Closedness of a point evaluation operator

Question

Given a Banach space $X=L^p(0;1)$ and a subspace $Z=\{u \in W^{2,p}(0;1);u'(0)=0\}$, we define the operator $C:Z\to \mathbb{R}$ by $Cf=f(0)-f(1)$. Now i dont know if the operator $C$ is closed or not? Thanks for help.