度量空間 (Metric space)

Metric space $(X,d)$ 是由 set $X$ 與 distance function (metric) $d$ 兩者所組成。
$d: X \times X \rightarrow [0, \infty)$ $d : X \times X \to [0, \infty)$ , $\forall x,y,z \in X$ $\forall x, y, z \in X$ satisfies
- (Non-negativity, 非負性) $d(x,y)=0 \Leftrightarrow x=y$ .
- (Symmetry, 距離有對稱性) $d(x,y)=d(y,x)$
- (Triangle inequality, 三角不等式) $d(x,y)+d(y,z) \leq d(x,z)$ .

度量與賦範的比較(Comparison between metric and norm)

Metric	Norm
binary operator $d: X \times X \rightarrow \mathbb{R}^{+}$	uniary operator: $\begin{Vmatrix} \cdot \end{Vmatrix}: X \rightarrow \mathbb{R}^{+}$
$d(x,y) = 0 \Leftrightarrow x=y$	$\begin{Vmatrix} x \end{Vmatrix} =0 \Leftrightarrow x=0$
$d(x,y) = d(y,x)$	$\begin{Vmatrix} cx\end{Vmatrix} =\begin{vmatrix}c\end{vmatrix}\begin{Vmatrix} x \end{Vmatrix}$
$d(x,y)+ d(y,z) \leq d(x,z)$	$\begin{Vmatrix} x+ y \end{Vmatrix} \leq \begin{Vmatrix} x \end{Vmatrix} + \begin{Vmatrix} y \end{Vmatrix}$ .

(Euclidean space) $d(x,y)=|x-y| = \sqrt{\sum_{n=1}^N (x_n - y_n)^2 }$ on $\mathbb{R}^n$ .
(Discrete space) $\phi \subset M$ , $d(x,y)=1$ if $x=y$ else $d(x,y)=0$ .
(Functional space) $f, g$ $f, g$ are continuous functions
- $d_{1}(f,g)=\int_{0}^{1}\left|f(x)-g(x)\right|dx$
- $d_{\infty}(f,g)=\sup_{0\leq x\leq1}\left|f(x)-g(x)\right|$

A sequence $\left\{ x_{n}\right\}$ in $\left(X,d\right)$ is called Cauchy if $d\left(x_{n},x_{m}\right)\rightarrow0$ as $n,m\rightarrow\infty$ .
A converge sequence is a Cauchy sequence. (收斂數列必為Cauchy數列，反之不一定成立，因為極限點不一定在 $X$ 中)
A Cauchy sequence in the complete space is a converge sequence.
Complete metric space是space中任意的Cauchy數列都會收斂, 且數列的極限值在 $X$ $X$ 中。
- E.g. Rational number $\mathbf{Q}$ with $d(x,y)=|x-y|$ 不是complete metric space，因為 $\left\{ 1.4,1.41,1.414,1.4142,\cdots\right\}$ 此Cauchy數列收斂，但極限點並非有理數。