Linear operator norm equality Let $A:\mathbb R^n \to \mathbb R^m$ be a linear operator.
How can I show this equality in the operator norm?
$$\sup_{x\in \mathbb R^n\setminus\{0\}} \frac{||Ax||}{||x||} = \max_{x\in \mathbb R^n , ||x||=1}||Ax||$$
I've tried to use the fact that for every $x\in \mathbb R^n$ we have $\frac{x}{||x||}\in S^{n-1}$, but didn't understand what to do next. Thanks.

If $x\neq0$, then $\left\|\frac x{\|x\|}\right\|=1$, and therefore$$\frac{\|Ax\|}{\|x\|}=\frac1{\|x\|}\|Ax\|=\left\|A\left(\frac x{\|x\|}\right)\right\|\leqslant\sup_{\|x\|=1}\|Ax\|.$$
On the other hand, if $\|x\|=1$, then$$\|Ax\|=\frac{\|Ax\|}{\|x\|}\leqslant\sup_{x\neq0}\frac{\|Ax\|}{\|x\|}.$$
