In one exercice my teacher asked us to prove the following:
Let $E,F$ be two normed spaces and $f:E\to F$ be a continuous linear map between them. Then $$\sup_{||x||\leq 1}||f(x)|| = \sup_{||x|| = 1} ||f(x)||$$
My approach was the following:
Let $a =\sup_{||x||\leq 1}||f(x)|| $ and $b = \sup_{||x|| = 1} ||f(x)||$. It's obvious that $b\leq a$ and I was trying to use the fact that: $$\exists M>0:\forall x\in E, ||f(x)||\leq M||x||$$ to show that $b$ is an upper bound of $\{||f(x)||, ||x||\leq 1\}$ and hence $a\leq b$, but I wasn't able to do so. How can this be done?
