A vector space $E$, generally over the field $\mathbb R$ or $\mathbb C$ with a map $\lVert \cdot\rVert\colon E\to\mathbb R_+$ is a normed space if the three conditions are fulfilled:

  1. $\lVert x\rVert =0\Rightarrow x=0$,
  2. For all $x\in E$ and for all $\lambda\in\mathbb R$, $\lVert\lambda x\rVert=|\lambda |\lVert x\rVert$,
  3. For all $x,y\in E$, $\lVert x+y\rVert\leq\lVert x\rVert+\lVert y\rVert$.
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