Norm is a function on a vector space $X$ which generalizes notion of length of vector in general vector spaces.

Norm on a vector space $X$ over field $K$ is a function $\Vert\cdot\Vert:X\to\mathbb{R}_+:x\mapsto \Vert x\Vert$, such that $$ \forall x\in X\quad\forall\alpha\in K\quad\Vert\alpha x\Vert\leq|\alpha|\Vert x\Vert $$ $$ \forall x_1\in X\quad\forall x_2\in X \quad\Vert x_1+x_2\Vert\leq\Vert x_1\Vert+\Vert x_2\Vert $$ $$ \Vert x\Vert=0\Longrightarrow x=0 $$ For example the function $\Vert x\Vert=\left(\sum\limits_{k=1}^n|x_k|^2\right)^{1/2}$ defines a norm on a vector space $\mathbb{C}^n$ over field $\mathbb{C}$.

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