I'm trying to show that $D:(X, \|\cdot\|_\infty) \rightarrow C[0,1]$ is a continuous map. $D$ is the differential operator and $X$ is a closed (proper) subset of $C^1[0,1]$.
The fact that $X$ is closed in $C^1[0,1]$ must be important in the proof because otherwise this result is obviously false. However, I don't know how to use this fact.
I need this result to apply Arzela-Ascoli theorem to show that unit ball of X is compact and then conclude that $X$ is finite dimensional.
Does anyone know how to tackle this problem ?