Let $A\subset C([0,1])$ a closed linear subspace with respect to the usual supremum norm satisfying $A\subset C^1([0,1])$. Is $D\colon A\rightarrow C([0,1]), \ f\rightarrow f'$ continuous iff $A$ is finite dimensional?
If $A$ is finite dimensional $D$ is continuous of course. But is the other implication true at all? Wouldn't something like $A:=\overline{\text{span}\{\sin{\left(t+\frac{1}{n}\right)} \ | \ n\in\mathbb{N}\}}$ be a counterexample?