hcju/proofwikips huggingface dataset:
Definition:Addition/Complex Numbers
The '''addition operation''' in the domain of complex numbers $\C$ is written $+$. Let $z = a + i b, w = c + i d$ where $a, b, c, d \in \R, i^2 = -1$. Then $z + w$ is defined as: :$\paren {a + i b} + \paren {c + i d} = \paren {a + c} + i \paren {b + d}$
Definition:Category of Vector Spaces
The '''category of vector spaces''', denoted $\mathbf{Vect}$, is the metacategory with: {{DefineCategory |ob = vector spaces |mor = linear transformations }} {{DefinitionWanted|Explicate dependence on base field}}
Definition:Class (Descriptive Statistics)
Let $D$ be a finite collection of $n$ data regarding some quantitative variable.
Definition:Class (Descriptive Statistics)/Integer Data
Let the data in $D$ be described by natural numbers or by integers. Let $d_{\min}$ be the value of the smallest datum in $D$. Let $d_{\max}$ be the value of the largest datum in $D$.   Let $P = \left\{{x_0, x_1, x_2, \ldots, x_{n-1}, x_n}\right\} \subseteq \Z$ be a subdivision of $\left[{a \,.\,.\, b}\right]$, where $a \le x_0 \le x_n \le b$. The integer interval $\left[{a \,.\,.\, b}\right]$, where $a \le d_{\min} \le d_\max \le b$, is said to be divided into '''classes''' of integer intervals of the forms $\left[{x_i \,.\,.\, x_{i+1}}\right]$ or $\left[{x_i \,.\,.\, x_i}\right]$ : : Every datum is assigned into exactly one class : Every class is disjoint from every other : The union of all classes contains the entire integer interval $\left[{x_0 \,.\,.\, x_n}\right]$ By convention, the first and last classes are not empty classes.
Definition:Class (Descriptive Statistics)/Real Data
Let the data in $D$ be described by rational numbers or by real numbers. Let $d_{\min}$ be the value of the smallest datum in $D$. Let $d_{\max}$ be the value of the largest datum in $D$.   Let $P = \left\{{x_0, x_1, x_2, \ldots, x_{n-1}, x_n}\right\} \subseteq \R$ be a subdivision of $\left[{a \,.\,.\, b}\right]$, where $a \le x_0 \le x_n \le b$. The closed real interval $\left[{a \,.\,.\, b}\right]$, where $a \le d_{\text{min}} \le d_{\text{max}} \le b$, is said to be divided into '''classes''' of real intervals with endpoints $x_i$ and $x_{i+1}$ : : Every datum is assigned into exactly one class : Every class is disjoint from every other : The union of all classes contains the entire real interval $\left[{x_0 \,.\,.\, x_n}\right]$ The classes may be any combination of open, closed, or half-open intervals that fulfill the above criteria, but usually: : Every class except the last is of the form $\left[{x_i \,.\,.\, x_{i+1}}\right)$ : The last class is of the form  $\left[{x_{n-1} \,.\,.\, x_n}\right]$  By convention, the first and last classes are not empty classes.
Definition:Pointwise Addition
The (binary) operation of '''pointwise addition''' is defined on $\mathbb F^S$ as: :$+: \mathbb F^S \times \mathbb F^S \to \mathbb F^S: \forall f, g \in \mathbb F^S:$ ::$\forall s \in S: \map {\paren {f + g} } s := \map f s + \map g s$ where the $+$ on the  is conventional arithmetic addition.
Definition:Pointwise Addition of Complex-Valued Functions
Let $f, g: S \to \C$ be complex-valued functions. Then the '''pointwise sum of $f$ and $g$''' is defined as: :$f + g: S \to \C:$ ::$\forall s \in S: \map {\paren {f + g} } s := \map f s + \map g s$ where $+$ on the  is complex addition.
Definition:Pointwise Operation on Complex-Valued Functions
Let $\C^S$ be the set of all mappings $f: S \to \C$, where $\C$ is the set of complex numbers. Let $\oplus$ be a binary operation on $\C$. Define $\oplus: \C^S \times \C^S \to \C^S$, called '''pointwise $\oplus$''', by: :$\forall f, g \in \C^S: \forall s \in S: \map {\paren {f \oplus g} } s := \map f s \oplus \map g s$ In the above expression, the operator on the  is the given $\oplus$ on the complex numbers.
Definition:Closed under Mapping
Let $f: S \to T$ be a mapping. Let $S' \subseteq S$. Then $S'$ is '''closed under $f$''' : :$f \sqbrk {S'} \subseteq S'$ where $f \sqbrk {S'}$ is the image of $S'$ under $f$. That is: :$x \in S' \implies \map f x \in S'$ === Arbitrary Product === {{:Definition:Closed under Mapping/Arbitrary Product}}
Definition:Pointwise Operation on Number-Valued Functions
Let $S$ be a non-empty set. Let $\mathbb F$ be one of the standard number sets: $\Z, \Q, \R$ or $\C$. Let $\mathbb F^S$ be the set of all mappings $f: S \to \mathbb F$. Let $\oplus$ be a binary operation on $\mathbb F$. The (binary) operation '''pointwise $\oplus$''' is defined on $\mathbb F^S$ as: :$\oplus: \mathbb F^S \times \mathbb F^S \to \mathbb F^S: \forall f, g \in \mathbb F^S:$ ::$\forall s \in S: \map {\paren {f \oplus g} } s := \map f s \oplus \map g s$
Definition:Pointwise Operation on Integer-Valued Functions
Let $\Z^S$ be the set of all mappings $f: S \to \Z$, where $\Z$ is the set of integers. Let $\oplus$ be a binary operation on $\Z$. Define $\oplus: \Z^S \times \Z^S \to \Z^S$, called '''pointwise $\oplus$''', by: :$\forall f, g \in \Z^S: \forall s \in S: \map {\paren {f \oplus g} } s := \map f s \oplus \map g s$ In the above expression, the operator on the  is the given $\oplus$ on the integers.
Definition:Pointwise Operation on Rational-Valued Functions
Let $\Q^S$ be the set of all mappings $f: S \to \Q$, where $\Q$ is the set of rational numbers. Let $\oplus$ be a binary operation on $\Q$. Define $\oplus: \Q^S \times \Q^S \to \Q^S$, called '''pointwise $\oplus$''', by: :$\forall f, g \in \Q^S: \forall s \in S: \map {\paren {f \oplus g} } s := \map f s \oplus \map g s$ In the above expression, the operator on the  is the given $\oplus$ on the rational numbers.
Definition:Pointwise Addition of Integer-Valued Functions
Let $f, g: S \to \Z$ be integer-valued functions. Then the '''pointwise sum of $f$ and $g$''' is defined as: :$f + g: S \to \Z:$ ::$\forall s \in S: \map {\paren {f + g} } s := \map f s + \map g s$ where the $+$ on the  is integer addition.
Definition:Integer-Valued Function
Let $f: S \to T$ be a function. Let $S_1 \subseteq S$ such that $f \left({S_1}\right) \subseteq \Z$. Then $f$ is said to be '''integer-valued on $S_1$'''. That is, $f$ is defined as integer-valued on $S_1$  the image of $S_1$ under $f$ lies entirely within the set of integers $\Z$. An '''integer-valued function''' is a function $f: S \to \Z$ whose codomain is the set of integers $\Z$. That is, $f$ is '''integer-valued'''  it is '''integer-valued''' over its entire domain.
Definition:Pointwise Addition of Rational-Valued Functions
Let $f, g: S \to \Q$ be rational-valued functions. Then the '''pointwise sum of $f$ and $g$''' is defined as: :$f + g: S \to \Q:$ ::$\forall s \in S: \map {\paren {f + g} } s := \map f s + \map g s$ where the $+$ on the  is integer addition.
Definition:Rational-Valued Function
Let $f: S \to T$ be a function. Let $S_1 \subseteq S$ such that $f \left({S_1}\right) \subseteq \Q$. Then $f$ is said to be '''rational-valued on $S_1$'''. That is, $f$ is defined as rational-valued on $S_1$ iff the image of $S_1$ under $f$ lies entirely within the set of rational numbers $\Q$. A '''rational-valued function''' is a function $f: S \to \Q$ whose codomain is the set of rational numbers $\Q$. That is, $f$ is '''rational-valued'''  it is rational-valued over its entire domain.
Definition:Pointwise Multiplication of Integer-Valued Functions
Let $f, g: S \to \Z$ be integer-valued functions. Then the '''pointwise product of $f$ and $g$''' is defined as: :$f \times g: S \to \Z:$ ::$\forall s \in S: \map {\paren {f \times g} } s := \map f s \times \map g s$ where the $\times$ on the  is integer multiplication.
Definition:Pointwise Multiplication of Rational-Valued Functions
Let $f, g: S \to \Q$ be rational-valued functions. Then the '''pointwise product of $f$ and $g$''' is defined as: :$f \times g: S \to \Q:$ ::$\forall s \in S: \map {\paren {f \times g} } s := \map f s \times \map g s$ where the $\times$ on the  is rational multiplication.
Definition:Pointwise Multiplication of Real-Valued Functions
Let $f, g: S \to \R$ be real-valued functions. Then the '''pointwise product of $f$ and $g$''' is defined as: :$f \times g: S \to \R:$ ::$\forall s \in S: \map {\paren {f \times g} } s := \map f s \times \map g s$ where $\times$ on the  denotes real multiplication.
Definition:Pointwise Multiplication of Complex-Valued Functions
Let $f, g: S \to \Z$ be complex-valued functions. Then the '''pointwise product of $f$ and $g$''' is defined as: :$f \times g: S \to \Z:$ ::$\forall s \in S: \map {\paren {f \times g} } s := \map f s \times \map g s$ where the $\times$ on the  is complex multiplication.
Definition:Pointwise Scalar Multiplication of Number-Valued Function
When one of the functions is the constant mapping $f_\lambda: S \to \mathbb F: \map {f_\lambda} s = \lambda$, the following definition arises: The (binary) operation of '''pointwise scalar multiplication''' is defined on $\mathbb F \times \mathbb F^S$ as: :$\times: \mathbb F \times \mathbb F^S \to \mathbb F^S: \forall \lambda \in \mathbb F, f \in \mathbb F^S:$ ::$\forall s \in S: \map {\paren {\lambda \times f} } s := \lambda \times \map f s$ where the $\times$ on the  is conventional arithmetic multiplication.
Definition:Pointwise Scalar Multiplication of Integer-Valued Function
Let $f: S \to \Z$ be an integer-valued function. Let $\lambda \in \Z$ be an integer. Then the '''pointwise scalar product of $f$ by $\lambda$''' is defined as: :$\lambda \times f: S \to \Z:$ ::$\forall s \in S: \map {\paren {\lambda \times f} } s := \lambda \times \map f s$ where $\times$ on the  is integer multiplication.
Definition:Pointwise Scalar Multiplication of Rational-Valued Function
Let $f: S \to \Q$ be an rational-valued function. Let $\lambda \in \Q$ be an rational number. Then the '''pointwise scalar product of $f$ by $\lambda$''' is defined as: :$\lambda \times f: S \to \Q:$ ::$\forall s \in S: \map {\paren {\lambda \times f} } s := \lambda \times \map f s$ where the $\times $ on the  is rational multiplication.
Definition:Pointwise Scalar Multiplication of Real-Valued Function
Let $f: S \to \R$ be an real-valued function. Let $\lambda \in \R$ be an real number. Then the '''pointwise scalar product of $f$ by $\lambda$''' is defined as: :$\lambda \times f: S \to \R:$ ::$\forall s \in S: \map {\paren {\lambda \times f} } s := \lambda \times \map f s$ where the $\times $ on the  is real multiplication.
Definition:Pointwise Scalar Multiplication of Complex-Valued Function
Let $f: S \to \C$ be an complex-valued function. Let $\lambda \in \C$ be an complex number. Then the '''pointwise scalar product of $f$ by $\lambda$''' is defined as: :$\lambda \times f: S \to \C:$ ::$\forall s \in S: \map {\paren {\lambda \times f} } s := \lambda \times \map f s$ where $\times$ on the  denotes complex multiplication.
Definition:Addition/Rational Numbers
The addition operation in the domain of rational numbers $\Q$ is written $+$. Let: :$a = \dfrac p q, b = \dfrac r s$ where: :$p, q \in \Z$ :$r, s \in \Z_{\ne 0}$ Then $a + b$ is defined as: :$\dfrac p q + \dfrac r s = \dfrac {p s + r q} {q s}$ This definition follows from the definition of and proof of existence of the field of quotients of any integral domain, of which the set of integers is an example.
Definition:Quaternion/Addition
The '''sum''' of two quaternions $\mathbf x_1 = a_1 \mathbf 1 + b_1 \mathbf i + c_1 \mathbf j + d_1 \mathbf k$ and $\mathbf x_2 = a_2 \mathbf 1 + b_2 \mathbf i + c_2 \mathbf j + d_2 \mathbf k$ is defined as: :$\mathbf x_1 + \mathbf x_2 := \paren {a_1 + a_2} \mathbf 1 + \paren {b_1 + b_2} \mathbf i + \paren {c_1 + c_2} \mathbf j + \paren {d_1 + d_2} \mathbf k$
Definition:Quaternion/Multiplication
The '''product''' of two quaternions $\mathbf x_1 = a_1 \mathbf 1 + b_1 \mathbf i + c_1 \mathbf j + d_1 \mathbf k$ and $\mathbf x_2 = a_2 \mathbf 1 + b_2 \mathbf i + c_2 \mathbf j + d_2 \mathbf k$ is defined as: {{begin-eqn}}      {{eqn | l = \mathbf x_1 \mathbf x_2       | o = :=       | r = \paren {a_1 a_2 - b_1 b_2 - c_1 c_2 - d_1 d_2} \mathbf 1       | c =  }} {{eqn | o =        | ro= +       | r = \paren {a_1 b_2 + b_1 a_2 + c_1 d_2 - d_1 c_2} \mathbf i       | c =  }} {{eqn | o =        | ro= +       | r = \paren {a_1 c_2 - b_1 d_2 + c_1 a_2 + d_1 b_2} \mathbf j       | c =  }} {{eqn | o =        | ro= +       | r = \paren {a_1 d_2 + b_1 c_2 - c_1 b_2 + d_1 a_2} \mathbf k       | c =  }} {{end-eqn}}
Definition:Quaternion/Algebra over Field
An algebra of quaternions can be defined over any field as follows: Let $\mathbb K$ be a field, and $a$, $b \in \mathbb K$. Define the '''quaternion algebra''' $\left\langle{ a,b }\right\rangle_\mathbb K$ to be the $\mathbb K$-vector space with basis $\{1, i, j, k\}$ subject to: : $i^2 = a$ : $j^2 = b$ : $ij = k = -ji$ Formally this could be achieved as a multiplicative presentation of a suitable group, or as a linear subspace of a finite extension of $\mathbb K$. Taking $\mathbb K = \R$ and $a = b = -1$ we see that this generalises  Hamilton's quaternions.
Definition:Field Extension/Degree
Let $E / F$ be a field extension. The '''degree of $E / F$''', denoted $\index E F$, is the dimension of $E / F$ when $E$ is viewed as a vector space over $F$. === Finite === {{:Definition:Field Extension/Degree/Finite}} === Infinite === {{:Definition:Field Extension/Degree/Infinite}}
Definition:Class/Proper Class
A '''proper class''' is a class which is not a set. That is, $A$ is a '''proper class''' : : $\neg \exists x: x = A$ where $x$ is a set. A class which is not a proper class is a small class.
Definition:Normal Series/Factor Group
Let $\sequence {G_i}_{i \mathop \in \closedint 0 n}$ be a normal series for $G$: :$\sequence {G_i}_{i \mathop \in \closedint 0 n} = \tuple {\set e = G_0 \lhd G_1 \lhd \cdots \lhd G_{n - 1} \lhd G_n = G}$ The '''factor groups''' of $\sequence {G_i}_{i \mathop \in \closedint 0 n}$: :$\set e = G_0 \lhd G_1 \lhd \cdots \lhd G_n = G$ are the quotient groups: :$G_1 / G_0, G_2 / G_1, \ldots, G_i / G_{i - 1}, \ldots, G_n / G_{n-1}$
Definition:Normal Series/Sequence of Homomorphisms
Let $\sequence {G_i}_{i \mathop \in \closedint 0 n}$ be a normal series for $G$: :$\sequence {G_i}_{i \mathop \in \closedint 0 n} = \tuple {\set e = G_0 \lhd G_1 \lhd \cdots \lhd G_{n - 1} \lhd G_n = G}$ whose factor groups are: :$H_1 = G_1 / G_0, H_2 = G_2 / G_1, \ldots, H_i = G_i / G_{i - 1}, \ldots, H_n = G_n / G_{n - 1}$ By Kernel of Group Homomorphism Corresponds with Normal Subgroup of Domain, such a series can also be expressed as a sequence $\phi_1, \ldots, \phi_n$ of group homomorphisms: :$\set e \stackrel {\phi_1} {\to} H_1 \stackrel {\phi_2} {\to} H_2 \stackrel {\phi_3} {\to} \cdots \stackrel {\phi_n} {\to} H_n$
Definition:Normal Series/Infinite
A normal series may or may not terminate at either end: :$\cdots \stackrel {\phi_{i - 1} } {\longrightarrow} H_{i - 1} \stackrel {\phi_i} {\longrightarrow} H_i \stackrel {\phi_{i + 1} } {\longrightarrow} H_{i + 1} \stackrel {\phi_{i + 2} } {\longrightarrow} \cdots$ Such a series is referred to as an '''infinite normal series'''. The context will determine which end, if either, it terminates.
Definition:Normal Series/Length
Let $\sequence {G_i}_{i \mathop \in \closedint 0 n}$ be a normal series for $G$: :$\sequence {G_i}_{i \mathop \in \closedint 0 n} = \tuple {\set e = G_0 \lhd G_1 \lhd \cdots \lhd G_{n-1} \lhd G_n = G}$ The '''length''' of $\sequence {G_i}_{i \mathop \in \closedint 0 n}$ is the number of (normal) subgroups which make it. In this context, the '''length''' of $\sequence {G_i}_{i \mathop \in \closedint 0 n}$ is $n$. If such a normal series is infinite, then its '''length''' is not defined.
Definition:Non-Trivial Group
A '''non-trivial group''' is a group which is not the trivial group. It is often used in arguments to exclude the sometimes erratic behaviour of the trivial group.
Definition:Exact Normal Series
Let $\sequence {H_i}_{i \mathop \in I \mathop \subseteq \Z}$ be a normal series: :$\cdots \stackrel {\phi_{i - 1} } {\longrightarrow} H_{i - 1} \stackrel {\phi_i} {\longrightarrow} H_i \stackrel {\phi_{i + 1} } {\longrightarrow} H_{i + 1} \stackrel {\phi_{i + 2} } {\longrightarrow} \cdots$ Suppose that, for some $i \in I$: :$\Img {\phi_i} = \map \ker {\phi_{i + 1} }$ That is, the image of one homomorphism is the kernel of the next. Then $\sequence {H_i}$ is referred to as '''exact at $H_i$'''. If $\sequence {H_i}$ is '''exact''' for all $i \in I$, then $\sequence {H_i}$ itself is an '''exact normal series'''.
Definition:Equalizer
Let $\mathbf C$ be a metacategory. Let $f, g: C \to D$ be morphisms with common domain and codomain. An '''equalizer''' for $f$ and $g$ is the limit of the diagram $C \underset f{\overset g\rightrightarrows} D$, that is, a morphism $e: E \to C$ such that: :$f \circ e = g \circ e$ and subject to the following UMP: :For any $a: A \to C$ such that $f \circ a = g \circ a$, there is a unique $u: A \to E$ such that: :::$\begin{xy}\xymatrix{  E    \ar[r]^*{e} &  C   \ar[r]<2pt>^*{f}   \ar[r]<-2pt>_*{g}  &  D \\  A   \ar@{.>}[u]^*{u}   \ar[ur]_*{a} }\end{xy}$ :is a commutative diagram. I.e., $a = e \circ u$.
Definition:Metric Space/Distance Function
The mapping $d: A \times A \to \R$ is referred to as a '''distance function on $A$''' or simply '''distance'''.
Definition:Metric Space/Triangle Inequality
Axiom $\text M 2$ is referred to as the '''triangle inequality''', as it is a generalization of the Triangle Inequality which holds on the real number line and complex plane.
Definition:Metric Space/Point
The elements of $A$ are called the '''points''' of the space.
Definition:Quasimetric/Quasimetric Space
A '''quasimetric space''' $M = \left({A, d}\right)$ is an ordered pair consisting of a set $A \ne \varnothing$ followed by a quasimetric $d: A \times A \to \R$ which acts on that set.
Definition:Distance/Points/Real Numbers
Let $x, y \in \R$ be real numbers. Let $\size {x - y}$ be the absolute value of $x - y$. Then the function $d: \R^2 \to \R$: :$\map d {x, y} = \size {x - y}$ is called the '''distance between $x$ and $y$'''.
Definition:Pseudometric/Pseudometric Space
A '''pseudometric space''' $M = \left({A, d}\right)$ is an ordered pair consisting of a set $A \ne \varnothing$ followed by a pseudometric $d: A \times A \to \R$ which acts on that set.
Definition:Operation Induced by Injection
Let $\struct {T, \circ}$ be an algebraic structure. Let $f: S \to T$ be an injection. Then the '''operation induced on $S$ by $f$ and $\circ$''' is the binary operation $\circ_f$ on $S$ defined by: :$\circ_f: S \times S \to S: x \circ_f y := \map {f^{-1} } {\map f x \circ \map f y}$
Definition:Continuous Real Function/Point
Let $A \subseteq \R$ be any subset of the real numbers. Let $f: A \to \R$ be a real function. Let $x \in A$ be a point of $A$. === Definition by Epsilon-Delta === {{:Definition:Continuous Real Function/Point/Definition by Epsilon-Delta}} === Definition by Neighborhood === {{:Definition:Continuous Real Function/Point/Definition by Neighborhood}}
Definition:Continuous Real Function/Subset
Let $A \subseteq \R$ be any subset of the real numbers. Let $f: A \to \R$ be a real function. Then '''$f$ is continuous on $A$'''  $f$ is continuous at every point of $A$.
Definition:Continuous Real Function/One Side
=== Continuity from the Left at a Point === {{Definition:Continuous Real Function/Left-Continuous/Point}} === Continuity from the Right at a Point === {{Definition:Continuous Real Function/Right-Continuous/Point}}
Definition:Continuous Mapping (Metric Space)/Point
{{Definition:Continuous Mapping (Metric Space)/Point/Definition 1}}
Definition:Continuous Mapping (Metric Space)/Space
{{Definition:Continuous Mapping (Metric Space)/Space/Definition 1}}
Definition:Frequency (Descriptive Statistics)
Let $S$ be a sample or a population. Let $\omega$ be a qualitative variable, or a class of a quantitative variable. The '''frequency''' of $\omega$ is the number of individuals in $S$ satisfying $\omega$.
Definition:Class (Descriptive Statistics)/Class Mark
The midpoint of a class is called the '''class mark'''.
Definition:Mean of Grouped Data
Let $\mathsf C_1, \mathsf C_2, \cdots, \mathsf C_n$ be classes of data. Let $m_i$ be the class mark of the $i$th class. Let $f_i$ be the frequency of the $i$th class. The '''mean of the grouped data''' for $\mathsf C_1, \mathsf C_2, \cdots, \mathsf C_n$ is defined as: :$\displaystyle \bar x_{\mathsf C} := \frac 1 n \sum_{i \mathop = 1}^n m_i f_i$
Definition:Class (Descriptive Statistics)/Empty Class
A class is '''empty''' if it is of frequency zero.
Definition:Euclidean Metric/Real Number Line
On the real number line, the Euclidean metric can be seen to degenerate to: :$\map d {x, y} := \sqrt {\paren {x - y}^2} = \size {x - y}$ where $\size {x - y}$ denotes the absolute value of $x - y$.
Definition:Open Ball/Radius
In $\map {B_\epsilon} a$, the value $\epsilon$ is referred to as the '''radius''' of the open $\epsilon$-ball.
Definition:Affine Space
=== Associativity Axioms === {{:Definition:Affine Space/Associativity Axioms}} === Group Action === {{:Definition:Affine Space/Group Action}} === Weyl's Axioms === {{:Definition:Affine Space/Weyl's Axioms}} === Addition === {{:Definition:Addition (Affine Geometry)}} === Subtraction === {{:Definition:Subtraction (Affine Geometry)}} === Tangent Space === {{:Definition:Tangent Space (Affine Geometry)}} === Vector === {{:Definition:Vector (Affine Geometry)}} === Point === {{:Definition:Point (Affine Geometry)}}
Definition:Sequentially Compact Space/In Itself
A subspace $H \subseteq S$ is '''sequentially compact in itself'''  every infinite sequence in $H$ has a subsequence which converges to a point in $H$. This is understood to mean that $H$ is sequentially compact when we consider it as a topological space with the induced topology of $T$.
Definition:P-Product Metric/General Definition
Let $M_{1'} = \left({A_{1'}, d_{1'}}\right), M_{2'} = \left({A_{2'}, d_{2'}}\right), \ldots, M_{n'} = \left({A_{n'}, d_{n'}}\right)$ be metric spaces. Let $\displaystyle \mathcal A = \prod_{i \mathop = 1}^n A_{i'}$ be the cartesian product of $A_{1'}, A_{2'}, \ldots, A_{n'}$. Let $p \in \R_{\ge 1}$. The '''$p$-product metric''' on $\mathcal A$ is defined as: : $\displaystyle d_p \left({x, y}\right) := \left({\sum_{i \mathop = 1}^n \left({d_{i'} \left({x_i, y_i}\right)}\right)^p}\right)^{\frac 1 p}$ where $x = \left({x_1, x_2, \ldots, x_n}\right), y = \left({y_1, y_2, \ldots, y_n}\right) \in \mathcal A$. The metric space $\mathcal M_p := \left({\mathcal A, d_p}\right)$ is the '''$p$-product (space)''' of $M_{1'}, M_{2'}, \ldots, M_{n'}$.
Definition:Affine Geometry
'''Affine geometry''' is the study of the geometry of affine spaces. It provides a modern axiomatic approach to the study of configurations of lines, planes and hypersurfaces. In particular an affine space can be thought of as a finite dimensional vector space with no distinguished origin, and its affine transformations are those that preserve collinearity.
Definition:Path-Connected/Topology/Points
Let $T = \left({S, \tau}\right)$ be a topological space. Let $a, b \in S$ be such that there exists a path from $a$ to $b$. That is, there exists a continuous mapping $f: \left[{0 \,.\,.\, 1}\right] \to S$ such that $f \left({0}\right) = a$ and $f \left({1}\right) = b$. Then $a$ and $b$ are '''path-connected in $T$'''.
Definition:Path-Connected/Topology/Set
Let $T = \left({S, \tau}\right)$ be a topological space. Let $U \subseteq S$ be a subset of $S$. Let $T' = \left({U, \tau_U}\right)$ be the subspace of $T$ induced by $U$. Then $U$ is a '''path-connected set in $T$'''  every two points in $U$ are path-connected in $T\,'$. That is, $U$ is a '''path-connected set in $T$''' : :for every $x, y \in U$, there exists a continuous mapping $f: \left[{0 \,.\,.\, 1}\right] \to U$ such that $f \left({0}\right) = x$ and $f \left({1}\right) = y$.
Definition:Path-Connected/Topology/Topological Space
Let $T = \struct {S, \tau}$ be a topological space. Then $T$ is a '''path-connected space'''  $S$ is a path-connected set of $T$. That is, $T$ is a '''path-connected space''' : :for every $x, y \in S$, there exists a continuous mapping $f: \closedint 0 1 \to S$ such that $\map f 0 = x$ and $\map f 1 = y$.
Definition:Path-Connected/Topology
=== Points in Topological Space === {{:Definition:Path-Connected/Topology/Points}} === Set of Topological Space === {{:Definition:Path-Connected/Topology/Set}} === Topological Space === {{:Definition:Path-Connected/Topology/Topological Space}}