TITLE: Norms on the reals
QUESTION [5 upvotes]: On the real numbers the absolute value is a norm on this vector space. We can also define the norm of $x$ to be $c|x|$, where $c>0$ is a constant. Are they the only norms on the real numbers? If not, what are other norms on the real numbers?

REPLY [7 votes]: Yes, $c|x|$ is  a norm and there are no others. If $\|x\|$ is any norm then $\|x\|=\|(x)(1)\|=|x|\|1\|$ so $\|x\|=c|x|$ where $c =\|1\|$.

REPLY [2 votes]: These are the only ones because of the homogeneity: $$\|x\| = |x|\|1\|.$$