Every norm on $\mathbb{R}^n$ is equivalent to $\|\cdot \|_\infty$?

Question

We have just come across the lemma that every norm on $\mathbb{R}^n$ is equivalent to $\|\cdot \|_\infty$. My question is that do all norms on $\mathbb{R}^n$ take the form $\|\cdot \|_1, \|\cdot \|_2, \dots, \|\cdot \|_\infty$ etc. ? And if not how can you possibly prove that any norm on $\mathbb{R}^n$ is equivalent to $\|\cdot \|_\infty$ if you don't know what 'form' the norm takes?

Answer 1

For the sake of completeness, here's an answer to your questions:

1) Consider, on $\mathbb{R}^2$, the norm $$ \|(x,y)\|=2|x|+3|y| $$ (of course, any other choice of positive coefficients will to, too). You can also do the same kind of variation with the $p$-norms. Notice that if you take any subadditive positive function on $\mathbb{R}$ ($f(t)>0$ for all $t\ne0$, $f(0)=0$, $f(s+t)\leq f(s)+f(t)$), then $$ \|(x,y)\|_f=f(x)+f(y) $$ defines a norm, and even (for $p\geq1$) $$ \|(x,y)\|_{f,p}=\left(f(x)^p+f(y)^p\right)^{1/p}. $$

2) The answer to your second question can be seen from the proof of Fabian quoted above: in a finite-dimensional space any norm is bounded above by a multiple of the infinity norm; this together with the compactness of any closed and bounded set (again a consequence of the finite-dimensionality) guarantees the equivalence of your norm with the infinity norm.