I have really been struggling to prove the following problem.
Problem: Let $v$ be a norm on a $\mathbb R^n$. Show that $v$ is equivalent to the Euclidean Norm.
Hint: Show that $v$ is continuous with respect to the taxicab norm $|x|_1$ and then apply the Extreme Value Theorem.
So far have been able to show that the $v$ is continuous with respect to the taxicab norm, but I am really not sure where to go from there.
Any help would be very much appreciated. Thank you!
Hint: you need to use (or show) that the unit sphere $S$ with respect to the $|\cdot|_1$ norm is compact. You have already shown that $\nu$ is continuous with respect to the $|\cdot|_1$ norm and since $\nu$ is a norm, it is non-zero on $S$. Therefore it achieves a minimum value, necessarily positive on $S$; call the minimum value $c$. Show that this implies $\nu(x) \ge c |x|_1$ for all $x$.