Let $E,F$ be normed vector spaces, and let $\mathcal{L}(E,F)$ be the set of linear maps from $E$ to $F$.
All the definitions of linear maps being continuous (and differentiable) that I have seen require that $E$ and $F$ be Banach spaces (i.e., complete normed vector spaces).
Why is this necessary? If $E$ or $F$ is not Banach, are there no linear maps from $E$ to $F$ that are continuous/differentiable?