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?