lacunary - Definition Index

Abelian groups are groups whose operation is commutative. For $a,b \in G, a * b = b * g$.

The subgroup of $S_n$ consisting of all even permutations of $n$ letters is the alternating group $A_n$ on $n$ letters. If $n \geq 2$, then this set forms a subgroup of $S_n$ of order $n!/2$.

The Archimedean property is that given two positive numbers $x$ and $y,$ there is an integer $n$ such that $nx > y.$

A set $A$ is said to be at most countable if $A$ is finite or countable.

Given $\vec{x} \in \mathbb{R}^k, r > 0,$ the open or closed ball with center $\vec{x}$ and radius $r$ is defined as the set of points $\vec{y}$ such that $|\vec{x} - \vec{y}| < r$ or $|\vec{x} - \vec{y}| \leq r,$ respectively.

A collection ${V_\alpha}$ of open subsets of $X$ is said to be a base for $X$ if the following is true: For every $x \in X$ and every open set $G \subset X$ such that $x \in G,$ we have $x \in V_\alpha \subset G$ for some $\alpha.$ In other words, every open set in $X$ is the union of a subcollection of ${V_\alpha}.$

$E$ is bounded if there is a real number $M$ and a point $q \in X$ such that $d(p, q) < M$ for all $p \in E.$

The sequence $\{p_n\}$ is said to be bounded if its range (sequence) is bounded.

Let $E_0$ be the interval $[0, 1].$ Remove the segment $(\frac{1}{3}, \frac{2}{3}),$ and let

$$ E_1 = [0, \frac{1}{3}] \cup [\frac{2}{3}, 1]. $$

Similarly, remove the middle thirds of these intervals, and let

$$E_2 = [0, \frac{1}{9}] \cup [\frac{2}{9}, \frac{3}{9}] \cup [\frac{6}{9}, \frac{7}{9}] \cup [\frac{8}{9}, 1]. $$

We can continue this forever, and we get a nested sequence $\{E_n\}$ of compact sets $E_n$ where:

(a) $E_{n+1} \subset E_n.$

(b) $E_n$ is the union of $2^n$ intervals, each of length $1/3^n.$

Finally, the set

$$ P = \bigcap_{n=1}^\infty E_n $$

is called the Cantor set.

Given two sets, $A$ and $B$, if there is a bijection (a one-to-one mapping of $A$ onto $B$) between $A$ and $B$, we say $A$ and $B$ have the same cardinal number, or that $A$ and $B$ are equivalent. We denote this as $A \sim B$.

A sequence $\{p_n\}$ in a metric space $X$ is said to be a Cauchy sequence if for every $\epsilon > 0$ there is an integer $N$ such that $d(p_n, p_m) < \epsilon$ is $n \geq N$ and $m \geq N.$

The center of a group $G$ is all the elements that commute with all elements of $G$:

$$ Z(G) = \{ z \in G | zg = gz \text{ for all } g \in G \}. $$

A set $E$ is closed if every limit point of $E$ is a point of $E.$

If $X$ is a metric space, $E \subset X,$ and $E'$ denotes the set of all limit points of $E$ in $X,$ then the closure of $E$ is the set $\closure{E} = E \cup E'.$

A ring in which multiplication is commutative is called a commutative ring.

The commutator subgroup of $G$ is the group $C$ generated by all elements of the set

$$ \{aba^{-1}b^{-1} | a,b \in G\}. $$

A subset $K$ of a metric space $X$ is said to be compact if every open cover of $K$ contains a finite subcover. More explicitly, the requirement is that if $\{G_\alpha\}$ is an open cover of $K,$ then there are finitely many indicies $\alpha_1, \dots, \alpha_n$ such that

$$ K \subset G_{\alpha_1} \cup \cdots \cup G_{\alpha_n}. $$

The complement of $E$ (denoted by $E^c$) is the set of all points $p \in X$ such that $p \notin E.$

A metric space in which every cauchy sequence converges is said to be complete.

Suppose $X, Y, Z$ are metric spaces, $E \subset X,$ $f: E \to Y,$ $g: f(E) \to Z,$ $h: E \to Z$ with

$$ h(x) = g(f(x)) \quad (x \in E). $$

The function $h$ is called the composition or the composite of $f$ and $g.$ The notation

$$ h = g \circ f $$

is frequently used.

A point $p$ in a metric space $X$ is said to be a condensation point of a set $E \subset X$ if every neighborhood of $p$ contains uncountably many points of $E.$

If $X$ is a metric space, a set $E \subset X$ is said to be connected if $E$ is not a union of two nonempty separated sets.

A set $S$ is said to be connected if every pair of points in $S$ can be joined by a finite number of line segments joined end to end that lie entirely within $S$.

Suppose $X$ and $Y$ are metric spaces, $E \subset X, p \in E,$ and $f : E \to Y.$ Then $f$ is said to be continuous at $p$ if for every $\epsilon > 0$ there exists a $\delta > 0$ such that

$$ d_Y(f(x), f(p)) < \epsilon $$

for all points $x \in E$ for which $d_X(x, p) < \delta.$

If $f$ is continuous at every point of $E,$ then $f$ is said to be continuous on $E$.

A sequence $\{p_n\}$ in a metric space $X$ is said to converge if there is a point $p \in X$ with the following property: For every $\epsilon > 0,$ there is an integer $N$ such that $n \geq N$ implies that $d(p_n, p) < \epsilon.$

If a sequence converges, it is said to be a convergent sequence.

A set $E \subset \mathbb{R}^k$ is said to be convex if

$$ \lambda \vec{x} + (1 - \lambda)\vec{y} \in E $$

whenever $\vec{x}, \vec{y} \in E,$ and $0 < \lambda < 1.$

In geometric terms, this means a set is convex if we can connect any two points in the set with a line segment whose points are all within the set.

Let $H$ be a subgroup of $G$. Given $a \in G$, the subset $aH = \{ah | h \in H\}$ of $G$ is the left coset of $H$ containing $a$, while the subset $Ha = \{ha | h \in H\}$ is the right coset of $H$ containing $a$.

A set $A$ is said to be countable if there exists a bijection between $A$ and the set of all positive integers $\mathbb{Z}_{>0}$, that is, if $A \sim \mathbb{Z}_{>0}.$

A cyclic group is a group where there exists some $g \in G$ such that every element in $G$ can be generated from the group operation applied to $g$.

That is, $G = \{g^N | n \in \mathbb{Z}\}$ when we think of the operation as multiplication, or $G = \{ng | n \in \mathbb{Z}\}$ when we think of the operation as addition.

Any subgroup of a cyclic group is also cyclic - a cyclic subgroup.

A permutation is a bijection $\phi : A \to A$, that is, a bijection from a set onto itself.

Let $\sigma$ be a permutation of $A$. The equivalence classes in $A$ determined by the equivalence relation "~" are the orbits of $\sigma$.

A permutation $\sigma \in S_n$ is a cycle if it has at most one orbit containing more than one elements. The length of a cycle is the number of elements in its largest orbit.

$E$ is dense in $X$ if every point of $X$ is a limit point of $E,$ or a point of $E$ (or both.)

The derivative of a function $f$ at a point $a$ is defined as: $$f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}$$ provided this limit exists.

Let $E$ be a nonempty subset of a metric space $X,$ and let $S$ be the set of all real numbers of the form $d(p,q),$ with $p, q \in E.$ The supremum of $S$ is called the diameter of $E.$

The discrete metric is defined as:

$$ d(x, y) = \begin{cases} 0 & \text{if } x = y \\ 1 & \text{if } x \neq y \end{cases} $$

Two or more than two cycles are disjoint if no element appears in more than one cycle.

If a sequence $\{p_n\}$ does not converge, it is said to diverge.

Let $R$ be a ring with unity. If every nonzero element of $R$ is a unit (has a multiplicative inverse), then $R$ is called a division ring.

If $f$ is a function from the set $A$ to the set $B,$ the set $A$ is called the domain of $f.$

The objects that make up a set are called its elements or its members.

A permutation of a finite set is even if it is the product of an even number of transpositions.

A commutative division ring is called a field.

A set $A$ is said to be finite if $A \sim \mathbb{N}_n$ for some $n.$

Consider two sets, $A$ and $B,$ whose elements may be any objects whatsoever, and suppose that with each element $x$ of $A$ there is associated, in some manner, any element of $B,$ which we denote by $f(x).$ Then $f$ is said to be a function from $A$ to $B.$

There is also a concept of a greatest lower bound, which is a lower bound that is greater than or equal to every other lower bound. The greatest lower bound is also known as the infimum.

A group, is a set $G$ together with a binary operation $*$ such that:

Closure: The set is closed under the binary operation. For all $a, b \in G, a * b \in G$.

Associativity: The binary operation is associative on the set. For all $a, b, c \in G, (a * b) * c = a * (b * c)$.

Identity: The set contains an identity element, denoted $e$. For all $a \in G, a * e = a$.

Inverses: All elements in the set have inverse elements in the set, denoted using $a^{-1}$. For all $a \in G$ there exists $a^{-1} \in G$ such that $a * a^{-1} = e$.

This set/operation combination $G$ is commonly denoted as the pair $(G, *)$.

A half-open interval $(a,b]$ or $[a, b)$ is the set of all real numbers such that $a < x \leq b$ or $a \leq x < b,$ respectively.

A homomorphism is a map $\phi : G \to G'$ between groups (not necessarily a bijection), $\langle G, * \rangle$ and $\langle G', *' \rangle,$ that satisfies the homomorphism property:

$$ \phi(a * b) = \phi(a) *' \phi(b), ~ \forall a, b \in G. $$

The number of left cosets of a subgroup $H$ in a group $G$ is the index $(G:H)$ of $H$ in $G$.

A set $A$ is said to be infinite it is not finite.

If $\vec{x}$ and $\vec{y}$ are vectors in $\mathbb{R}^n,$ then their inner product is defined as

$$ \vec{X} \cdot \vec{y} = \sum_{i = 1}^n x_i y_i. $$

A point $p$ is an interior point of $E$ if there is a neighborhood $N$ of $p$ such that $N \subset E.$

A interval $[a, b]$ is the set of all real numbers $x$ such that $a \leq x \leq b.$

If $p \in E$ and $p$ is not a limit point of $E,$ then $p$ is called an isolated point of E.

Given $\vec{a}, \vec{b} \in \mathbb{R}^k,$ if $\vec{a}_i < \vec{b}_i$ for all $i = 1, 2, \dots, k,$ then the set of all points $\vec{x}$ who satisfy $\vec{a}_i \leq \vec{x}_i \leq \vec{b}_i,$ $i = 1, 2, \dots, k,$ is called a k-cell. So, a 1-cell is an interval, a 2-cell is a rectangle, and so on.

The kernel of a homomorphism $\phi$ is the set of elements that $\phi$ sends to $e'$, and it is denoted by $\ker{(\phi)}$. It is a normal subgroup of $G$.

The number $L$ is said to be the least upper bound of the set $A$ if it's an upper bound of $A$ and if $L \leq m$ for all upper bounds $m$ of $A$.

The least upper bound is also known as the supremum.

Let $X$ and $Y$ be metric spaces; suppose $E \subset X,$ $f : E \to Y,$ and $p$ is a limit point of $E.$ We write $f(x) \to q$ as $x \to p,$ or

$$ \lim_{x \to p} f(x) = q $$

if there is a point $q \in Y$ with the following property: For every $\epsilon > 0,$ there exists a $\delta > 0$ such that

$$ d_Y(f(x), q) < \epsilon $$

for all points $x \in E$ for which

$$ 0 < d_X(x,p) < \delta. $$

If a sequence $\{p_n\}$ converges to $p,$ we say that $p$ is the limit of $\{p_n\},$ denoted as:

$$ \lim_{n \to \infty} p_n = p. $$

A point $p$ is a limit point of the set $E$ if every neighborhood of $p$ contains a point $q \neq p$ such that $q \in E.$

The number $m$ is said to be a lower bound of a nonempty set $A$ if $x \geq m$ for all $x \in A.$

Let $\vec{x} = (x_1, x_2, \cdots, x_n) \in \mathbb{R}^n$. The magnitude, or length, $\vec{x}$ is denoted as $|| \vec{x} ||$ and is defined as:

$$ ||\vec{x}|| = \sqrt{ {x_1}^2 + {x_2}^2 + \cdots + {x_n}^2 } = \sqrt{\vec{x} \cdot \vec{x}} = \sqrt{\sum_{i=1}^n x_i^2} $$

The membership criterion for a set $X$ is a statement of the form $x \in X \iff P(x),$ where $P(x)$ is a proposition that is true for precisely for those objects $x$ that are elements of $X,$ and no others.

A set $X,$ whose elements we'll call points, together with a distance function $d: X \times X \to \mathbb{R}$ is called a metric space and the distance function $d$ is called a metric, if the following conditions, called the metric axioms, hold for $p, q, r \in X:$

• If $p \neq q, d(p,q) > 0.$ (distance is always positive between two distinct points.)

• $d(p,p) = 0.$ (distance is always zero between a point and itself.)

• $d(p,q) = d(q,p).$ (the distance from $p$ to $q$ is the same as the distance from $q$ to $p$.)

• $d(p,q) \leq d(p,r) + d(r,p)$ (triangle inequality.)

We can denote a metric space on set $X$ with metric $d$ as the tuple $(X, d).$

For some element $a$ in a ring with unity $R$ where $1 \neq 0$, if $a^{-1} \in R$ such that $aa^{-1} = a^{-1}a = 1$, $a^{-1}$ is said to be the multiplicative inverse of $a.$

A neighborhood, or r-neighborhood of $p$ is a set $N_r(p)$ consisting of all $q$ such that $d(p, q) < r$ for some $r > 0.$ This subset of $X$ is all the points within a circle of radius $r$ - the open ball of radius $r$ centered at $p.$

A subgroup $H$ of a group $G$ is normal if its left and right cosets coincide, that is, if $gH = Hg$, i.e. $gHg^{-1} = H$, for all $g \in G$.

A permutation of a finite set is odd if it is the product of an odd number of transpositions.

A set $E$ is open if every point of $E$ is an interior point of $E.$

An open cover of a set $E$ in a metric space $X$ is a collection $\{G_\alpha\}$ of open subsets of $X$ such that $E \subset \bigcup_\alpha G_\alpha.$

Suppose $E \subset Y \subset X,$ and $X$ is a metric space. We say that $E$ is open relative to $Y$ if to each $p \in E$ there is associated an $r > 0$ such that $q \in E, q \in Y$ whenever $d(p, q) < r.$

The order of a finite group is the number of its elements. The order of group $G$ is denoted as $\ord{(G)}$ or $\|G\|$. The order of an element $a$ (also called period length or period) is the number of elements in the subgroup generated by $a$, and is denoted by $\ord{(a)}$ or $\|a\|$.

$E$ is perfect if $E$ is closed and if every point of $E$ is a limit point of $E.$

The set of all values of a function $f$ is called the range of $f.$

The set of all points $p_n$ of a sequence $\{p_n\} (n = 1, 2, 3, \dots)$ is the range of $\{p_n\}.$

The real numbers $\mathbb{R}$ are a set of objects along with two binary operations $+ : \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ and $\cdot : \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ that satisfy the 9 field axioms along with the Order axiom and the Completeness axiom. That is, the reals are an ordered, complete field.

A real sequence of numbers if a function $f$ from $\mathbb{N}$ to $\mathbb{R}$.

A ring is a set together with two binary operations $+$ and $\cdot$, which we will call addition and multiplication, such that the following axioms are satisfied:

1. $\langle R, + \rangle$ is an abelian group.

2. Multiplication is associative.

3. For all $a,b,c \in R$, the left distributive law and the right distribute law hold, i.e.

$$ a \cdot (b + c) = (a \cdot b) + (a \cdot c), \quad (a + b) \cdot c = (a \cdot c) + (b \cdot c). $$

A ring homomorphism $\phi : R \to R'$ must satisfy the following two properties:

1. $\phi{(a+b)} = \phi{(a)} + \phi{(b)}.$

2. $\phi{(ab)} = \phi{(a)}\phi{(b)}.$

A ring that has a multiplicative identity element is called a ring with unity.

A segment $(a, b)$ is the set of all real numbers $x$ such that $a < x < b.$

A metric space is called separable if it contains a countable dense subset.

Two subsets $A$ and $B$ of a metric space $X$ are said to be separated if both $A \cap \closure{B}$ and $\closure{A} \cap B$ are empty, i.e., if no point of $A$ lies in the closure of $B$ and no point of $B$ lies in the closure of $A.$

A sequence is a function $f$ defined on the set $J$ of all positive integers.

A set is a collection of objects, considered as a whole.

A group $G$ is simple if it has no proper nontrivial normal subgroups, that is, if $|G| > 1$ and the only normal subgroups of $G$ are $\{e\}$ and $G$ itself.

A subgroup $H$ of a group $G$ is a subset of $G$ group together with the same operation as $G$ that still forms a group. The identity element of $G$ must also be the identity element of $H$.

Given a sequence $\{p_n\},$ consider a sequence $\{n_k\}$ of positive integers, such that $n_1 < n_2 < n_3 < \cdots.$ Then the sequence $\{p_{n_i}\}$ is called a subsequence of $\{p_n\}.$

If a subsequence $\{p_{n_i}\}$ of $\{p_n\}$ converges, its limit (sequence) is called a subsequential limit of $\{p_n\}.$

If $X$ and $Y$ are sets such that every element of $X$ is also an element of $Y,$ then we say $X$ is a subset of $Y,$ denoted as $X \subset Y.$ Formally,

$$X \subset Y \iff (\forall x)(x \in X \implies x \in Y)$$

If $X$ and $Y$ are sets such that every element of $Y$ is also an element of $X,$ then we say $X$ is a superset of $Y,$ denoted as $X \supset Y.$ This is the same as $Y \subset X.$

If $f$ is a sequence denoted as $\{x_n\},$ the values of $f,$ that is, the elements $x_n,$ are called the terms of the sequence.

A cycle of length 2 is a transposition.

A set $A$ is said to be uncountable if it is neither finite nor countable.

If $a$ has a multiplicative inverse in $R,$ $a$ is said to be a unit in $R$.

The element $1$ is also called unity.

The number $m$ is said to be an upper bound of a nonempty set $A$ if $x \leq m$ for all $x \in A$.

If $f$ is a function from the set $A$ to the set $B,$ the elements $f(x) \in B$ are called the values of $f.$

A vector space over a field $F$ is a set $V$ together with two operations: - Vector addition: $V \times V \to V$ - Scalar multiplication: $F \times V \to V$

satisfying the following axioms: 1. $(V, +)$ is an abelian group 2. Scalar multiplication is associative: $a(bv) = (ab)v$ 3. Distributive laws hold 4. Identity: $1v = v$ for all $v \in V$