p-norm

For vector $a=(a_1,\dots ,a_k{)}^{\prime }$ , some classic p-norms are:

$$||a|{|}_1=\sum _{j=1}^k|a_j|$$ $$||a|{|}_2={\left(\sum _{j=1}^k{a}_j^2\right)}^{1/2}$$ $$||a|{|}_{\infty }=\underset{1\le j\le k}{\max }|a_j|$$ $$||a|{|}_0=\sum _{j=1}^k\mathbb{I}\{a_j\ne 0\}$$