hcju/proofwikips huggingface dataset:
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'''  $\operatorname{cl} \left({T \left({\operatorname{ball} H}\right) }\right)$ 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''',  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'''  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 $\alpha$''' : :$h \in \map \ker {A - \alpha I}$ That is,  $Ah = \alpha h$. === 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''' : :$\forall s \in S: s \preceq \map \phi s$
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}}
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.
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''' : :$\forall a, b \in S: x \circ a = x \circ b \implies a = b$
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''' : :$\exists m \in T: \forall i \in \N: m \preceq x_i$
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''' : :$\exists M \in T: \forall i \in \N: x_i \preceq M$
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.
Definition:Unbounded Divergent Sequence/Real Sequence
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 $\infty$''' : :$\forall H > 0: \exists N: \forall n > N: \cmod {z_n} > H$ 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.
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 $S$)'''  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: :$f^{-1} \left({\Sigma'}\right) := \left\{{f^{-1} \left({E'}\right): E' \in \Sigma'}\right\}$
Definition:Sigma-Algebra Generated by Collection of Subsets
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 $\Sigma$)'''  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'''  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:Limit of Vector-Valued Function
=== Definition 1 === {{:Definition:Limit of Vector-Valued Function/Definition 1}} === Definition 2 === {{:Definition:Limit of Vector-Valued Function/Definition 2}}
Definition:Derivative/Vector-Valued Function
=== 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'''  it is not reducible. That is,  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 $X$)'''  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,  $\MM$ is closed under countable unions and countable intersections.