Definitions

Table of contents

  1. Metric Space
  2. Convergent sequence
  3. Cauchy sequence
  4. Open Set
  5. Continuous functions
  6. Bounded
  7. Vector Space
  8. Norm
  9. Compact Support
  10. Support
  11. Covers
  12. Compactness

Metric Space

A metric space is a set $X$ with a metric $d:X\times X \to [0,\infty)$ such that $\forall x,y,z\in X$, we have that $d$ satisfies the following properties:

  1. Positive definite: $d(x,y) \geq 0$ and $d(x,y) = 0 \iff x=y$.
  2. Symmetric: $d(x,y) = d(y,x)$.
  3. Triangle Inequality: $d(x,z) \leq d(x,y) + d(y,z)$.

Convergent sequence

A sequence ${x_n}$ in a metric space $(X,d)$ converges to $x\in X$ if and only if $\forall \epsilon>0$ there exists an $N\in \mathbb{N}$ such that $\forall n \geq N$, \(d(x_n, x) \leq \epsilon.\)

Cauchy sequence

A sequence ${x_n}$ in $(X,d)$ is a Cauchy sequence if and only if $\forall \epsilon>0$ there exists an $N\in \mathbb{N}$ such that $\forall n,m \geq N$, \(d(x_n, x_m) \leq \epsilon.\)

Open Set

A set $A\subseteq X$ is open if and only if $\forall x\in A$, there exists an $\epsilon >0$ such that \(B(x,\epsilon) := \{y\in X \mid d(x,y) < \epsilon\} \subset A.\)

We say that $B(x,\epsilon)$ is a ball of radius $\epsilon$ centered at $x$.

Continuous functions

Let $X$ and $Y$ be metric spaces with metrics $d_X,d_Y$ respectively. Then, a function $f: X \supset A \to Y$ is continuous if and only if given $x\in A$, $\forall \epsilon >0$ there exists a $\delta >0$ such that \(d_X(x,y) \leq \delta \implies d_Y(f(x), f(y)) \leq \epsilon.\)

Bounded

A sequence ${x_n}$ in $(X,d)$ is bounded if and only if there exists a $p\in X$ and a $B\in \mathbb{R}$ such that \(d(x_n,p) \leq B \quad \forall n\in \mathbb{N}.\)

Similarly, a subset $A\subseteq X$ is bounded if and only if there exists a $p\in X$ and a $B\in \mathbb{R}$ such that \(d(x,p) \leq B \quad \forall x\in A.\)

Vector Space

A vector space $V$ over a field $k$ is a set of vectors which come with addition $(+: V\times V \to V)$ and scalar multiplication $(\cdot: k\times V\to V)$ along with some classic axioms: commutativity, associativity, identity and inverse of addition, identity of multiplication, and distributivity.

Norm

A norm on a vector space $V$ over the real numbers is a function $\lVert \cdot\rVert:V\to [0,\infty)$ satisfying the following three properties:

  1. Positive Definite: $\lVert v\rVert \geq 0$ and $\lVert v\rVert = 0 \iff v = 0$.
  2. Homogeneity: $\lVert \lambda v\rVert = \vert \lambda\vert \lVert v\rVert$ for all $v\in V$ and $\lambda \in \mathbb{R}$.
  3. Triangle Inequality: $\lVert x+y\rVert \leq \lVert x\rVert + \lVert y\rVert$.

A vector space with a norm on it is defined as a normed space.

Compact Support

A function $f\in C^0(\mathbb{R})$ has compact support if $f = 0$ outside of some interval $[-n,n]$ for a finite $n.$

Support

The support of a function $f\in C^0(\mathbb{R})$ is the closure of the set \(\{x\in \mathbb{R} \mid f(x) \neq 0\}.\)

Covers

Let $A\subset X$ where $X$ is a metric space. Then, \(\{U_i\}_{i\in I}\) is an open cover of $A$ if \(A \subset \bigcup_{i\in I} U_i\) and $U_i$ is open for each $i$.

A subcover of an open cover is a subcollection of the sets $U_i$ that still cover $A$.

A finite subcover of an open cover is a finite subcollection of the sets $U_i$ that still cover $A$.

Compactness

Let $(X,d)$ be a metric space.
A set $A\subset X$ is sequentially compact if and only if every sequence in $A$ has a convergent subsequence in $A$.

A set $A\subset X$ is compact or topologically compact if every open cover of $A$ has a finite subcover.