[Math] is bounded linear operator necessarily continuous

Question

Let $U, V$ be separable Banach spaces.

Suppose we have a bounded, linear operator
$C : U\to V$.

Questions are the following

*) Shall $C$ be continuous since $V$ is a Banach space?

*) In general, is a bounded linear operator necessarily continuous (I guess the answer is no, but what would be a counter example?)

*) Are things in Banach spaces always continuous?

Answer 1

An operator $C$ is bounded iff the set {$\|Cx\|:\|x\|\leq 1$} is bounded $\Leftrightarrow$ there is a $M<\infty:\|Cx\|\leq M\|x\|$ for every $x\in U$.

Let $ε>0$. If $x,y\in U:\|x-y\|<ε/M$, then $\|Cx-Cy\|\leq M\|x-y\|<ε$. Thus $C$ is not only continuous but uniformly continuous also.

So, a bounded operator is always continuous on norm-spaces. Banach space is a norm-space which is complete, thus things are not different there.