l-p metrics

https://en.wikipedia.org/wiki/Norm_(mathematics)#p-norm

norms of a vector space

the $p$-norm of a vector $\mathbf{x}$ is something like: $||x|{|}_p:={\left({\sum }_{i=1}^n|x_i{|}^p\right)}^{1/p}$

links to the notion of a metric space, where we’re defining a distance function for different vector spaces

ex:

the euclidean metric is the $l_2$ norm, $||x|{|}_2=\sqrt{x_1^2+x_2^2+\cdots +x_n^2}$

in particular, the $l_{\infty }$ metric is particularly useful. when the limit is taken, the biggest component of the vector dominates, so it goes to $(x_i^p{)}^{\frac{1}{p}}=x_i$, i.e. $\max x$