This is purely a question about nomenclature/terminology.
Let $X$ be a normed vector space and $S$ a subset of $X$.
Question: Is there a reasonably standard way to notate the quantity $$\sup_{x\in S}\{\|x\|\}?$$
Thoughts: Such a notation would seem very useful for example in spectral theory for operators in a Banach algebra $A$, since the spectral radius of $a\in A$ is defined as $r(a)=\sup_{x\in\sigma(a)}\{|x|\}$? (In this case $X=\mathbb C$ and $\sigma(a)$ is the spectrum of $a$.)
It is seems natural to define a map $\|\cdot\|\colon\mathcal P(X)\to[0,\infty]$, where $\mathcal{P}(X)$ is the power set of $X$, by $$\| S\|:=\sup_{x\in S}\{\|x\|\}.$$ This would then reduce with the usual norm on $X$ when $S$ is a singleton. In this notation, we can write simply $r(a)=\|\sigma(a)\|$. (However, I figured that before I start using this notation I should check whether there is a shorthand already in use.)