Norms on the reals

Question

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?

score 7 · Accepted Answer · answered Feb 07 '19 at 00:36

7

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 $c =\|1\|$.

answered Feb 07 '19 at 00:36

Kavi Rama Murthy

311,013

score 2 · Answer 2 · answered Feb 07 '19 at 00:36

2

These are the only ones because of the homogeneity: $$\|x\| = |x|\|1\|.$$

answered Feb 07 '19 at 00:36

Klaus

10,578

Norms on the reals

2 Answers2