賦範空間 (Normed space)

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

Norm

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

Normed space

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

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