norms and lp norms

norm

• a real-valued function maps and $x\in R^n$ into a real number
• denote: $||x||$, norm = length of the vector (vector of norm 1 = unit vec)
• $||x||$ is a norm if:
  • $\Vert x\Vert \ge 0\forall x\in X,||x||=0$ 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 $x\in X$

lp norms

• defined as $\Vert x{\Vert }_p\doteq {\left({\sum }_{k=1}^n|x_k{|}^p\right)}^{1/p},1\le p<\infty$
• p = 2: standard Euclidean length
  • $\Vert x{\Vert }_2\doteq \sqrt{{\sum }_{k=1}^n{x}_k^2}$
• p = 1: sum-of-absolute-values length
  • $\Vert x{\Vert }_1\doteq {\sum }_{k=1}^n|x_k|$
• p = $\infty$: max absolute value norm / Chebyshev norm
  • $\Vert x\Vert \infty \doteq {\max }_{k=1,\dots ,n}|x_k|$
• p = 0: pseudo norm / the cardinality (number of non-zero elements) / ${\ell }_p$

unit balls associated with popular lp norms

• 
• $I_2$ measures ordinary distance
• $I_1$ measures distance in a rectangular grid
• $I_{\infty }$ measures peak (absolute) values