08 - Norms

Norms

Let $V$ be any vector space (over $\mathbb{R}$)

Function $||\cdot ||:V\to [0,\infty )$ is called a norm if

1. $||x||=0$ if and only if $x=0$
2. $||\lambda x||=|\lambda |||x||$ for all $\lambda \in \mathbb{R}$, $x\in V$
3. $||x+y||\le ||x||+||y||$ for all $x,y\in V$

Relation between norm and metric spaces

Let $V$ be a vector space over the reals
Let $||\cdot ||$ be a norm on $V$
Define

$$d:V\times V\to [0,\infty )\text{ by }d(x,y):=||x-y||$$

Then

$$(V,d)\text{ is a metric space}$$

Examples of Different Type of Norms

1. For $d_1$

$$||x|{|}_1:=\sum _{i=1}^n|x_i|$$

2. For $d_{\infty }$

$$||x|{|}_{\infty }:=\underset{i=1,\cdots ,n}{\max }|x_i|$$

3. For $l^p$ norms

$$||x|{|}_p:={\left(\sum _{i=1}^n|x_i{|}^p\right)}^{1/p}$$

Normed Space

Vector space endowed with norm $||\cdot ||$