Example of two norms and ONE linear operator that is bounded and unbounded in a norm. I am looking for an example of a linear operator that is bounded as well as unbounded depending on which norm you take. Since I do not have much experience with Functional Analysis, I do not know many examples.
 A: Assume $\varphi : (E, \| \cdot \|_{E}) \to (F, \| \cdot \|_F)$ is an unbounded linear operator, i.e.
$$
\sup_{x \neq 0} \frac{\|\varphi(x)\|_F}{\|x\|_E} = \infty.
$$
Then define
$$
\forall x \in E, \qquad \|x\| \overset{def}= \|x\|_E + \|\varphi(x)\|_F.
$$
You can check that this gives a well defined norm on $E$ under which $\varphi$ is bounded. You don't need $\varphi$ to be unbounded for that to work, but if it is unbounded, then you have an example. The reason why $\varphi$ becomes bounded is because
$$
\sup_{x \neq 0} \frac{\|\varphi(x)\|_F}{\|x\|} = \sup_{x \neq 0} \frac{\|\varphi(x)\|_F}{\|x\|_E + \|\varphi(x)\|_F} \le 1.
$$
This is an example that happens in practice, it is not just a counter-example ; for instance, if $E = C^1[a,b]$ and $F = C[a,b]$, if we equip both of them with the supremum norm $\| \cdot \|_{\infty}$, then $\varphi : E \to F$ defined by $\varphi(f) = f'$ is not a bounded linear operator. Thus we define the new norm on $E$, namely $\|f\| = \|f\|_{\infty} + \|f'\|_{\infty}$, so that differentiation becomes continuous.
Hope that helps,
