1. ## norms on R

Describe all norms on $\mathbb{R}^1$. Does anyone know how to solve it?

2. ## Re: norms on R

Originally Posted by Camille91
Describe all norms on $\mathbb{R}^1$. Does anyone know how to solve it?
Think about it like this, let $f:\mathbb{R}\to\mathbb{R}$ be a norm, then we know that $f(ar)=|a|f(r)$ for every $a\in\mathbb{R}$, and so evidently $f(r)=|r|f(1)$ for all $r\in\mathbb{R}$. Thus, if $f_\lambda:\mathbb{R}\to\mathbb{R}$ denotes the function $f_\lambda(x)=|x|\lambda$ then evidently every norm is a $f_\lambda$. Now, $\lambda|x+y|\leqslant \lambda|x|+\lambda|y|$ holds for all $x,y\in\mathbb{R}$ if and only if $\lambda\geqslant 0$. Noting finally that $f_\lambda(0)=0$ we may combine this to conclude that $\left\{\text{norms}\right\}=\left\{f_\lambda :\lambda\geqslant 0\right\}$.

3. ## Re: norms on R

I think you are right. Just using simple definiton of norms. Thanks a lot!