Common Vector Norms (p-norms)

A mapping $||\cdot ||:{\mathbb{R}}^n\to {\mathbb{R}}^{+}$ that satisfies

$$\begin{array}{rl}& 1.||x||=0\\ & 2.||x+z||\le ||x||+||z||,\forall x,z\in {\mathbb{R}}^n\\ & 3.||\alpha x||=|\alpha |||x||,\forall \alpha \in \mathbb{R}\text{and}x\in {\mathbb{R}}^n\end{array}$$

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1-norm 2-norm ∞-norm