Vector Norm

Definition:

• A norm on ${\mathbb{R}}^n$ is a real-valued function with special properties that maps any element $x\in {\mathbb{R}}^n$ into a real number (denoted by $||x||$)
• Must satisfy 3 conditions:
  • $\Vert x\Vert \ge 0\forall x\in X,||x||=0$ IFF $x=0$
  • $\Vert x+y\Vert \le \Vert x\Vert +\Vert y\Vert ,$ for any $x,y\in X$ (triangle inequality)
  • $||\alpha x||=|\alpha |.||x||$, for any $\alpha$ scalar and $x\in X$

$l_p$ norm:

• $||.|{|}_p={\mathbb{R}}^n\to \mathbb{R}$
• Defined as $|x{|}_p\doteq \left(\sum _{k=1}^n|x_k{|}^p\right){}^{1/p},1\le p<\infty$

$p=2$:

• Euclidean Length

$p=1$:

• sum-of-absolute-values length
• $|x{|}_1\doteq {\sum }_{k=1}^n|x_k|$
• $||x|{|}_1=1$ forms a diamond

$p=0$:

• pseudo norm/the cardinality (number of non-zero elements)
• not a norm as it doesnt satisfy the last condition

$p=\infty$:

• max absolute value norm / Chebyshev Norm
• $|x{|}_{\infty }\doteq {\max }_{k=1,\dots ,n}|x_k|$
• $||x|{|}_{\infty }=1$ forms a square

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