Denote $AC[0,1]$, the set of all absolutely continuous trajectories from $[0,1]$ to $R^n$, with it's usual norm. Then $X:=C^{2} [0,1]$ is obviously a vector-norm subspace of $AC[0,1]$ when we impose the norm from $AC[0,1]$ to $C^{2} [0,1]$.
My question is: What is the (topological) dual spaces of $X$? In another words, I would like to identify the members of $X^*$.
I am also interested to know the dual of the space $C^{\infty} [0,1]$ under same norm.
Thanks in advance !