Is the dual to $C^1[0,1]$ separable?

Question

$C^1[0,1]$ is endowed with the norm $\|f\| = \sup_{t \in [0,1]}|f| + \sup_{t \in [0,1]}|f'| $. I need to check if its dual $(C^1[0,1])^*$ is separable (I hope it is not). I am asking for the answer and the idea of proof.

Answer 1

For $x\in[0,1]$ define $\lambda_x\in(C^1)^*$ by $\lambda_x f=f'(x)$.

If $x\ne y$ you can construct $f$ with $|f|\le1$, $|f'|\le 1$, $f'(x)=1$ and $f'(y)=0$. Hence $$||\lambda_x-\lambda_y||\ge1/2\quad(x\ne y)...$$