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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 $X$.