What does it mean to write $|||x|||$ rather than $||x||$?

Question

I am familiar with the notation $||x||$ meaning some norm of $x$. I have just come across the notation $|||x|||$ (in a text that also uses the former for norms). What is the difference?

Answer 1

Usually the triple bar notation is used for the subordinate norm for the linear transformation i.e. if $f: (E,||.||_E)\to (F,||.||_F)$ is a linear transformation then we definite the subordinate norm of $f$ by

$$|||f|||=\sup_{x\in E\setminus\{0_E\}}\frac{||f(x)||_F}{||x||_E}$$