norm

norm $V$ 이란, 벡터 공간에 존재하는 벡터의 크기를 측정하기 위한 함수를 의미한다.

$$\begin{array}{rl}\Vert \cdot \Vert :V& \to \mathbb{R}\\ \mathbf{x}& \mapsto \Vert \mathbf{x}\Vert \end{array}$$

• $\mathbf{x},\mathbf{y}\in V$
• each vector $\mathbf{x}$ its length $\Vert \mathbf{x}\Vert \in \mathbb{R}$
• $\lambda \in \mathbb{R}$

norm 은 아래와 같은 조건을 만족한다.

• Absolutely homogeneous: $\Vert \lambda \mathbf{x}\Vert =\mid \lambda \Vert \Vert \mathbf{x}\Vert$
• Triangle inequality: $\Vert \mathbf{x}+\mathbf{y}\Vert ⩽\Vert \mathbf{x}\Vert +\Vert \mathbf{y}\Vert$
• Positive definite: $\Vert \mathbf{x}\Vert ⩾0$ and $\Vert \mathbf{x}\Vert =0⟺\mathbf{x}=0$

Related

Manhattan Norm

Euclidean distance

$L^p$ norm

$$\Vert \mathbf{x}{\Vert }_p={\left(\sum _i{\left|x_i\right|}^p\right)}^{\frac{1}{p}}\text{for}p\in \mathbb{R},p\ge 1$$

• properties: norm 함수는 다음과 같은 특징들을 따른다.
  • $f(\mathbf{x})=0\Rightarrow \mathbf{x}=0$
  • $f(\mathbf{x}+\mathbf{y})\le f(\mathbf{x})+f(\mathbf{y})$ (the triangle inequality)
  • $\forall \alpha \in \mathbb{R},f(\alpha \mathbf{x})=|\alpha |f(\mathbf{x})$
• max norm: $L^{\infty }$
  • $\Vert x{\Vert }_{\infty }={\max }_i\left|x_i\right|$
• unit norm
  • unit vector is a vector with unit norm: $\Vert \mathbf{x}{\Vert }_2=1$
  • related with orthonormal