org/wiki/dubious dubious], premises that are equally dubious provide no reason to support the conclusion.


Definition:Idempotent Semigroup

An '''idempotent semigroup''' is a semigroup whose operation is idempotent. That is, a semigroup $\struct {S, \circ}$ is '''idempotent''' {{iff}}: :$\forall x \in S: x \circ x = x$


Definition:Commutative Semigroup with respect to Equivalence Relation

Let $C$ be a class. Let $\thickapprox$ be an equivalence relation on $C$. Let $\left({C, \cdot}\right)$ be a large magma. Then $\left({C, \cdot}\right)$ is a '''commutative semigroup with respect to $\thickapprox$''' iff: :it is a semigroup with respect to $\thickapprox$ :$\forall x, y \in C: x \cdot y \thickapprox y \cdot x$


Definition:Bounded Linear Transformation

Let $H, K$ be Hilbert spaces. Let $A: H \to K$ be a linear transformation. Then $A$ is a '''bounded linear transformation''' {{iff}} :$\exists c > 0: \forall h \in H: \left\Vert{A h}\right\Vert_K \le c \left\Vert{h}\right\Vert_H$


Definition:Space of Bounded Linear Transformations

Let $H, K$ be Hilbert spaces. Let $\Bbb F \in \set {\R, \C}$ be the ground field of $K$. Then the '''space of bounded linear transformations from $H$ to $K$''', $\map B {H, K}$, is the set of all bounded linear transformations: :$\map B {H, K} := \set {A: H \to K: A \text{ linear}, \norm A < \infty}$ endowed with pointwise addition and ($\F$)-scalar multiplication. Then $\map B {H, K}$ is a vector space over $\Bbb F$. Furthermore, let $\norm {\,\cdot\,}$ denote the norm on bounded linear transformations. Then $\norm{\,\cdot\,}$ is a norm on $\map B {H, K}$, and it even turns $\map B {H, K}$ into a Banach space. These results are proved in Space of Bounded Linear Transformations is Banach Space. === Space of Bounded Linear Operators === When $H = K$, one denotes $\map B H$ for $\map B {H, K}$. In line with the definition of linear operator, $\map B H$ is called the '''space of bounded linear operators on $H$'''.


Definition:Semigroup with respect to Equivalence Relation

Let $C$ be a class. Let $\thickapprox$ be an equivalence relation on $C$. Let $\left({C, \cdot}\right)$ be a large magma. Then $\left({C, \cdot}\right)$ is a '''semigroup with respect to $\thickapprox$'''  iff: :$\forall x, y,z \in C: \left( x \cdot y \right)\cdot z \thickapprox x \cdot \left(y \cdot z\right)$


Definition:Ground Field

Let $K$ be a field. Let $V$ be a vector space over $K$. Then $K$ is referred to as the '''ground field''' of $V$. Category:Definitions/Abstract Algebra 3uzqk57qlqusrsghl3rh9u4fi0j70wj


Definition:Pointwise Scalar Multiplication of Mappings

Let $\struct {R, +_R, \times_R}$ be a ring, and let $\struct {S, \circ}_R$ be an $R$-algebraic structure. Let $X$ be a non-empty set, and let $S^X$ be the set of all mappings from $X$ to $S$. Then '''pointwise ($R$)-scalar multiplication''' on $S^X$ is the binary operation $\circ: R \times S^X \to S^X$ (the $\circ$ is the same as for $S$) defined by: :$\forall \lambda \in R: \forall f \in S^X: \forall x \in X: \map {\paren {\lambda \circ f} } x := \lambda \circ \map f x$


Definition:Degenerate Case

A '''degenerate case''' is a specific manifestation of a particular type of object being included in another, usually simpler, type of object. {{WIP|Include links to the various types of degenerate objects that we have already defined.}}


Definition:Sesquilinear Form

Let $H, K$ be Hilbert spaces over $\Bbb F \in \left\{{\R, \C}\right\}$. A '''sesquilinear form''' is a function $u: H \times K \to \Bbb F$ subject to: :$(1): \qquad \forall \alpha \in \Bbb F, h_1, h_2 \in H, k \in K: u \left({\alpha h_1 + h_2, k}\right) = \alpha u \left({h_1, k}\right) + u \left({h_2, k}\right)$ :$(2): \qquad \forall \alpha \in \Bbb F, h \in H, k_1, k_2 \in K: u \left({h, \alpha k_1 + k_2}\right) = \bar{\alpha} u \left({h, k_1}\right) + u \left({h, k_2}\right)$ That is, $u$ is linear in the first argument, and conjugate linear in the second. If $\Bbb F = \R$, then a sesquilinear form is the same as a bilinear map.


Definition:Equidistance

Define the following coordinates in the $xy$-plane: :$a = \tuple {x_1, x_2}$ :$b = \tuple {y_1, y_2}$ :$c = \tuple {z_1, z_2}$ :$d = \tuple {u_1, u_2}$ where $a, b, c, d \in \R^2$ 400px :$a b \equiv c d \dashv \vdash \paren {\paren {x_1 - y_1}^2 + \paren {x_2 - y_2}^2 = \paren {z_1 - u_1}^2 + \paren {z_2 - u_2}^2}$


Definition:Bounded Sesquilinear Form

Let $H, K$ be Hilbert spaces over $\Bbb F \in \left\{{\R, \C}\right\}$. Let $u: H \times K \to \Bbb F$ be a sesquilinear form. Then $u$ is said to be a '''bounded sesquilinear form''', or to be '''bounded''', iff: :$\exists M \in \R: \forall h \in H, k \in K: \left\vert{u \left({h, k}\right)}\right\vert \le M \left\Vert{h}\right\Vert_H \left\Vert{k}\right\Vert_K$ A constant $M$ satisfying the above is called a '''bound''' for $u$.


Definition:Adjoint Linear Transformation

Let $H, K$ be Hilbert spaces. Let $A \in \map B {H, K}$ be a bounded linear transformation. Let $B \in \map B {K, H}$ be the unique bounded linear transformation provided by Existence and Uniqueness of Adjoint. Then $B$ is called the '''adjoint''' of $A$, and denoted $A^*$.


Definition:Upper Semilattice on Classical Set

Let $\struct {S, \le}$ be an ordered set with the property that: :$\forall x, y \in S: \sup \set {x, y} \in S$ where $\sup$ denotes supremum. Then $\struct {S, \vee}$ is called an '''upper semilattice''', where $\vee: S \times S \to S$ is defined by: :$x \vee y := \sup \set {x, y}$ An '''upper semilattice''' hence is a particular kind of algebraic structure.


Definition:Tarski's Axioms

'''Tarski's Axioms''' are a series of axioms whose purpose is to provide a rigorous basis for the definition of Euclidean geometry entirely within the framework of first order logic. In the following: : $\equiv$ denotes the relation of equidistance. : $\mathsf{B}$ denotes the relation of betweenness. : $=$ denotes the relation of equality. The axioms are as follows:


Definition:Directed Line Segment

A '''directed line segment''' is a line segment endowed with the additional property of direction. It is often used in the context of applied mathematics to represent a vector quantity. {{expand|Perhaps the above statement should also be expanded to allow a D.L.S. to be defined as a '''vector quantity''' applied at a particular point. There is a danger (as pointed out on the Definition:Vector Quantity page) of implying  / believing that a vector, in general, is applied at a particular point, for example usually the origin. Thus, this page allows the opportunity to consider a definition of an object which consists of a vector "rooted" at a particular point, as a convenient fiction for what is actually happening in the context of physics.}} {{stub|needs a picture It may be worthwhile to point out that this can be formalized with an ordered pair. Establish connection with Definition:Affine Space}}


Definition:Vector (Euclidean Space)

A vector is defined as an element of a vector space. We have that $\R^n$, with the operations of vector addition and scalar multiplication, form a real vector space. Hence a '''vector in $\R^n$''' is defined as any element of $\R^n$. {{expand|While it is possible to identify a vector by a tuple in $\R^n$ we need to explain that the vector is not the point.}} === $\R^2$: Plane Vector === {{:Definition:Vector (Euclidean Space)/Plane Vector}} === $\R^3$: Space Vector === {{:Definition:Vector (Euclidean Space)/Space Vector}}


Definition:Engineering Notation

=== Euclidean 2-space === Define the ordered 2-tuples: {{begin-eqn}} {{eqn | l = \mathbf i       | r = \left\langle{1, 0}\right\rangle }} {{eqn | l = \mathbf j       | r = \left\langle{0, 1}\right\rangle }} {{end-eqn}} From Standard Ordered Basis is Basis, we have that any vector in $\R^2$ can be represented by: :$c_1 \mathbf i + c_2 \mathbf j$ where $c_1, c_2 \in \R$. This way of presenting vectors is called '''engineering notation'''. === Euclidean 3-space === Define the ordered 3-tuples: {{begin-eqn}} {{eqn | l = \mathbf i       | r = \left\langle{1, 0, 0}\right\rangle }} {{eqn | l = \mathbf j       | r = \left\langle{0, 1, 0}\right\rangle }} {{eqn | l = \mathbf k       | r = \left\langle{0, 0, 1}\right\rangle }} {{end-eqn}} By the same logic as the above definition, we can write any vector in $\R^3$ as: :$c_1 \mathbf i + c_2 \mathbf j + c_3 \mathbf k$ where $c_1, c_2, c_3 \in \R$. Note that $\mathbf i$ and $\mathbf j$ take on a different meaning in $3$-space than in $2$-space. === Euclidean $n$-space === In higher dimensions, rather than writing $\mathbf l, \mathbf m, \mathbf n$, and so on, the convention is to use: {{begin-eqn}} {{eqn | l = \mathbf e_1       | r = \left\langle{1, 0, 0, \ldots, 0, 0}\right\rangle }} {{eqn | l = \mathbf e_2       | r = \left\langle{0, 1, 0, \ldots, 0, 0}\right\rangle }} {{eqn | o = \vdots }} {{eqn | l = \mathbf e_n       | r = \left\langle{0, 0, 0, \ldots, 0, 1}\right\rangle }} {{end-eqn}} Then any vector in $\R^n$ can be expressed as: :$c_1 \mathbf e_1 + c_2 \mathbf e_2 + \cdots + c_n \mathbf e_n$ where $c_1, c_2, \cdots, c_n \in \R$. This convention is also frequently seen for $2$-space and $3$-space.


Definition:Vector (Linear Algebra)

Let $V$ be a vector space. Any element $v$ of $V$ is called a '''vector'''.


Definition:Self-Adjoint Operator

Let $H$ be a Hilbert space. Let $A \in B \left({H}\right)$ be a bounded linear operator. Then $A$ is said to be '''self-adjoint''' or '''hermitian''' iff: :$A = A^*$ That is, if it equals its adjoint $A^*$. {{namedfor|Charles Hermite}}


Definition:Normal Operator

Let $H$ be a Hilbert space. Let $A \in B \left({H}\right)$ be a bounded linear operator. Then $A$ is said to be '''normal''' {{iff}}: :$A^* A = A A^*$ where $A^*$ denotes the adjoint of $A$.


Definition:Real Part (Linear Operator)

Let $H$ be a Hilbert space over $\C$. Let $A \in B \left({H}\right)$ be a bounded linear operator. Then the '''real part of $A$''' is the self-adjoint operator: :$\operatorname{Re} A := \dfrac 1 2 \left({A + A^*}\right)$ The real part of $A$ may be denoted by $\operatorname{Re} \left({A}\right)$, $\operatorname{re} \left({A}\right)$ or $\Re \left({A}\right)$. This resembles the notation for the real part of a complex number.


Definition:Imaginary Part (Linear Operator)

Let $H$ be a Hilbert space over $\C$. Let $A \in B \left({H}\right)$ be a bounded linear operator. Then the '''imaginary part of $A$''' is the self-adjoint operator: :$\operatorname{Im} A := \dfrac 1 {2i} \left({A - A^*}\right)$ The imaginary part of $A$ may be denoted by $\operatorname{Im} \left({A}\right)$, $\operatorname{im} \left({A}\right)$ or $\Im \left({A}\right)$. This resembles the notation for the imaginary part of a complex number.


Definition:Absurd

'''Absurd''' is a word used mainly in logic meaning either '''meaningless''', '''contradictory''' or '''internally inconsistent'''.


Definition:Idempotent Operator

Let $H$ be a Hilbert space. Let $A \in B \left({H}\right)$ be a bounded linear operator. Then $A$ is said to be '''idempotent''', or an '''idempotent operator''', iff $A^2 = A$.


Definition:Projection (Hilbert Spaces)

Let $H$ be a Hilbert space. Let $P \in \map B H$ be an idempotent operator. Then $P$ is said to be a '''projection''' {{iff}}: :$\ker P = \paren {\Img P}^\perp$ where: :$\ker P$ denotes the kernel of $P$ :$\Img P$ denotes the image of $P$ :$\perp$ denotes orthocomplementation.


Definition:Complementary Idempotent

Let $H$ be a Hilbert space. Let $A \in B \left({H}\right)$ be an idempotent operator. Then the '''complementary idempotent (operator) of $A$''' is the bounded linear operator $I - A$, where $I$ is the identity operator on $H$. The name is appropriate, by Complementary Idempotent is Idempotent and the fact that $I - \left({I - A}\right) = A$. === Complementary Projection === If $A$ is a projection, $I - A$ is called the '''complementary projection'''. This name is justified by Complementary Projection is Projection.


Definition:Closed Linear Subspace

Let $V$ be a topological vector space. A linear subspace of $V$ that is closed is called a '''closed linear subspace'''.


Definition:Internal Hilbert Space Direct Sum

Let $H$ be a Hilbert space. Let $\left\{{M_i : i \in I}\right\}$ be an $I$-indexed collection of pairwise orthogonal subspaces of $H$. Then the '''internal (Hilbert space) direct sum''' of the $M_i$ is their closed linear span $\vee_i M_i$. It is denoted by $\bigoplus_i M_i$, or $\displaystyle \bigoplus_{i \in I} M_i$ if the set $I$ is to be stressed. When $I$ is finite, by Closed Linear Subspaces Closed under Setwise Addition, have that: :$\displaystyle \bigoplus_{i \in I} M_i = \sum_{i \in I} M_i$, where $\displaystyle \sum$ signifies setwise addition.


Definition:Orthogonal Difference

Let $H$ be a Hilbert space. Let $M, N$ be closed linear subspaces of $H$. Then the '''orthogonal difference of $M$ and $N$''', denoted $M \ominus N$, is the set $M \cap N^\perp$. It is in fact a closed linear subspace of $H$, as proven on Orthogonal Difference is Closed Linear Subspace.


Definition:Maximal/Set

Let $S$ be a set. Let $\powerset S$ be the power set of $S$. Let $\mathcal T \subseteq \powerset S$ be a subset of $\powerset S$. Let $\struct {\mathcal T, \subseteq}$ be the ordered set formed on $\mathcal T$ by $\subseteq$ considered as an ordering. Then $T \in \mathcal T$ is a '''maximal set''' of $\mathcal T$ {{iff}} $T$ is a maximal element of $\struct {\mathcal T, \subseteq}$. That is: :$\forall X \in \mathcal T: T \subseteq X \implies T = X$


Definition:Invariant Subspace

Let $H$ be a Hilbert space. Let $A \in B \left({H}\right)$ be a bounded linear operator. Let $M$ be a closed linear subspace of $H$. Then $M$ is said to be an '''invariant subspace for $A$''' iff $h \in M \implies Ah \in M$. That is, if $AM \subseteq M$.


Definition:Reducing Subspace

Let $H$ be a Hilbert space. Let $A \in B \left({H}\right)$ be a bounded linear operator. Let $M$ be a closed linear subspace of $H$; denote by $M^\perp$ its orthocomplement. Then $M$ is said to be a '''reducing subspace for $A$''' iff both $M$ and $M^\perp$ are invariant subspaces for $A$. That is, if $AM \subseteq M$ and $A M^\perp \subseteq M^\perp$.


Definition:Matrix Notation (Bounded Linear Operator)

Let $H$ be a Hilbert space. Let $A \in B \left({H}\right)$ be a bounded linear operator on $H$. Let $M$ be a closed linear subspace of $H$; denote by $M^\perp$ its orthocomplement. Observe that $M \oplus M^\perp \cong H$ by Direct Sum of Subspace and Orthocomplement. The associated isomorphism $U$ induces a bounded linear operator $A'$ on $M \oplus M^\perp$. $A'$ can be written as the matrix:  :$\begin{pmatrix}  W & X \\  Y & Z \end{pmatrix}$  where $W \in B \left({M}\right)$, $X \in B \left({M^\perp, M}\right)$, $Y \in B \left({M, M^\perp}\right)$, $Z \in B \left({M^\perp}\right)$. In the above matrix, linear transformations $W, X, Y, Z$ are uniquely determined by the requirement that: :$\begin{pmatrix} Wm + Xm^\perp \\ Ym + Zm^\perp\end{pmatrix} = \begin{pmatrix}  W & X \\  Y & Z \end{pmatrix} \begin{pmatrix}m \\m^\perp \end{pmatrix} = U^{-1} \left({Ah}\right)$ whenever $h = U \left({m, m^\perp}\right)$. By abuse of notation, one writes $A$ instead of $A'$ and calls it the '''matrix notation for $A$ (with respect to $M$)'''.


Definition:Inverse (Bounded Linear Transformation)

Let $H, K$ be Hilbert spaces. Let $A \in \map B {H, K}$ be a bounded linear transformation. An '''inverse''' for $A$ is a bounded linear transformation $A^{-1} \in \map B {K, H}$ satisfying: :$AA^{-1} = I_K$ :$A^{-1}A = I_H$ where $I_K, I_H$ denote the identity operators on $K, H$, respectively. If such a $A^{-1}$ exists, $A$ is said to be an '''invertible (bounded) linear transformation'''. The operation assigning $A^{-1}$ to $A$ is referred to as '''inverting'''. === Invertible Bounded Linear Operator === When $H = K$, the notation simplifies considerably, and $A$ is said to be a '''invertible (bounded) linear operator'''.


Definition:Coprime/GCD Domain

Let $\struct {D, +, \times}$ be a GCD domain. Let $U \subseteq D$ be the group of units of $D$. Let $a, b \in D$ such that $a \ne 0_D$ and $b \ne 0_D$ Let $d = \gcd \set {a, b}$ be the greatest common divisor of $a$ and $b$. Then $a$ and $b$ are '''coprime''' {{iff}} $d \in U$. That is, two elements of a GCD domain are '''coprime''' {{iff}} their greatest common divisor is a unit of $D$.


Definition:Coprime/Euclidean Domain

Let $\struct {D, +, \times}$ be a Euclidean domain. Let $U \subseteq D$ be the group of units of $D$. Let $a, b \in D$ such that $a \ne 0_D$ and $b \ne 0_D$ Let $d = \gcd \set {a, b}$ be the greatest common divisor of $a$ and $b$. Then $a$ and $b$ are '''coprime''' {{iff}} $d \in U$. That is, two elements of a Euclidean domain are '''coprime''' {{iff}} their greatest common divisor is a unit of $D$.


Definition:Coprime/Integers

Let $a$ and $b$ be integers such that $b \ne 0$ and $a \ne 0$ (that is, they are both non-zero). Let $\gcd \set {a, b}$ denote the greatest common divisor of $a$ and $b$. Then $a$ and $b$ are '''coprime''' {{iff}} $\gcd \set {a, b} = 1$.


Definition:Monotonicity

Let $\left({S, \preceq_1}\right)$ and $\left({T, \preceq_2}\right)$ be ordered sets. Let $\phi: \left({S, \preceq_1}\right) \to \left({T, \preceq_2}\right)$ be a monotone mapping. Then whether $\phi$ is increasing or decreasing is known as the '''monotonicity of $\phi$'''.  That is: :if $\phi$ is such that $x \mathop{\preceq_1} y \implies \phi \left({x}\right) \mathop{\preceq_2} \phi \left({y}\right)$, then the '''monotonicity''' of $\phi$ is increasing :if $\phi$ is such that $x \mathop{\prec_1} y \implies \phi \left({x}\right) \mathop{\prec_2} \phi \left({y}\right)$, then the '''monotonicity''' of $\phi$ is strictly increasing :if $\phi$ is such that $x \mathop{\preceq_1} y \implies \phi \left({y}\right) \mathop{\preceq_2} \phi \left({x}\right)$, then the '''monotonicity''' of $\phi$ is decreasing :if $\phi$ is such that $x \mathop{\prec_1} y \implies \phi \left({y}\right) \mathop{\prec_2} \phi \left({x}\right)$, then the '''monotonicity''' of $\phi$ is strictly decreasing.


Definition:Continuation of Woset

Let $\left({S, \preccurlyeq}\right)$ be a woset. Let $\left({T, \preccurlyeq}\right)$ be a set with an ordering such that: :$(1): \quad T$ is an initial segment of $S$ :$(2): \quad$ The ordering of the elements of $T$ is the same as their ordering in $S$. Then $S$ is a '''continuation''' of $T$.


Definition:Addition/Summand

Let $a + b$ denote the operation of addition on two objects. The objects $a$ and $b$ are known as the '''summands''' of $a + b$.


Definition:Between (Geometry)

Define the following coordinates in the $xy$-plane: {{begin-eqn}} {{eqn | l = a       | r = \left({x_1, x_2}\right) }} {{eqn | l = b       | r = \left({y_1, y_2}\right) }} {{eqn | l = c       | r = \left({z_1, z_2}\right) }} {{end-eqn}} where $a, b, c \in \R^2$. Let: {{begin-eqn}} {{eqn | l = \Delta x_1       | r = x_3 - x_2 }} {{eqn | l = \Delta x_2       | r = x_2 - x_1 }} {{eqn | l = \Delta y_1       | r = y_2 - y_1 }} {{eqn | l = \Delta y_2       | r = y_3 - y_2 }} {{end-eqn}} Then: :File:Betweenness(Analytic Def'n).png :$\mathsf{B}abc \dashv \vdash \left({\Delta x_1 \Delta y_1 = \Delta x_2 \Delta y_2}\right) \land$ :$\left({0 \le \Delta x_1 \Delta y_1 \land 0 \le \Delta x_2 \Delta y_2}\right)$


Definition:Ordering Induced by Injection

Let $\left({T, \le}\right)$ be an ordered set, and let $S$ be a set. Let $f: S \to T$ be an injection. Define $\le_f$ as the '''ordering induced by $f$ on $S$''' by: :$\forall s_1, s_2 \in S: s_1 \le_f s_2 \iff f \left({s_1}\right) \le f \left({s_2}\right)$ That $\le_f$ is in fact an ordering is shown on Ordering Induced by Injection is Ordering.


Definition:Dominate (Analysis)

Let $\left \langle {a_n} \right \rangle$ be a sequence in $\R$. Let $\left \langle {z_n} \right \rangle$ be a sequence in $\C$. Then $\left \langle {a_n} \right \rangle$ '''dominates''' $\left \langle {z_n} \right \rangle$ iff: :$\forall n \in \N: \left|{z_n}\right| \le a_n$ {{SUBPAGENAME}} {{SUBPAGENAME}} dpeqz6xo6qbc2m40xp0hhgcrjj7wbro


Definition:Vertex Cut

Let $G$ be a graph. A '''vertex cut''' of $G$ is a set of vertices $W \subseteq V \left({G}\right)$ such that the vertex deletion $G \setminus W$ is disconnected.


Definition:K-Connected

Let $G$ be a graph. Let $k \in \Z_{>0}$. Then $G$ is '''$k$-connected''' {{iff}}: : $\left\lvert{V \left({G}\right) }\right\rvert > k$ : $G$ is connected : $G$ has no vertex cut $W$ with $\left\lvert{W}\right\rvert < k$ Informally: : $G$ is connected and has more than $k$ vertices : There is no vertex cut of $G$ containing less than $k$ vertices.


Definition:Connectivity

Let $G$ be a graph. Then $\kappa \left({G}\right)$, the '''connectivity''' of $G$, is the smallest $k \in \Z_{>0}$ such that $G$ is $k$-connected.  If $G$ is disconnected, or if $G$ has fewer than 2 vertices, then $\kappa \left({G}\right) = 0$. {{SUBPAGENAME}} 8n9tlxeii5l0iw6ad6hzbwkzdkislw1


Definition:Vertex Deletion

Let $G = \left({V, E}\right)$ be an (undirected) graph. Let $W \subseteq V$ be a set of vertices of $G$. Then the '''graph obtained by deleting $W$ from $G$''', denoted by $G - W$, is the subgraph induced by $V \setminus W$. Alternatively: : $G - W = \left({V \setminus W, \left\{{e \in E : e \cap W = \varnothing}\right\}}\right)$ Informally, $G - W$ is the graph obtained from $G$ by removing all vertices in $W$ and all edges incident to those vertices. If $W$ is a singleton such that $W = \left\{{v}\right\}$, then $G - W$ may be expressed $G - v$.


Definition:Minimum Degree

Let $G = \left({V, E}\right)$ be a graph. Then the '''minimum degree''' of $G$ is: : $\delta\left({G}\right) = \min\left\{{\deg_G\left({v}\right) : v \in V}\right\}$ That is, it is the minimum degree of all the vertices of $G$. {{SUBPAGENAME}} pdfmm4z4qjqku87z11orh4durbra7v7


Definition:Neighborhood (Graph Theory)

Let $G = \struct {V, E}$ be a graph. Let $v \in V$ be a vertex of $G$. The '''neighborhood of $v$ in $G$''' is: :$\map {\Gamma_G} v = \set {u \in V : u v \in E}$ That is, it is the set of all vertices which are adjacent to $v$. {{Languages|Neighborhood}} {{Language|Dutch|nabuurschap}} {{Language|French|voisinage}} {{End-languages}}


Definition:Halmos Symbol

The '''Halmos symbol''' is the character: $\blacksquare$ used to indicate the end of a proof. It replaces the old-fashioned and embarrassingly uncool Q.E.D. which muggles sometimes use when pretending to be clever.


Definition:Included Set Topology

Let $S$ be a set which is non-null. Let $H \subseteq S$ be some subset of $S$. We define a subset $\tau_H$ of the power set $\powerset S$ as: :$\tau_H = \set {A \subseteq S: H \subseteq A} \cup \set \O$ that is, all the subsets of $S$ which are supersets of $H$, along with the empty set $\O$. Then $\tau_H$ is a topology called the '''included set topology on $S$ by $H$''', or just '''an included set topology'''. The topological space $T = \struct {S, \tau_H}$ is called the '''included set space on $S$ by $H$''', or just '''an included set space'''.


Definition:Compact Linear Transformation

Let $H, K$ be Hilbert spaces. Let $T: H \to K$ be a linear transformation. Let $\operatorname{ball} H$ be the closed unit ball of $H$. Then $T$ is said to be a '''compact linear transformation''', or simply '''compact''' {{iff}} $\operatorname{cl} \left({T \left({\operatorname{ball} H}\right) }\right)$ is compact in $K$, where $\operatorname{cl}$ denotes closure. === Compact Operator === When $H$ and $K$ are equal, one speaks about '''compact (linear) operators''' instead. This is in line with the definition of a linear operator.


Definition:Closed Unit Ball

Let $\struct {X, \norm {\, \cdot \,}}$ be a normed vector space. Let $a \in X$. The '''closed unit ball''' of $X$, denoted $\operatorname{ball} X$, is the set: :$\map {B_1^-} a := \set {x \in X: \norm {x - a} \le 1}$


Definition:Space of Compact Linear Transformations

Let $H, K$ be Hilbert spaces. Let $\Bbb F \in \left\{{\R, \C}\right\}$ be the ground field of $K$. The '''space of compact linear transformations from $H$ to $K$''',  $B_0 \left({H, K}\right)$, is the set of all compact linear transformations: :$B_0 \left({H, K}\right):= \left\{{T: H \to K: T \text{ compact}}\right\}$ endowed with pointwise addition and ($\F$)-scalar multiplication. It is a Banach space, as proven on Space of Compact Linear Transformations is Banach Space. The notation resembles that for the space of bounded linear transformations $B \left({H, K}\right)$. This is appropriate as a Compact Linear Transformation is Bounded; i.e., $B_0 \left({H, K}\right) \subseteq B \left({H, K}\right)$. === Space of Compact Linear Operators === When $H$ is equal to $K$, one speaks about the '''space of compact (linear) operators''' instead. One writes $B_0 \left({H}\right)$ for $B_0 \left({H, H}\right)$.


Definition:Finite Rank Operator

Let $H, K$ be Hilbert spaces. Let $T: H \to K$ be a linear transformation. Then $T$ is said to be a '''finite rank operator''', or of '''finite rank''', {{iff}} its range, $\Rng T$, is finite dimensional. Note that a finite rank operator is not necessarily bounded.


Definition:Condensed Series

Let $\left \langle {a_n} \right \rangle: n \mapsto a\left({n}\right)$ be a decreasing sequence of strictly positive terms in $\R$ which converges with a limit of zero. That is, for every $n$ in the domain of $\left \langle {a_n} \right \rangle$: $a_n > 0, a_{n+1} \le a_n$, and $a_n \to 0$ as $n \to +\infty$. The series: :$\displaystyle \sum_{n \mathop = 1}^\infty 2^n a\left({2^n}\right)$ is called the '''condensed''' form of the series: :$\displaystyle \sum_{n \mathop = 1}^\infty a_n$


Definition:Space of Continuous Finite Rank Operators

Let $H, K$ be Hilbert spaces. Then the '''space of continuous finite rank operators from $H$ to $K$''', denoted $B_{00} \left({H, K}\right)$, is the set: :$B_{00} \left({H, K}\right) := \left\{{A \in B \left({H, K}\right): A \text{ is of finite rank} }\right\}$ of all bounded linear transformations of finite rank. By definition, it is a subset of the space of bounded linear transformations $B \left({H, K}\right)$. In fact, by Finite Rank Operator is Compact, it is contained in $B_0 \left({H, K}\right)$, the space of compact linear transformations.


Definition:Elementary Matrix

=== Elementary Row Matrix === {{:Definition:Elementary Matrix/Row Operation}} === Elementary Column Matrix === {{:Definition:Elementary Matrix/Column Operation}}


Definition:Diagonalizable Operator

Let $H$ be a Hilbert space over $\Bbb F \in \set {\R, \C}$. Let $A:H \to H$ be a linear operator on $H$. The following two definitions of '''diagonalizable operator''' are equivalent: === By a Basis === $A$ is said to be '''diagonalizable''' {{iff}} there exist: :a basis $E$ for $H$ :an indexed set $\family {\alpha_e}_{e \mathop \in E} \subseteq \Bbb F$ of scalars (with $E$ as indexing set) such that: :$\forall e \in E: Ae = \alpha_e e$ ==== Value Set ==== The indexed set $\family {\alpha_e}_{e \mathop \in E}$ may be called the '''value set of $A$ (with respect to the basis $E$)'''. === By a Partition of Unity === $A$ is said to be '''diagonalizable''' iff there exist: :a partition of unity $\family {P_i}_{i \mathop \in I}$ on $H$ :an indexed set $\family {\alpha_i}_{i \mathop \in I} \subseteq \Bbb F$ of scalars (with the same $I$ as indexing set) such that: :$\forall i \in I: \forall h \in \Rng {P_i}: A h = \alpha_i h$ To express that $A$ is diagonalizable, one writes $A = \displaystyle \sum_{i \in I} \alpha_i P_i$ or $A = \displaystyle \bigoplus_{i \in I} \alpha_i P_i$. ==== Value Set ==== The indexed set $\family {\alpha_i}_{i \mathop \in I}$ may be called the '''value set of $A$ (with respect to the partition of unity $\family {P_i}_{i \mathop \in I}$)'''.


Definition:Row Equivalence

Two matrices $\mathbf A = \sqbrk a_{m n}, \mathbf B = \sqbrk b_{m n}$ are '''row equivalent''' if one can be obtained from the other by a finite sequence of elementary row operations. This relationship can be denoted $\mathbf A \sim \mathbf B$.


Definition:Parity Group

: The group $\struct {\Z_2, +_2}$ : $C_2$, the cyclic group of order 2 : The group $\struct {\set {1, -1}, \times}$ : The quotient group $\dfrac {S_n} {A_n}$ of the symmetric group of order $n$ with the alternating group of order $n$ etc.


Definition:Eigenvector

Let $H$ be a Hilbert space over $\Bbb F \in \set {\R, \C}$. Let $A \in \map B H$ be a bounded linear operator. Let $\alpha \in \Bbb F$ be an eigenvalue of $A$. A nonzero vector $h \in H$ is said to be an '''eigenvector for $\alpha$''' {{iff}}: :$h \in \map \ker {A - \alpha I}$ That is, {{iff}} $Ah = \alpha h$. === Eigenspace === {{refactor|Separate page for this}} The '''eigenspace''' for an eigenvalue $\alpha$ is the set $\map \ker {A - \alpha I}$. By Kernel of Linear Transformation is Closed Linear Subspace, it is a closed linear subspace of $H$.


Definition:Partition of Unity (Hilbert Space)

Let $H$ be a Hilbert space. A '''partition of unity''' or '''partition of identity''' on $H$ is a family $\family {P_i}_{i \mathop \in I}$ of projections, subject to: :If $i \ne j$, then $P_i P_j = P_j P_i = 0$ :$\vee \set {\Img {P_i}: i \in I} = H$, where $\vee$ signifies closed linear span One may encounter the notations $1 = \sum_i P_i$ and $1 = \bigoplus_i P_i$. Here, $1$ signifies the identity operator on $H$.


Definition:Inflationary Mapping

Let $\struct {S, \preceq}$ be an ordered set. Let $\phi: S \to S$ be a mapping. Then $\phi$ is '''inflationary''' {{iff}}: :$\forall s \in S: s \preceq \map \phi s$


Definition:Hartogs Number

Let $S$ be a set. The '''Hartogs number''' of $S$ is the smallest ordinal $\alpha$ such that there exists no injection from $\alpha$ to $S$.


Definition:Lyndon Word

A '''Lyndon word''' is a string that is strictly smaller in lexicographic order than all of its rotations. {{NamedforDef|Roger Conant Lyndon|cat = Lyndon}} Category:Definitions/Combinatorics og9zhmasfrj91zns892whlznip07t25


Definition:Rotation (Permutation Theory)

Let $\tuple {a_1, \ldots, a_n}$ be a string over an alphabet $A$. A '''rotation''' is a mapping $r: A^n \to A^n$ given by: :$\tuple {a_1, \ldots, a_n} \mapsto \tuple {a_{\map \phi 1}, \cdots, a_{\map \phi n} }$ where $\phi$ is a permutation on n letters. Category:Definitions/Permutation Theory 4tkxx9i8aqrxcshizqgmrkh3c3jbwir


Definition:Isomorphism (Category Theory)

Let $\mathbf C$ be a category, and let $X, Y$ be objects of $\mathbf C$. A morphism $f: X \to Y$ is an '''isomorphism''' if there exists a morphism $g: Y \to X$ such that: :$g \circ f = I_X$ :$f \circ g = I_Y$ where $I_X$ denotes the identity morphism on $X$. It can be seen that this is equivalent to $g$ being both a retraction and a section of $f$.


Definition:Isomorphism (Abstract Algebra)/Ordered Structure Isomorphism

An '''ordered structure isomorphism''' from an ordered structure $\struct {S, \circ, \preceq}$ to another $\struct {T, *, \preccurlyeq}$ is a mapping $\phi: S \to T$ that is both: :$(1): \quad$ An isomorphism, that is a bijective homomorphism, from the structure $\struct {S, \circ}$ to the structure $\struct {T, *}$ :$(2): \quad$ An order isomorphism from the ordered set $\struct {S, \preceq}$ to the ordered set $\struct {T, \preccurlyeq}$.


Definition:Symmetry Group

Let $P$ be a geometric figure. Let $S_P$ be the set of all symmetries of $P$. Let $\struct {S_P, \circ}$ be the algebraic structure such that $\circ$ denotes the composition of mappings. Then $\struct {S_P, \circ}$ is called the '''symmetry group of $P$'''.


Definition:Cancellable Element/Left Cancellable

An element $x \in \struct {S, \circ}$ is '''left cancellable''' {{iff}}: :$\forall a, b \in S: x \circ a = x \circ b \implies a = b$


Definition:Cancellable Element/Right Cancellable

An element $x \in \left ({S, \circ}\right)$ is '''right cancellable''' iff: :$\forall a, b \in S: a \circ x = b \circ x \implies a = b$


Definition:Circle Group

Let $K$ be the set of all complex numbers of unit modulus: :$K = \set {z \in \C: \cmod z = 1}$ Consider the algebraic structure $\struct {K, \times}$ where $\times$ denotes the operation of complex multiplication. Then $\struct {K, \times}$ is called the '''circle group'''.


Definition:Power Series/Real Domain

Let $\xi \in \R$ be a real number. Let $\sequence {a_n}$ be a sequence in $\R$. The series $\displaystyle \sum_{n \mathop = 0}^\infty a_n \paren {x - \xi}^n$, where $x \in \R$ is a variable, is called a '''power series in $x$ about the point $\xi$'''.


Definition:Power Series/Complex Domain

Let $\xi \in \C$ be a complex number. Let $\sequence {a_n}$ be a sequence in $\C$. The series $\displaystyle \sum_{n \mathop = 0}^\infty a_n \paren {z - \xi}^n$, where $z \in \C$ is a variable, is called a '''(complex) power series in $z$ about the point $\xi$'''.


Definition:Radius of Convergence/Real Domain

Let $\xi \in \R$ be a real number. Let $\displaystyle \map S x = \sum_{n \mathop = 0}^\infty a_n \paren {x - \xi}^n$ be a power series about $\xi$. Let $I$ be the interval of convergence of $\map S x$. Let the endpoints of $I$ be $\xi - R$ and $\xi + R$. (This follows from the fact that $\xi$ is the midpoint of $I$.) Then $R$ is called the '''radius of convergence''' of $\map S x$. If $\map S x$ is convergent over the whole of $\R$, then $I = \R$ and thus the radius of convergence is infinite.


Definition:Radius of Convergence/Complex Domain

Let $\xi \in \C$ be a complex number. For $z \in \C$, let: :$\displaystyle \map f z = \sum_{n \mathop = 0}^\infty a_n \paren {z - \xi}^n$ be a power series about $\xi$. The '''radius of convergence''' is the extended real number $R \in \overline \R$ defined by: :$R = \displaystyle \inf \set {\cmod {z - \xi}: z \in \C, \sum_{n \mathop = 0}^\infty a_n \paren {z - \xi}^n \text{ is divergent} }$ where a divergent series is a series that is not convergent. As usual, $\inf \O = +\infty$.


Definition:Operation Induced by Restriction

Let $\struct {S, \circ}$ be a magma. Let $\struct {T, \circ} \subseteq \struct {S, \circ}$. That is, let $T$ be a subset of $S$ such that $\circ$ is closed in $T$. Then the restriction of $\circ$ to $T$, namely $\circ {\restriction_T}$, is called the '''(binary) operation induced on $T$ by $\circ$'''.


Definition:Euclidean Norm

Let $\mathbf v = \tuple {v_1, v_2, \ldots, v_n}$ be a vector in the Euclidean $n$-space $\R^n$. The '''Euclidean norm''' of $\mathbf v$ is defined as: :$\displaystyle \norm {\mathbf v} = \paren {\sum_{k \mathop = 1}^n v_k^2}^{1/2}$


Definition:Complex Sequence

A '''complex sequence''' is a sequence (usually infinite) whose codomain is the set of complex numbers $\C$.


Definition:Length of Sequence

The '''length''' of a finite sequence is the number of terms it contains, or equivalently, the cardinality of its domain. === Sequence of $n$ Terms === {{:Definition:Sequence of n Terms}}


Definition:Sequence of n Terms

A '''sequence of $n$ terms''' is a (finite) sequence whose length is $n$.


Definition:Sequence/Empty Sequence

An '''empty sequence''' is a (finite) sequence containing no terms. Thus an '''empty sequence''' is a mapping from $\varnothing$ to $S$, that is, the empty mapping.


Definition:Bounded Below Sequence

Let $\struct {T, \preceq}$ be an ordered set. Let $\sequence {x_n}$ be a sequence in $T$. Then $\sequence {x_n}$ is '''bounded below''' {{iff}}: :$\exists m \in T: \forall i \in \N: m \preceq x_i$


Definition:Bounded Above Sequence

Let $\struct {T, \preceq}$ be an ordered set. Let $\sequence {x_n}$ be a sequence in $T$. Then $\sequence {x_n}$ is '''bounded above''' {{iff}}: :$\exists M \in T: \forall i \in \N: x_i \preceq M$


Definition:Continuous Mapping

The concept of '''continuity''' makes precise the intuitive notion that a function has no "jumps" at a given point. Loosely speaking, in the case of a real function, continuity at a point is defined as the property that the graph of the function does not have a "break" at the point. This concept appears throughout mathematics and correspondingly has many variations and generalizations.


Let $\sequence {x_n}$ be a sequence in $\R$. === Divergent to Positive Infinity === {{:Definition:Unbounded Divergent Sequence/Real Sequence/Positive Infinity}} === Divergent to Negative Infinity === {{:Definition:Unbounded Divergent Sequence/Real Sequence/Negative Infinity}} === Divergent to Infinity === Consider the case where $\sequence {x_n}$ is both unbounded above and unbounded below. {{:Definition:Unbounded Divergent Sequence/Real Sequence/Infinity}}


Definition:Unbounded Divergent Sequence/Complex Sequence

Let $\sequence {z_n}$ be a sequence in $\C$. Then $\sequence {z_n}$ '''tends to $\infty$''' or '''diverges to $\infty$''' {{iff}}: :$\forall H > 0: \exists N: \forall n > N: \cmod {z_n} > H$ where $\cmod {z_n}$ denotes the modulus of $z_n$. We write: :$x_n \to \infty$ as $n \to \infty$.


Definition:Strictly Convex Real Function

Let $f$ be a real function which is defined on a real interval $I$. === Definition 1 === {{:Definition:Convex Real Function/Definition 1/Strictly}} === Definition 2 === {{:Definition:Convex Real Function/Definition 2/Strictly}} === Definition 3 === {{:Definition:Convex Real Function/Definition 3/Strictly}}


Definition:Hölder Mean

Let $x_1, x_2, \ldots, x_n \in \R_{\ge 0}$ be positive real numbers. Let $p$ be an extended real number. The '''Hölder mean with exponent $p$ of $x_1, x_2, \ldots, x_n$''' is denoted $\map {M_p} {x_1, x_2, \ldots, x_n}$. For real $p \ne 0$, it is defined as: :$\displaystyle \map {M_p} {x_1, x_2, \ldots, x_n} = \paren {\frac 1 n \sum_{k \mathop = 1}^n x_k^p}^{1/p}$ whenever the above expression is defined. For $p = 0$, it is defined as: :$\map {M_0} {x_1, x_2, \ldots, x_n} = \paren {x_1 x_2 \cdots x_n}^{1/n}$ the geometric mean of $x_1, x_2, \ldots, x_n$. For $p = \infty$, it is defined as: :$\map {M_\infty} {x_1, x_2, \ldots, x_n} = \max {\set {x_1, x_2, \ldots, x_n} }$ For $p = -\infty$, it is defined as: :$\map {M_{-\infty} } {x_1, x_2, \ldots, x_n} = \min {\set {x_1, x_2, \ldots, x_n} }$


Definition:Vector Addition

=== Vector Addition on Module === {{:Definition:Vector Addition/Module}} === Vector Addition on Vector Space === {{:Definition:Vector Addition/Vector Space}} === Vector Sum === {{:Definition:Vector Sum}}


Definition:Inductive Ordered Set

An '''inductive ordered set''' is an ordered set in which every chain has an upper bound.


Definition:P-Norm

Let $p \ge 1$ be a real number. Let $\ell^p$ denote the $p$-sequence space. Let $\mathbf x = \sequence {x_n} \in \ell^p$. Then the '''$p$-norm''' of $\mathbf x$ is defined as: :$\displaystyle \norm {\mathbf x}_p = \paren {\sum_{n \mathop = 0}^\infty \size {x_n}^p}^{1/p}$


Definition:Left-Hand Derivative

Let $B$ be a Banach space over the set of real numbers $\R$. Let $f: \R \to B$ be a mapping from $\R$ to $B$. The '''left-hand derivative''' of $f$ is defined as the left-hand limit: :$\displaystyle \map {f'_-} x = \lim_{h \mathop \to 0^-} \frac {\map f {x + h} - \map f x} h$ If the '''left-hand derivative exists''', then $f$ is said to be '''left-hand differentiable''' at $x$.


Definition:Right-Hand Derivative

Let $B$ be a Banach space over the set of real numbers $\R$. Let $f: \R \to B$ be a mapping from $\R$ to $B$. The '''right-hand derivative''' of $f$ is defined as the right-hand limit: :$\displaystyle \map {f'_+} x = \lim_{h \mathop \to 0^+} \frac {\map f {x + h} - \map f x} h$ If the '''right-hand derivative''' exists, then $f$ is said to be '''right-hand differentiable''' at $x$. === Real Functions === {{:Definition:Right-Hand Derivative/Real Function}}


Definition:One-Sided Derivative

A '''one-sided derivative''' is a right-hand derivative or a left-hand derivative. Category:Definitions/Differential Calculus f96ff9uj9rfx4inafni66xo7pjwpr7z


Definition:Difference Quotient

Let $V$ be a vector space over the real numbers $\R$. Let $f: \R \to V$ be a function. A '''difference quotient''' is an expression of the form: :$\dfrac {\map f {x + h} - \map f x} h$ where $h \ne 0$ is a real number.


Definition:Left Difference Quotient

Let $V$ be a vector space over the real numbers $\R$. Let $f: \R \to V$ be a function. A '''left difference quotient''' is an expression of the form: :$\dfrac {\map f {x + h} - \map f x} h$ where $h < 0$ is a strictly negative real number.


Definition:Right Difference Quotient

Let $V$ be a vector space over the real numbers $\R$. Let $f: \R \to V$ be a function. A '''right difference quotient''' is an expression of the form: :$\dfrac {\map f {x + h} - \map f x} h$ where $h > 0$ is a strictly positive real number.


Definition:Strictly Concave Real Function

Let $f$ be a real function which is defined on a real interval $I$. === Definition 1 === {{:Definition:Concave Real Function/Definition 1/Strictly}} === Definition 2 === {{:Definition:Concave Real Function/Definition 2/Strictly}} === Definition 3 === {{:Definition:Concave Real Function/Definition 3/Strictly}}


Definition:Co-Countable Set

Let $S$ be a set, and let $A \subseteq S$. Then $A$ is said to be '''co-countable (in $S$)''' {{iff}} its relative complement $\complement_S \left({A}\right)$ is countable.


Definition:Trace Sigma-Algebra

Let $X$ be a set, and let $\Sigma$ be a $\sigma$-algebra on $X$. Let $E \subseteq X$ be a subset of $X$. Then the '''trace $\sigma$-algebra (of $E$ in $\Sigma$)''', $\Sigma_E$, is defined as: :$\Sigma_E := \set {E \cap S: S \in \Sigma}$ It is a $\sigma$-algebra on $E$, as proved on Trace Sigma-Algebra is Sigma-Algebra.


Definition:Pre-Image Sigma-Algebra/Domain

Let $X, X'$ be sets, and let $f: X \to X'$ be a mapping. Let $\Sigma'$ be a $\sigma$-algebra on $X'$. Then the '''pre-image $\sigma$-algebra (of $\Sigma'$) on the domain of $f$''' is defined as: :$f^{-1} \left({\Sigma'}\right) := \left\{{f^{-1} \left({E'}\right): E' \in \Sigma'}\right\}$


Let $X$ be a set. Let $\GG \subseteq \powerset X$ be a collection of subsets of $X$. === Definition 1 === {{:Definition:Sigma-Algebra Generated by Collection of Subsets/Definition 1}} === Definition 2 === {{:Definition:Sigma-Algebra Generated by Collection of Subsets/Definition 2}} === Generator === {{:Definition:Sigma-Algebra Generated by Collection of Subsets/Generator}}


Definition:Open Rectangle

Let $n \ge 1$ be a natural number. Let $a_1, \ldots, a_n, b_1, \ldots, b_n$ be real numbers. The Cartesian product: :$\displaystyle \prod_{i \mathop = 1}^n \openint {a_i} {b_i} = \openint {a_1} {b_1} \times \cdots \times \openint {a_n} {b_n} \subseteq \R^n$ is called an '''open rectangle in $\R^n$''' or '''open $n$-rectangle'''. The collection of all '''open $n$-rectangles''' is denoted $\JJ_o$, or $\JJ_o^n$ if the dimension $n$ is to be emphasized. === Degenerate Case === In case $a_i \ge b_i$ for some $i$, the rectangle is taken to be the empty set $\O$. This is in accordance with the result Cartesian Product is Empty iff Factor is Empty for general Cartesian products.


Definition:Atom of Sigma-Algebra

Let $\struct {X, \Sigma}$ be a measurable space. Let $E \in \Sigma$ be non-empty. $E$ is said to be an '''atom (of $\Sigma$)''' {{iff}} it satisfies: :$\forall F \in \Sigma: F \subsetneq E \implies F = \O$ Thus, '''atoms''' are the minimal non-empty sets in $\Sigma$ with respect to the subset ordering.


Definition:Character (Representation Theory)

Let $\left({G, \cdot}\right)$ be a finite group. Let $V$ be a finite dimensional $k$-vector space. Consider a linear representation $\rho: G \to \operatorname{GL}\left({V}\right)$ of $G$. The '''character associated with $\rho$''' is defined as: : $\chi:G \to k$ where $\chi \left({g}\right) = \operatorname{Tr} \left({\rho\left({g}\right)}\right)$, the trace of $\rho\left({g}\right)$; which is a linear automorphism of $V$.


Definition:Reducible Linear Representation

Let $\rho: G \to \operatorname{GL} \left({V}\right)$ be a linear representation. $\rho$ is '''reducible''' {{iff}} there exists a non-trivial proper vector subspace $W$ of $V$ such that: :$\forall g \in G: \rho \left({g}\right) \left({W}\right) \subseteq W$ That is, such that $W$ is invariant for every linear operator in the set $\left\{{\rho \left({g}\right): g \in G}\right\}$.


Definition:Vector-Valued Function

Let $f_1, f_2, \ldots, f_n$ be real functions of $t$. Let $\mathbb T \subseteq \R, \mathbb Y \subseteq \R^n$ (where usually $n \ge 2$). Let $\mathbf r$ be a mapping from $\mathbb T \to \mathbb Y$ that maps each $t \in \mathbb T$ to a vector $\left \langle{f_1 \left({t}\right), f_2 \left({t}\right), \ldots, f_n \left({t}\right)}\right \rangle \in \mathbb Y$. Then $\mathbf r$ is said to be a '''vector-valued function''' (of the '''parameter''' $t$). If $\mathbb T$ is not explicitly defined, it is taken to be the intersection of all the domains of $f_1 ,f_2, \cdots, f_n$. === Component Function === {{:Definition:Vector-Valued Function/Component Function}}


=== Definition 1 === {{:Definition:Limit of Vector-Valued Function/Definition 1}} === Definition 2 === {{:Definition:Limit of Vector-Valued Function/Definition 2}}


=== Derivative at a Point === {{:Definition:Derivative/Vector-Valued Function/Point}} === Derivative on an Open Set === {{:Definition:Derivative/Vector-Valued Function/Open Set}}


Definition:Irreducible (Representation Theory)/Linear Representation

Let $\rho: G \to \operatorname{GL} \left({V}\right)$ be a linear representation. Then $\rho$ is '''irreducible''' {{iff}} it is not reducible. That is, {{iff}} there exists '''no''' non-trivial proper vector subspace $W$ of $V$ such that: : $\forall g \in G: \rho \left({g}\right) \left({W}\right) \subseteq W$


Definition:Monotone Class

Let $X$ be a set, and let $\powerset X$ be its power set. Let $\MM \subseteq \powerset X$ be a collection of subsets of $X$. Then $\MM$ is said to be a '''monotone class (on $X$)''' {{iff}} for every countable, nonempty, index set $I$, it holds that: :$\displaystyle \family {A_i}_{i \mathop \in I} \in \MM \implies \bigcup_{i \mathop \in I} A_i \in \MM$ :$\displaystyle \family {A_i}_{i \mathop \in I} \in \MM \implies \bigcap_{i \mathop \in I} A_i \in \MM$ that is, {{iff}} $\MM$ is closed under countable unions and countable intersections.


Definition:Monotone Class Generated by Collection of Subsets

Let $X$ be a set. Let $\mathcal G \subseteq \mathcal P \left({X}\right)$ be a collection of subsets of $X$. Then the '''monotone class generated by $\mathcal G$''', $\mathfrak m \left({\mathcal G}\right)$, is the smallest monotone class on $X$ that contains $\mathcal G$. That is, $\mathfrak m \left({\mathcal G}\right)$ is subject to: :$(1):\quad \mathcal G \subseteq \mathfrak m \left({\mathcal G}\right)$ :$(2):\quad \mathcal G \subseteq \mathcal M \implies \mathfrak m \left({\mathcal G}\right) \subseteq \mathcal M$ for any monotone class $\mathcal M$ on $X$ In fact, $\mathfrak m \left({\mathcal G}\right)$ always exists, and is unique, as proved on Existence and Uniqueness of Monotone Class Generated by Collection of Subsets. === Generator === One says that $\mathcal G$ is a '''generator''' for $\mathfrak m \left({\mathcal G}\right)$.


Definition:Topology Induced by Metric

Let $M = \left({A, d}\right)$ be a metric space. === Definition 1 === {{:Definition:Topology Induced by Metric/Definition 1}} === Definition 2 === {{:Definition:Topology Induced by Metric/Definition 2}}


Definition:Pre-Measure

Let $X$ be a set. Let $\mathcal S \subseteq \mathcal P \left({X}\right)$ be a collection of subsets of $X$. Let $\varnothing \in \mathcal S$. Let $\mu: \mathcal S \to \overline{\R}_{\ge 0}$ be a mapping, where $\overline{\R}_{\ge 0}$ denotes the set of positive extended real numbers. Then $\mu$ is said to be a '''pre-measure''' iff it satisfies the following conditions: :$(1):\quad \mu \left({\varnothing}\right) = 0$ :$(2):\quad$ For every sequence $\left({A_n}\right)_{n \in \N}$ of pairwise disjoint sets in $\mathcal S$ with $\displaystyle \bigcup_{n \mathop \in \N} A_n \in \mathcal S$: ::$\displaystyle \mu \left({\bigcup_{n \mathop \in \N} A_n}\right) = \sum_{n \mathop \in \N} \mu \left({A_n}\right)$ :that is, that $\mu$ is countably additive.


Definition:Finite Measure

Let $\mu$ be a measure on a measurable space $\struct {X, \Sigma}$. Then $\mu$ is said to be a '''finite measure''' {{iff}}: :$\map \mu X < \infty$


Definition:Finite Measure Space

A measure space $\struct {X, \Sigma, \mu}$ is said to be '''finite''' {{iff}} $\mu$ is a finite measure.


Definition:Exhausting Sequence of Sets

Let $\left\langle{S_k}\right \rangle_{k \in \N}$ be a nested sequence of subsets of $S$ such that: :$(1):\quad \forall k \in \N: S_k \subseteq S_{k + 1}$ :$(2):\quad \displaystyle \bigcup_{k \mathop \in \N} S_k = S$ Then $\left\langle{S_k}\right \rangle_{k \in \N}$ is an '''exhausting sequence of sets (in $\mathcal S$)'''. That is, it is an increasing sequence of subsets of $S$, whose union is $S$. It is common to write $\left\langle{S_k}\right\rangle_{k \in \N} \uparrow S$ to indicate an '''exhausting sequence of sets'''. Here, the $\uparrow$ denotes a limit of an increasing sequence.


Definition:Increasing Sequence of Sets

Let $\sequence {S_k}_{k \mathop \in \N}$ be a nested sequence of subsets of $S$ such that: :$\forall k \in \N: S_k \subseteq S_{k + 1}$ Then $\sequence {S_k}_{k \mathop \in \N}$ is an '''increasing sequence of sets (in $\SS$)'''.


Definition:Sigma-Finite Measure

Let $\mu$ be a measure on a measurable space $\left({X, \Sigma}\right)$. Then $\mu$ is said to be a '''$\sigma$-finite''' (or '''sigma-finite''') '''measure''' iff there exists an exhausting sequence $\left({E_n}\right)_{n \in \N}$ in $\Sigma$, subject to: :$\forall n \in \N: \mu \left({E_n}\right) < \infty$


Definition:Sigma-Finite Measure Space

A measure space $\left({X, \Sigma, \mu}\right)$ is said to be '''$\sigma$-finite''' (or '''sigma-finite''') iff $\mu$ is a $\sigma$-finite measure.


Definition:G-Submodule

Let $\left({G, \cdot}\right)$ be a finite group. Let $\left({V, \phi}\right)$ be a $G$-module. Let $W$ be a vector subspace of $V$. If $\phi$ is a linear group action when restricted to $G \times W \subseteq G \times V$. Then $(W,\phi_W)$; where $\phi_W$ is the restriction of $\phi$ on $G\times W$, is called a $G$-submodule of $(V,\phi)$.


Definition:G-Module Homomorphism

Let $\left({G, \cdot}\right)$ be a group. Let $\left({V, \phi}\right)$ and $\left({W, \mu}\right)$ be $G$-modules. Then a linear mapping $f: V \to W$ is called a '''$G$-module homomorphism''' {{iff}}: :$\forall g \in G: \forall v \in V: f \left({\phi \left({g, v}\right)}\right) = \mu \left({g, f \left({v}\right)}\right)$


Definition:Null Space

Let: :$ \mathbf A_{m \times n} = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \\ \end{bmatrix}$,  $\mathbf x_{n \times 1} = \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix}$, $\mathbf 0_{m \times 1} = \begin{bmatrix} 0 \\ 0 \\ \vdots \\ 0 \end{bmatrix}$ be matrices where each column is a member of a real vector space. The set of all solutions to $\mathbf A \mathbf x = \mathbf 0$: :$\map {\operatorname N} {\mathbf A} = \set {\mathbf x \in \R^n : \mathbf {A x} = \mathbf 0}$ is called the '''null space''' of $\mathbf A$.


Definition:G-Module

Let $\left({V, +, \cdot}\right)$ be a vector space over a field $\left({k, \oplus, \circ}\right)$. Let $G$ be a group. Let $\phi : G \times V \to V$ be an linear group action of $G$ on $V$. Then $(V,\phi)$ is called a $G$-module. Category:Definitions/Group Actions {{SUBPAGENAME}} bof7sirmz3j9f62dk4a07bdtg1h1qf7


Definition:Limit of Increasing Sequence of Sets

Let $\left({S_n}\right)_{n \in \N}$ be an increasing sequence of sets. Let $S = \displaystyle \bigcup_{n \mathop \in \N} S_n$. Then $S$ is said to be the '''limit''' of $\left({S_n}\right)_{n \in \N}$, and one writes $S_n \uparrow S$.


Definition:Decreasing Sequence of Sets

Let $\sequence {S_k}_{k \mathop \in \N}$ be a nested sequence of subsets of $S$ such that: :$\forall k \in \N: S_k \supseteq S_{k + 1}$ Then $\sequence {S_k}_{k \mathop \in \N}$ is a '''decreasing sequence of sets (in $\SS$)'''.


Definition:Limit of Decreasing Sequence of Sets

Let $\left({S_n}\right)_{n \in \N}$ be a decreasing sequence of sets. Let $S = \displaystyle \bigcap_{n \mathop \in \N} S_n$. Then $S$ is said to be the '''limit''' of $\left({S_n}\right)_{n \in \N}$, and one writes $S_n \downarrow S$.


Definition:Reducible G-Module

Let $M$ be a $G$-module. Then $M$ is '''reducible''' {{iff}} the corresponding linear representation is reducible.


Definition:Edge Contraction

Let $G$ be an undirected graph. Let $e \in \map E G$ be an edge of $G$. Then the '''graph obtained by contracting $e$ in $G$''', denoted by $G / e$, is the graph $H$ defined by: :$\map V H = \paren {\map V G \setminus e} \cup \set v$ :$\map E H = \set {f \in \map E G: f \cap e = \O} \cup \set {u v: \exists f \in \map E G: u \in f \setminus e, f \cap e \ne \O}$ where $v \notin \map V G$ is a fresh object. {{explain|A lot of the above - see talk page (in prep)}} Informally, it is the graph obtained from $G$ by replacing the vertices incident to $e$ with a single vertex adjacent to all their neighbors.


Definition:Dirac Measure

Let $\left({X, \Sigma}\right)$ be a measurable space. Let $x \in X$ be any point in $X$. Then the '''Dirac measure at $x$''', denoted $\delta_x$ is the measure defined by: :$\delta_x: \Sigma \to \overline \R, \ \delta_x \left({E}\right) := \begin{cases}0 & \text{if } x \notin E \\ 1 & \text{if } x \in E\end{cases}$ where $\overline \R$ denotes the extended set of real numbers. That $\delta_x$ actually is a measure is shown on Dirac Measure is Measure. In fact, Dirac measure is a probability measure.


Definition:Enumeration

=== Finite Sets === {{:Definition:Enumeration/Finite}} === Countably Infinite Sets === {{:Definition:Enumeration/Countably Infinite}}


Definition:Discrete Probability Measure

Let $\Omega$ be a countable set. Let $\mathcal P \left({\Omega}\right)$ be its power set, regarded as a $\sigma$-algebra. Let $\left({p_\omega}\right)_{\omega \in \Omega} \subseteq \left[{0 \,.\,.\, 1}\right]$ be a subset of the closed unit interval in $\R$, indexed by $\Omega$. Suppose that $\displaystyle \sum_{\omega \mathop \in \Omega} p_\omega = 1$. {{explain|link to definition of sum over countable set}} The '''discrete probability measure on $\Omega$''', denoted $P$, is the mapping defined by: :$\displaystyle P: \mathcal P \left({\Omega}\right) \to \overline \R, \ P \left({S}\right) = \sum_{\omega \mathop \in \Omega} p_\omega \delta_\omega \left({S}\right)$ where $\overline \R$ denotes the extended real numbers, and $\delta_\omega$ is the Dirac measure at $\omega$. From this definition, it is seen that the name '''discrete probability measure''' is compatible with the notion of discrete measure, as $\Omega$ is countable. {{MissingLinks|to eg. Definition:Discrete Sample Space and connected stuff}} === Discrete Probability Space === The measure space $\left({\Omega, \mathcal P \left({\Omega}\right), P}\right)$ is called '''discrete probability space'''.


Definition:Proper G-Submodule

Let $\left({V, \phi}\right)$ be a $G$-module. A $G$-submodule of $V$ is called '''proper''' iff it is a proper vector subspace of $V$. {{SUBPAGENAME}} hu0luq718d6r29qzya71z71pi3o2iro


Definition:Trivial G-Module

Let $\left({V, \phi}\right)$ be a $G$-module. Then $\left({V, \phi}\right)$ is said to be '''trivial''' iff $V = \left\{{\mathbf 0}\right\}$. {{SUBPAGENAME}} bhlwwt3n0a6zqpyzdyhlznjnsw4gdk3


Definition:Irreducible (Representation Theory)/G-Module

A $G$-module is '''irreducible''' {{iff}} the corresponding linear representation is irreducible.


Definition:Equivalent Linear Representations

Let $ \struct {G, \cdot}$ be a group. Consider two linear representations $\rho: G \to \GL V$ and $\rho': G \to \GL W$ of $G$. Then $\rho$ and $\rho'$ are called '''equivalent (linear representations)''' {{iff}} their correspondent $G$-modules using Correspondence between Linear Group Actions and Linear Representations are isomorphic. {{rewrite|above line needs rewriting, but I can't come up with a suitable replacement}} {{explain|The "isomorphic" link goes to a generic "abstract algebra" page, but I believe no actual definition has been made for an isomorphism between two G-modules.}} Category:Definitions/Representation Theory i9p93ofh5e70d1hkt0tg49v26zq2kgm


Definition:Infinite Measure

Let $\left({X, \Sigma}\right)$ be a measurable space. Then the '''infinite measure''' is the measure defined by: :$\mu: \Sigma \to \overline \R, \ \mu \left({E}\right) := \begin{cases} 0 & : \text{if $E = \varnothing$} \\ +\infty & : \text{otherwise}\end{cases}$ where $\overline \R$ denotes the extended real numbers.


Definition:Null Measure

Let $\struct {X, \Sigma}$ be a measurable space. Then the '''null measure''' is the measure defined by: :$\mu: \Sigma \to \overline \R: \map \mu E := 0$ where $\overline \R$ denotes the extended real numbers.


Definition:Preimage/Relation

Let $\RR \subseteq S \times T$ be a relation. Let $\RR^{-1} \subseteq T \times S$ be the inverse relation to $\RR$, defined as: :$\RR^{-1} = \set {\tuple {t, s}: \tuple {s, t} \in \RR}$ === Preimage of Element === {{:Definition:Preimage/Relation/Element}} === Preimage of Subset === {{:Definition:Preimage/Relation/Subset}} === Preimage of Relation === {{:Definition:Preimage/Relation/Relation}}


Definition:Preimage/Mapping

Let $f: S \to T$ be a mapping. Let $f^{-1} \subseteq T \times S$ be the inverse of $f$, considered as a relation: :$f^{-1} = \set {\tuple {t, s}: \map f s = t}$ === Preimage of Element === {{:Definition:Preimage/Mapping/Element}} === Preimage of Subset === {{:Definition:Preimage/Mapping/Subset}} === Preimage of Mapping === {{:Definition:Preimage/Mapping/Mapping}}


Definition:Preimage/Relation/Element

Every $s \in S$ such that $\tuple {s, t} \in \RR$ is called '''a preimage of $t$'''. In some contexts, it is not individual elements that are important, but ''all'' elements of $S$ which are of interest. Thus '''the preimage of $t \in T$''' is defined as: :$\map {\RR^{-1} } t := \set {s \in S: \tuple {s, t} \in \RR}$ This can also be written: :$\map {\RR^{-1} } t := \set {s \in \Img {\RR^{-1} }: \tuple {t, s} \in \RR^{-1} }$ That is, '''the preimage of $t$ under $\RR$''' is the image of $t$ under $\RR^{-1}$.


Definition:Preimage/Relation/Subset

Let $Y \subseteq T$. The '''preimage of $Y$ under $\RR$''' is defined as: :$\RR^{-1} \sqbrk Y := \set {s \in S: \exists t \in Y: \tuple {s, t} \in \RR}$ That is, the '''preimage of $Y$ under $\RR$''' is the image of $Y$ under $\RR^{-1}$: :$\RR^{-1} \sqbrk Y := \set {s \in S: \exists t \in Y: \tuple {t, s} \in \RR^{-1} }$ If no element of $Y$ has a '''preimage''', then $\RR^{-1} \sqbrk Y = \O$.


Definition:Preimage/Relation/Relation

The '''preimage''' of $\RR \subseteq S \times T$ is: :$\Preimg \RR := \RR^{-1} \sqbrk T = \set {s \in S: \exists t \in T: \tuple {s, t} \in \RR}$


Definition:Preimage/Mapping/Element

Every $s \in S$ such that $\map f s = t$ is called '''a preimage''' of $t$. '''The preimage''' of an element $t \in T$ is defined as: :$\map {f^{-1} } t := \set {s \in S: \map f s = t}$ This can also be expressed as: :$\map {f^{-1} } t := \set {s \in \Img {f^{-1} }: \tuple {t, s} \in f^{-1} }$ That is, the '''preimage of $t$ under $f$''' is the image of $t$ under $f^{-1}$.


Definition:Preimage/Mapping/Subset

Let $Y \subseteq T$.


Definition:Preimage/Mapping/Mapping

The '''preimage of $f$''' is defined as: :$\Preimg f := \set {s \in S: \exists t \in T: f \paren s = t}$ That is: :$\Preimg f := f^{-1} \sqbrk T$ where $f^{-1} \sqbrk T$ is the image of $T$ under $f^{-1}$. In this context, $f^{-1} \subseteq T \times S$ is the the inverse of $f$. It is a relation but not necessarily itself a mapping.


Definition:Image (Set Theory)/Relation/Relation

The '''image''' of $\RR$ is the set: :$\Img \RR := \RR \sqbrk S = \set {t \in T: \exists s \in S: \tuple {s, t} \in \RR}$


Let $f: S \to T$ be a mapping. <section notitle="true" name="tc"> === Definition 1 === {{:Definition:Image (Set Theory)/Mapping/Mapping/Definition 1}} === Definition 2 === {{:Definition:Image (Set Theory)/Mapping/Mapping/Definition 2}} </section>


Definition:Arbitrarily Large

Let $P$ be a property of real numbers. We say that '''$P \left({x}\right)$ holds for arbitrarily large $x$''' (or '''there exist arbitrarily large $x$ such that $P \left({x}\right)$ holds''') {{iff}}: :$\forall a \in \R: \exists x \in \R: x \ge a: P \left({x}\right)$ That is: :''For any real number $a$, there exists a (real) number not less than $a$ such that the property $P$ holds.'' or, more informally and intuitively: :''However large a number you can think of, there will be an even larger one for which $P$ still holds.''


Definition:Valuation

Let $\left({R, +, \cdot}\right)$ be a ring. A '''valuation on $R$''' is a mapping: : $\nu: R \to \Z \cup \left\{{+\infty}\right\}$ which fulfils the '''valuation axioms''': {{:Definition:Valuation Axioms}}


Definition:P-adic Valuation

Let $p \in \N$ be a prime number. === Integers === {{:Definition:P-adic Valuation/Integers}} === Rational Numbers === {{:Definition:P-adic Valuation/Rational Numbers}}


Definition:Angle Between Vectors

Let $\mathbf v, \mathbf w$ be two non-zero vectors in $\R^n$. === Case 1 === Suppose that $\mathbf v$ and $\mathbf w$ are not scalar multiples of each other: :$\neg \exists \lambda \in \R: \mathbf v = \lambda \mathbf w$ Then the '''angle between $\mathbf v$ and $\mathbf w$''' is defined as follows: Describe a triangle with lengths corresponding to: :$\norm {\mathbf v}, \norm {\mathbf w}, \norm {\mathbf v - \mathbf w}$ where $\norm {\, \cdot \,}$ denotes vector length: :600px The angle formed between the two sides with lengths $\norm {\mathbf v}$ and $\norm {\mathbf w}$ is called the '''angle between vectors $\mathbf v$ and $\mathbf w$'''. By convention, the angle is taken between $0$ and $\pi$. === Case 2 === Suppose that $\mathbf v$ and $\mathbf w$ ''are'' scalar multiples of each other: :$\exists \lambda \in \R: \mathbf v = \lambda \mathbf w$ As $\mathbf v$ and $\mathbf w$ as non-zero, $\lambda \ne 0$. If $\lambda > 0$, then the angle between $\mathbf v$ and $\mathbf w$ is defined as a zero angle, that is: :$\theta = 0$ If $\lambda < 0$, then the angle between $\mathbf v$ and $\mathbf w$ is defined as a straight angle, that is: :$\theta = \pi$


Definition:P-adic Number

Let $p$ be any prime number. Let $\norm {\,\cdot\,}_p$ be the p-adic norm on the rationals $\Q$. By P-adic Norm is Non-Archimedean Norm then $\norm {\,\cdot\,}_p$ is a non-archimedean norm on $\Q$ and the pair $\struct {\Q, \norm {\,\cdot\,}_p}$ is a valued field. === $p$-adic Norm Completion of Rational Numbers === {{:Definition:P-adic Number/P-adic Norm Completion of Rational Numbers}} === Quotient of Cauchy Sequences in $p$-adic Norm === {{:Definition:P-adic Number/Quotient of Cauchy Sequences in P-adic Norm}} === $p$-adic Norm on $\Q_p$ === {{:Definition:P-adic Norm on P-adic Numbers}}


Definition:Darboux Integral/Geometric Interpretation

The expression $\displaystyle \int_a^b \map f x \rd x$ can be (and frequently is) interpreted as '''the area under the graph'''.  This follows from the definition of the definite integral as a sum of the product of the lengths of intervals and the "height" of the function being integrated in that interval and the formula for the area of a rectangle. A depiction of the lower and upper sums illustrates this: :350px  350px It can intuitively be seen that as the number of points in the subdivision increases, the more "accurate" the lower and upper sums become. Also note that if the graph is below the $x$-axis, the signed area under the graph becomes negative.


Definition:Intersection Measure

Let $\left({X, \Sigma, \mu}\right)$ be a measure space. Let $F \in \Sigma$. Then the '''intersection measure (of $\mu$ by $F$)''' is the mapping $\mu_F: \Sigma \to \overline{\R}$, defined by: :$\mu_F \left({E}\right) = \mu \left({E \cap F}\right)$ It is in fact a measure on $\left({X, \Sigma}\right)$, as shown on Intersection Measure is Measure. {{expand|verify name, Schilling gives none}}


Definition:Perpendicular (Linear Algebra)

Let $\mathbf u$, $\mathbf v$ be non-zero vectors in the Euclidean space $\R^n$. :$\mathbf u$ and $\mathbf v$ are '''perpendicular''' {{iff}} the angle between them is a right angle.


Definition:Complete Measure Space

Let $\left({X, \Sigma, \mu}\right)$ be a measure space. Suppose that the family of $\mu$-null sets $\mathcal{N}_{\mu}$ satisfies the following condition: :$\forall N \in \mathcal{N}_{\mu}: \forall M \subseteq N: M \in \mathcal{N}_{\mu}$ That is, any subset of a $\mu$-null set is again a $\mu$-null set. Then $\left({X, \Sigma, \mu}\right)$ is said to be a '''complete measure space'''.


Definition:Completion (Measure Space)

Let $\left({X, \Sigma, \mu}\right), \left({\tilde X, \Sigma^*, \bar \mu}\right)$ be measure spaces. Then '''$\left({\tilde X, \Sigma^*, \bar \mu}\right)$ is a completion of $\left({X, \Sigma, \mu}\right)$''' or '''$\left({\tilde X, \Sigma^*, \bar \mu}\right)$ completes $\left({X, \Sigma, \mu}\right)$''' iff the following conditions hold: :$(1):\quad \left({\tilde X, \Sigma^*, \bar \mu}\right)$ is a complete measure space :$(2):\quad \tilde X = X$ :$(3):\quad \Sigma$ is a sub-$\sigma$-algebra of $\Sigma^*$ :$(4):\quad \forall E \in \Sigma: \bar \mu \left({E}\right) = \mu \left({E}\right)$, i.e. $\bar \mu \restriction_{\Sigma} = \mu$


Definition:Restricted Measure

Let $\left({X, \Sigma, \mu}\right)$ be a measure space. Let $\Sigma'$ be a sub-$\sigma$-algebra of $\Sigma$. Then the '''restricted measure on $\Sigma'$''' or the '''restriction of $\mu$ to $\Sigma'$''' is the mapping $\nu: \Sigma \to \overline{\R}$ defined by: :$\forall E' \in \Sigma': \nu \left({E'}\right) = \mu \left({E'}\right)$ That is, $\nu$ is the restriction $\mu \restriction_{\Sigma'}$.


Definition:Dynkin System

Let $X$ be a set, and let $\mathcal D \subseteq \mathcal P \left({X}\right)$ be a collection of subsets of $X$. Then $\mathcal D$ is called a '''Dynkin system (on $X$)''' {{iff}} it satisfies the following conditions: :$(1):\quad X \in \mathcal D$ :$(2):\quad \forall D \in \mathcal D: X \setminus D \in \mathcal D$ :$(3):\quad$ For all pairwise disjoint sequences $\left({D_n}\right)_{n \in \N}$ in $\mathcal D$, $\displaystyle \bigcup_{n \mathop \in \N} D_n \in \mathcal D$


Definition:Dynkin System Generated by Collection of Subsets

Let $X$ be a set. Let $\mathcal G \subseteq \mathcal P \left({X}\right)$ be a collection of subsets of $X$. Then the '''Dynkin system generated by $\mathcal G$''', denoted $\delta \left({\mathcal G}\right)$, is the smallest Dynkin system on $X$ that contains $\mathcal G$. That is, $\delta \left({\mathcal G}\right)$ is subject to: :$(1):\quad \mathcal G \subseteq \delta \left({\mathcal G}\right)$ :$(2):\quad \mathcal G \subseteq \mathcal D \implies \delta \left({\mathcal G}\right) \subseteq \mathcal D$ for any Dynkin system $\mathcal D$ on $X$ In fact, $\delta \left({\mathcal G}\right)$ always exists, and is unique, as proved on Existence and Uniqueness of Dynkin System Generated by Collection of Subsets. === Generator === One says that $\mathcal G$ is a '''generator''' for $\delta \left({\mathcal G}\right)$.


Definition:Central Product

Let $G$ and $H$ be groups. Let $Z$ and $W$ be central subgroups of $G$ and $H$ respectively. Let: :$Z \cong W$ where $\cong$ denotes isomorphism. Let such a group isomorphism be $\theta: Z \to W$. Let $X$ be the set defined as: :$X = \set {\tuple {x, \map \theta x^{-1} }: x \in Z}$ Then the quotient group $\struct {G \times H} / X$ is denoted $\struct {G \times_\theta H}$ and is called the '''central product of $G$ and $H$ via $\theta$'''.


Definition:Coset Product

Let $\struct {G, \circ}$ be a group. Let $N$ be a normal subgroup of $G$. Let $a, b \in G$. The '''coset product''' of $a \circ N$ and $b \circ N$ is defined as: :$\paren {a \circ N} \circ \paren {b \circ  N} = \paren {a \circ b} \circ N$ where $a \circ N$ and $b \circ N$ are the left cosets of $a$ and $b$ by $N$.


Definition:Translation-Invariant Measure

Let $\mu$ be a measure on $\R^n$ equipped with the Borel $\sigma$-algebra $\mathcal B \left({\R^n}\right)$. Then $\mu$ is said to be '''translation-invariant''' or '''invariant under translations''' iff: :$\forall x \in \R^n, \forall B \in \mathcal B: \mu \left({x + B}\right) = \mu \left({B}\right)$ where $x + B$ is the set $\left\{{x + b: b \in B}\right\}$. {{SUBPAGENAME}} nid2a2b6pzp4gwqrs3jjup0l3wgnkzz


Definition:Invariant Measure

Let $\left({X, \Sigma, \mu}\right)$ be a measure space. Let $\theta: X \to X$ be an $\Sigma / \Sigma$-measurable mapping. Then $\mu$ is said to be a '''$\theta$-invariant measure''' or to be '''invariant under $\theta$''' iff: :$\forall E \in \Sigma: \mu \left({\theta^{-1} \left({E}\right) }\right) = \mu \left({E}\right)$ In terms of a pushforward measure, this can be concisely formulated as: :$\theta_* \mu = \mu$


Definition:Independent Sigma-Algebras

Let $\struct {\Omega, \EE, \Pr}$ be a probability space. Let $\Sigma$ and $\Sigma'$ be sub-$\sigma$-algebras of $\EE$. Then $\Sigma$ and $\Sigma'$ are said to be '''($\Pr$-)independent''' {{iff}}: :$\forall E \in \Sigma, E' \in \Sigma': \map \Pr {E \cap E'} = \map \Pr E \map \Pr {E'}$


Definition:Stable under Intersection

Let $X$ be a set, and let $\SS \subseteq \powerset X$ be a collection of subsets of $X$. Then $\SS$ is said to be '''stable under intersection(s)''', or simply '''$\cap$-stable''', {{iff}}: :$\forall S, T \in \SS: S \cap T \in \SS$ Category:Definitions/Set Systems pnx3xh2mygzcfi5yx75kaq9xnpjf63e


Definition:Sub-Sigma-Algebra

Let $X$ be a set, and let $\mathcal A, \mathcal B$ be $\sigma$-algebras on $X$. Then $\mathcal B$ is said to be a '''sub-sigma-algebra''' or '''sub-$\sigma$-algebra''' of $\mathcal A$ iff $\mathcal B \subseteq \mathcal A$. {{SUBPAGENAME}} 501i2ym6ibfjwwspmzl7ljcqmez1jpm


Definition:Lebesgue Pre-Measure

Let $\mathcal J_{ho}$ be the collection of half-open $n$-rectangles. '''$n$-dimensional Lebesgue pre-measure''' is the mapping $\lambda^n: \mathcal J_{ho} \to \overline \R_{\ge 0}$ given by: :$\displaystyle \lambda^n \left({ \left[[{\mathbf a \,.\,.\, \mathbf b}\right)) }\right) = \prod_{i \mathop = 1}^n \left({b_i - a_i}\right)$ where $\overline \R_{\ge 0}$ denotes the set of positive extended real numbers.


Definition:Diffuse Measure

Let $\left({X, \Sigma, \mu}\right)$ be a measure space. The measure $\mu$ is said to be '''diffuse''' or '''non-atomic''' {{iff}} there are no $\mu$-atoms.


Definition:Half-Open Rectangle

Let $a_1, \ldots, a_n, b_1, \ldots, b_n$ be real numbers. The set: :$\ds \prod_{i \mathop = 1}^n \hointr {a_i} {b_i} = \hointr {a_1} {b_1} \times \cdots \times \hointr {a_n} {b_n} \subseteq \R^n$ is called an '''half-open rectangle in $\R^n$''' or '''half-open $n$-rectangle'''. Here, $\times$ denotes Cartesian product. The collection of all '''half-open $n$-rectangles''' is denoted $\JJ_{ho}$, or $\JJ_{ho}^n$ if the dimension $n$ is to be emphasized. In case $a_i \ge b_i$ for some $i$, the rectangle is taken to be the empty set $\O$. This is in accordance with the result Cartesian Product is Empty iff Factor is Empty for general Cartesian products.


Definition:Discrete Measure

Let $\struct {X, \Sigma, \mu}$ be a measure space. Then $\mu$ is said to be a '''discrete measure''' {{iff}} it is a series of Dirac measures. That is, {{iff}} there exist: :A sequence $\sequence {x_n}_{n \mathop \in \N}$ in $X$ :A sequence $\sequence {\lambda_n}_{n \mathop \in \N}$ in $\R$ such that: :$(1):\quad \forall E \in \Sigma: \map \mu E = \displaystyle \sum_{n \mathop \in \N} \lambda_n \, \map {\delta_{x_n} } E$ where $\delta_{x_n}$ denotes the Dirac measure at $x_n$. By Series of Measures is Measure, defining $\mu$ by $(1)$ yields a measure.


Definition:Series of Measures

Let $\left({X, \Sigma}\right)$ be a measurable space. Let $\left({\mu_n}\right)_{n \in \N}$ be a sequence of measures on $\left({X, \Sigma}\right)$. Let $\left({\lambda_n}\right)_{n \in \N}$ be a sequence of positive real numbers. Then the mapping $\mu: \Sigma \to \overline \R$, defined by: :$\displaystyle \mu \left({E}\right) := \sum_{n \mathop \in \N} \lambda_n \mu_n \left({E}\right)$ is called a '''series of measures'''.


Definition:Induced Outer Measure

Let $\mathcal S$ be a collection of subsets of a set $X$, and suppose that $\varnothing \in \mathcal S$. Let $\mu$ be a pre-measure on $\mathcal S$. The '''outer measure induced by the pre-measure $\mu$''' is the mapping $\mu^* : \mathcal P \left({X}\right) \to \overline \R_{\ge 0}$ defined as: :$\displaystyle \mu^* \left({S}\right) = \inf \ \left\{ {\sum_{n \mathop = 1}^\infty \mu \left({A_n}\right) : \forall n \in \N: A_n \in \mathcal S, \ S \subseteq \bigcup_{n \mathop = 1}^\infty A_n} \right\}$ Here, $\mathcal P \left({X}\right)$ denotes the power set of $X$, and $\overline \R_{\ge 0}$ denotes the set of positive extended real numbers. The infimum of the empty set is taken to be $+\infty$. It follows immediately by Construction of Outer Measure that the induced outer measure is an outer measure.


Definition:Homomorphism (Abstract Algebra)/Cartesian Product

Let $\phi: \struct {S, \circ} \to \struct {T, *}$ be a mapping from one algebraic structure $\struct {S, \circ}$ to another $\struct {T, *}$. We define the cartesian product $\phi \times \phi: S \times S \to T \times T$ as: :$\forall \tuple {x, y} \in S \times S: \map {\paren {\phi \times \phi} } {x, y} = \tuple {\map \phi x, \map \phi y}$ Hence we can state that $\phi$ is a homomorphism {{iff}}: :$\map \ast {\map {\paren {\phi \times \phi} } {x, y} } = \map \phi {\map \circ {x, y} }$ using the notation $\map \circ {x, y}$ to denote the operation $x \circ y$. The point of doing this is so we can illustrate what is going on in a commutative diagram: ::$\begin{xy} \xymatrix@L+2mu@+1em{  S \times S \ar[r]^*{\circ}      \ar[d]_*{\phi \times \phi} &  S \ar[d]^*{\phi} \\  T \times T  \ar[r]_*{\ast} &  T }\end{xy}$ Thus we see that $\phi$ is a homomorphism {{iff}} both of the composite mappings from $S \times S$ to $T$ have the same effect on all elements of $S \times S$.


Definition:Epimorphism (Category Theory)

Let $\mathbf C$ be a metacategory. An '''epimorphism''' is a morphism $f \in \mathbf C_1$ such that: : $g \circ f = h \circ f \implies g = h$ for all morphisms $g, h \in \mathbf C_1$ for which these compositions are defined. That is, an '''epimorphism''' is a morphism which is right cancellable.  One writes $f: C \twoheadrightarrow D$ to denote that $f$ is an '''epimorphism'''.


Definition:Ordered Structure Monomorphism

Let $\left({S, \circ, \preceq}\right)$ and $\left({T, *, \preccurlyeq}\right)$ be ordered structures. An '''ordered structure monomorphism''' from $\left({S, \circ, \preceq}\right)$ to $\left({T, *, \preccurlyeq}\right)$ is a mapping $\phi: S \to T$ that is both: :$(1): \quad$ A monomorphism, i.e. an injective homomorphism, from the structure $\left({S, \circ}\right)$ to the structure $\left({T, *}\right)$ :$(2): \quad$ An order embedding from the ordered set $\left({S, \preceq}\right)$ to the ordered set $\left({T, \preccurlyeq}\right)$.


Definition:Monomorphism (Category Theory)

Let $\mathbf C$ be a metacategory. A '''monomorphism''' is a morphism $f \in \mathbf C_1$ such that: :$f \circ g = f \circ h \implies g = h$ for all morphisms $g, h \in \mathbf C_1$ for which these compositions are defined. That is, a '''monomorphism''' is a morphism which is left cancellable.  One writes $f: C \rightarrowtail D$ to denote that $f$ is a '''monomorphism'''.


Definition:Complex Number/Real Part

Let $z = a + i b$ be a complex number. The '''real part''' of $z$ is the coefficient $a$. The '''real part''' of a complex number $z$ is usually denoted on {{ProofWiki}} by $\map \Re z$ or $\Re z$.


Definition:Complex Number/Imaginary Part

Let $z = a + i b$ be a complex number. The '''imaginary part''' of $z$ is the coefficient $b$ ('''note:''' not $i b$). The '''imaginary part''' of a complex number $z$ is usually denoted on {{ProofWiki}} by $\map \Im z$ or $\Im z$.


Definition:Strict Lower Closure

=== Strict Lower Closure of Element === {{:Definition:Strict Lower Closure/Element}} === Strict Lower Closure of Subset === {{:Definition:Strict Lower Closure/Set}}


Definition:Strict Upper Closure

=== Strict Upper Closure of Element === {{:Definition:Strict Upper Closure/Element}} === Strict Upper Closure of Subset === {{:Definition:Strict Upper Closure/Set}}


Definition:Upper Closure/Element

Let $\struct {S, \preccurlyeq}$ be an ordered set. Let $a \in S$. The '''upper closure of $a$ (in $S$)''' is defined as: :$a^\succcurlyeq := \set {b \in S: a \preccurlyeq b}$ That is, $a^\succcurlyeq$ is the set of all elements of $S$ that succeed $a$.


Definition:Order Topology

=== Definition 1 === {{:Definition:Order Topology/Definition 1}} === Definition 2 === {{:Definition:Order Topology/Definition 2}}


Definition:Upper Closure

{{:Definition:Upper Closure/Element}}


Definition:Lower Closure

{{:Definition:Lower Closure/Element}}


Definition:Projection (Mapping Theory)/First Projection

The '''first projection on $S \times T$''' is the mapping $\pr_1: S \times T \to S$ defined by: :$\forall \tuple {x, y} \in S \times T: \map {\pr_1} {x, y} = x$


Definition:Projection (Mapping Theory)/Second Projection

The '''second projection on $S \times T$''' is the mapping $\pr_2: S \times T \to T$ defined by: :$\forall \tuple {x, y} \in S \times T: \map {\pr_2} {x, y} = y$


Definition:Ordering on Extended Real Numbers

Let $\overline \R$ denote the extended real numbers. Extend the natural ordering $\le_\R$ on $\R$ to $\overline \R = \R \cup \set {+\infty, -\infty}$ by imposing: :$\forall x \in \overline \R: -\infty \le x$ :$\forall x \in \overline \R: x \le +\infty$ That is, considering the relations $\le$ and $\le_\R$ as subsets of $\overline \R \times \overline \R$: :${\le} := {\le_\R} \cup \set {\tuple {x, +\infty}: x \in \overline \R} \cup \set {\tuple {-\infty, x}: x \in \overline \R}$ where $\tuple {x, +\infty}$ and $\tuple {-\infty, x}$ denote ordered pairs in $\overline \R \times \overline \R$. The ordering $\le$ is called the '''(usual) ordering on $\overline \R$'''.


Definition:Topology on Extended Real Numbers

Let $\overline \R$ denote the extended real numbers. The '''(standard) topology on $\overline \R$''' is the order topology $\tau$ associated to the ordering on $\overline \R$.


Definition:External Direct Product/General Definition

Let $\left({S_1, \circ_1}\right), \left({S_2, \circ_2}\right), \ldots, \left({S_n, \circ_n}\right)$ be algebraic structures. Let $\displaystyle \mathcal S_n = \prod_{k \mathop = 1}^n S_k$ be the cartesian product of $S_1, S_2, \ldots, S_n$. Let $\circledcirc_n$ be the operation induced on $\mathcal S_n$ by $\circ_1, \ldots, \circ_n$ defined as: :$\left({s_1, s_2, \ldots, s_n}\right) \circledcirc_n \left({t_1, t_2, \ldots, t_n}\right) := \begin{cases} s_1 \circ_1 t_1 & : n = 1 \\ \left({s_1 \circ_1 t_1, s_2 \circ_2 t_2}\right) & : n = 2 \\ \left({\left({s_1, s_2, \ldots, s_{n-1} }\right) \circledcirc_{n-1} \left({t_1, t_2, \ldots, t_{n-1}}\right), s_n \circ_n t_n}\right) & : n > 2 \end{cases}$ for all ordered $n$-tuples in $\mathcal S_n$. That is: :$\left({s_1, s_2, \ldots, s_n}\right) \circledcirc_n \left({t_1, t_2, \ldots, t_n}\right) := \left({s_1 \circ_1 t_1, s_2 \circ_2 t_2, \ldots, s_n \circ_n t_n}\right)$ The algebraic structure $\left({\mathcal S_n, \circledcirc_n}\right)$  is called the '''(external) direct product''' of $\left({S_1, \circ_1}\right), \left({S_2, \circ_2}\right), \ldots, \left({S_n, \circ_n}\right)$.


Definition:External Direct Product/Structures with Two Operations

Let $\left({S_1, +_1 ,\circ_1}\right), \left({S_2, +_2 ,\circ_2}\right), \ldots, \left({S_n, +_n ,\circ_n}\right)$ be algebraic structures with two operations. Let $\displaystyle S = \prod_{k \mathop = 1}^n S_k$ be as defined in cartesian product. The operation $+$ induced on $S$ by $+_1, \ldots, +_n$ is defined as: :$\left({s_1, s_2, \ldots, s_n}\right) + \left({t_1, t_2, \ldots, t_n}\right) = \left({s_1 +_1 t_1, s_2 +_2 t_2, \ldots, s_n +_n t_n}\right)$ The operation $\circ $ induced on $S$ by $\circ_1, \ldots, \circ_n$ is defined as: :$\left({s_1, s_2, \ldots, s_n}\right) \circ \left({t_1, t_2, \ldots, t_n}\right) = \left({s_1 \circ_1 t_1, s_2 \circ_2 t_2, \ldots, s_n \circ_n t_n}\right)$ for all ordered $n$-tuples in $S$. The algebraic structure $\left({S, +, \circ}\right)$  is called the '''(external) direct product''' of $\left({S_1, +_1 ,\circ_1}\right), \left({S_2, +_2 ,\circ_2}\right), \ldots, \left({S_n, +_n ,\circ_n}\right)$.


Definition:Cartesian Product/Finite

Let $\sequence {S_n}$ be a sequence of sets.  The '''cartesian product''' of $\sequence {S_n}$ is defined as: :$\displaystyle \prod_{k \mathop = 1}^n S_k = \set {\tuple {x_1, x_2, \ldots, x_n}: \forall k \in \N^*_n: x_k \in S_k}$ It is also denoted $S_1 \times S_2 \times \cdots \times S_n$. Thus $S_1 \times S_2 \times \cdots \times S_n$ is the set of all ordered $n$-tuples $\tuple {x_1, x_2, \ldots, x_n}$ with $x_k \in S_k$.


Definition:Canonical Injection (Abstract Algebra)/General Definition

Let $\struct {S_1, \circ_1}, \struct {S_2, \circ_2}, \dotsc, \struct {S_j, \circ_j}, \dotsc, \struct {S_n, \circ_n}$ be algebraic structures with identities $e_1, e_2, \dotsc, e_j, \dotsc, e_n$ respectively. Then the '''canonical injection''' $\displaystyle \inj_j: \struct {S_j, \circ_j} \to \prod_{i \mathop = 1}^n \struct {S_i, \circ_i}$ is defined as: :$\map {\inj_j} x = \tuple {e_1, e_2, \dotsc, e_{j - 1}, x, e_{j + 1}, \dotsc, e_n}$


Definition:Sturm-Liouville Theory

'''Sturm-Liouville theory''' is the branch of mathematical physics concerned with the eigenvalues and eigenfunctions arising from the Sturm-Liouville equation: :$\map {\dfrac \d {\d x} } {\map p x \dfrac {\d y} {\d x} } + \map q x y = -\lambda \map w x y$


Definition:Sturm-Liouville Equation

A classical '''Sturm-Liouville equation''' is a real second order ordinary linear differential equation of the form: :$ (1): \quad \displaystyle - \frac {\mathrm d} {\mathrm d x} \left({p \left({x}\right) \frac {\mathrm d y} {\mathrm d x}}\right) + q \left({x}\right) y = \lambda w \left({x}\right) y$ where $y$ is a function of the free variable $x$. The functions $p \left({x}\right)$, $q \left({x}\right)$ and $w \left({x}\right)$ are specified. In the simplest cases they are continuous on the closed interval $\left[{a \,.\,.\, b}\right]$. In addition: :$(1a): \quad p \left({x}\right) > 0$ has a continuous derivative :$(1b): \quad w \left({x}\right) > 0$ :$(1c): \quad y$ is typically required to satisfy some boundary conditions at $a$ and $b$. === Weight Function === The function $w \left({x}\right)$, which is sometimes called $r \left({x}\right)$, is called the '''weight function''' or '''density function'''.  === Eigenvalues === The value of $\lambda$ is not specified in the equation. Finding the values of $\lambda$ for which there exists a non-trivial solution of $(1)$ satisfying the boundary conditions is part of the problem called the Sturm-Liouville problem (S-L).  Such values of $\lambda$ when they exist are called the '''eigenvalues''' of the boundary value problem defined by $(1)$ and the prescribed set of boundary conditions. The corresponding solutions (for such a $\lambda$) are the '''eigenfunctions''' of this problem.


Definition:Column Space

Let $R$ be a ring. Let: :$\mathbf A_{m \times n} = \begin{bmatrix} a_{1 1} & a_{1 2} & \cdots & a_{1 n} \\ a_{2 1} & a_{2 2} & \cdots & a_{2 n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m 1} & a_{m 2} & \cdots & a_{m n} \\ \end{bmatrix}$ be a matrix over $R$ such that every column is defined as a vector: :$\forall i: 1 \le i \le m: \begin {bmatrix} a_{1 i} \\ a_{2 i} \\ \vdots \\ a_{m i} \end {bmatrix} \in \mathbf V$ where $\mathbf V$ is some vector space. Then the '''column space of $\mathbf A$''' is the linear span of all such column vectors: :$\map {\mathrm C} {\mathbf A} = \map {\mathrm {span} } {\begin {bmatrix} a_{1 1} \\ a_{2 1} \\ \vdots \\ a_{m 1} \end {bmatrix}, \begin {bmatrix} a_{1 2} \\ a_{2 2} \\ \vdots \\ a_{m 2} \end {bmatrix}, \cdots, \begin {bmatrix} a_{1 n} \\ a_{2 n} \\ \vdots \\ a_{ mn} \end {bmatrix} }$


Definition:Row Space

Let $R$ be a ring. Let $\mathbf A$ be a matrix over $R$. Let $\mathbf A^\intercal$ be the transpose of $\mathbf A$.   Let the columns of $\mathbf A^\intercal$ be members of a vector space. The '''row space of $\mathbf A$''' is defined as the column space of $\mathbf A^\intercal$.


Definition:Simultaneous Linear Equations/Matrix Representation

A '''system of simultaneous linear equations''' can be expressed as: :$\mathbf A \mathbf x = \mathbf b$ where: :$\mathbf A = \begin {bmatrix} \alpha_{1 1} & \alpha_{1 2} & \cdots & \alpha_{1 n} \\ \alpha_{2 1} & \alpha_{2 2} & \cdots & \alpha_{2 n} \\ \vdots & \vdots & \ddots & \vdots \\ \alpha_{m 1} & \alpha_{m 2} & \cdots & \alpha_{m n} \\ \end {bmatrix}$,  $\mathbf x = \begin {bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix}$, $\mathbf b = \begin {bmatrix} \beta_1 \\ \beta_2 \\ \vdots \\ \beta_m \end {bmatrix}$ are matrices.