Let $X$ be a vector space over a field $\mathbb{K}$.

A norm on $X$ is a map $\| . \| : X \rightarrow [0, \infty)$ that satisfies the following properties:

1. $\| x \|=0$ if and only if $x=0$
2. $\| \lambda x \| = |\lambda| \|x \|$ for all $x \in X$ and $\lambda \in \mathbb{K}$
3. $\|x+y \| \leq \|x\| + \|y\|$ for all $x,y \in X$

A normed space is a pair ($X, \| .\|$), where $X$ is a vector space and $\| . \|$ is a norm on $X$.