[Def]: Metric and Metric Space

Let $X$ be a nonempty set

A mapping

$d: X \times X \rightarrow \mathbb{R}$

is said to be a metric of $X$ if it satisfies:

1. $d(x,y) \ge0$, $\forall x,y \in X$, $d(x,y)=0 \iff x=y$
2. $d(x,y)=d(y,x)$, $\forall x,y \in X$
3. for any $x,y,z\in X$, we have $d(x,z) \le d(x,y)+d(y,z)$

We denote $(X,d)$ as a metric space

[例]: Euclidean metric

$x=(x_1,x_2,...,x_n)\in\mathbb{R}^n$, $y=(y_1,y_2,...,y_n)\in\mathbb{R}^n$.

Define $d_e(x,y)=\sqrt{\sum_{k=1}^n(x_k-y_k)^2}$

則 $d_e$ 是 metric on $\mathbb{R}^n$, 稱 Euclidean metric

[例]: Discrete metric

Let $X$ be a nonempty set. On $X$, define

$$ d(a,b)=\left\{ \begin{array}{cl} 0, & \text{if } a=b \\ 1, & \text{else } a\neq b \\ \end{array}\right. $$

則 $d$ 是 metric on $X$, 稱 discrete metric

[例]: Taxical norm

On $\mathbb{R}^2$, for $\overrightarrow{\alpha_1}=(x_1,y_1)$, $\overrightarrow{\alpha_2}=(x_2,y_2)$, define

$$ d_1(\overrightarrow{\alpha_1},\overrightarrow{\alpha_2})=\|\overrightarrow{\alpha_1}-\overrightarrow{\alpha_2}\|_1=|x_1-x_2|+|y_1-y_2| $$

則 $d_1$ 是 metric on $\mathbb{R}^2$ (很好證)

[Def]: Equivalent Metric

Let $d_1$ and $d_2$ be two metrics on a nonempty set $X$.

We say that $d_1$ and $d_2$ are equivalent $d_1\sim d_2$ if for any $x_0\in X$ and $\varepsilon>0$

1. There exists $\delta_1>0$ s.t. if $d_1(x_0,y)<\delta_1$, then $d_2(x_0,y)<\varepsilon$
2. There exists $\delta_2>0$ s.t. if $d_2(x_0,z)<\delta_2$, then $d_1(x_0,z)<\varepsilon$