Let $C^2 [0,1]$ be the space of all twice continuously differentiable functions on the compact interval $[0,1]$. Equip $C^2 [0,1]$ by the norm $W^{1,1}$, that is $$ \|f\| := \int_{0}^{1} |f(t)| \; dt + \int_{0}^{1} |f'(t) | \; dt $$
Under this norm, clearly $C^2 [0,1]$ becomes a normed subspace of $W^{1,1}$, which is equivalent to $AC[0,1]$, however it is no longer a banach subspace.
My question is that what would be the dual of the space $C^2 [0,1]$ under above norm?