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