賦範空間 (Normed space)

• 一般常用的vector space $X$為real number vector, matrix以及continuous function。
• 而field $F$為real numbers $\mathbb{R}$ or complex numbers $\mathbb{C}$.

Norm

• A norm on a vector space $X$ is a function: $\begin{Vmatrix} \cdot \end{Vmatrix}: X \rightarrow \mathbb{R}^{+}$, that satisfies $\forall x, y \in X$, and $c \in \mathbb{F}$
  1. $\begin{Vmatrix}x\end{Vmatrix} = 0$ $\Leftrightarrow$ $x=0$.
  2. $\begin{Vmatrix} cx \end{Vmatrix} = \begin{vmatrix} c \end{vmatrix} \begin{Vmatrix} x \end{Vmatrix}$.
  3. $\begin{Vmatrix} x+y \end{Vmatrix} \leq \begin{Vmatrix} x \end{Vmatrix} + \begin{Vmatrix} y \end{Vmatrix}$.

Normed space

• A normed vector space $(X, \begin{Vmatrix} \cdot \end{Vmatrix})$ is a vector space $X$　with a norm $\begin{Vmatrix} \cdot \end{Vmatrix}$, and it usually denoted the norm on space $X$ by $\begin{Vmatrix} \cdot \end{Vmatrix}_X$.
• A Banach space is a complete normed space.
  • Complete指的是metric space $(X,d)$中的every Cauchy sequence in $X$ is converent in $X$, $d(x,y) = \begin{Vmatrix} x- y\end{Vmatrix}$.