Is this a well-defined norm on the space of linear maps?

Question

I'm reading linear maps in my lecture note:

In Ex 1.2.2, we are required to prove that $||| \cdot |||$ is indeed a norm on $L(E,F)$.

I'm struggling for two hours to prove that $\{||f(x)||_{F} \mid ||x||_{E}=1\}$ is bounded from above but to no avail. If $\{||f(x)||_{F} \mid ||x||_{E}=1\}$ is not bounded from above, $\sup _{\|x\|_{E}=1}\|f(x)\|_{F}$ is not finite and such $||| \cdot |||$ is not a function from $L(E,F)$ to $\mathbb R$.

My question: Is the exercise correctly written or I miss something?

Thank you so much!

Answer 1

The statement is false in general, that is, it can happen that $\left\{\bigl\lVert f(x)\bigr\rVert\,\middle|\,\lVert x\rVert=1\right\}$ is unbounded. You have to assume that $f$ is continuous. This will automatically hold if $\dim E<\infty$.