sequence in $H_0^1(0,1)\setminus D(0,1)$

Question

Consider $D(0,1)=\{f\in C^{\infty}(0,1); supp(f)\subset (0,1)\;\text{compakt}\}$ and $H_0^1(0,1)=\overline{D(0,1)}^{\|\cdot\|_{1,2}}$, with the norm $\|u\|_{1,2}=(\|u\|_{L^2}^2+\sum\limits_{|\alpha|\le 1}\|D^{\alpha}u\|_{L^2}^2)^{\frac{1}{2}}$. I'm searching for a sequence $(u_n)\subseteq H_0^1(0,1)$ which is not in $D(0,1)$ because I want to prove that the unbounded differential $i\frac{\partial}{\partial x}$ operator defined on $H_0^1(0,1)$ is not closed (in the sense: the graph is not closed). Do you know such a sequence to show, that $D(0,1)\not=H_0^1(0,1)$? Regards

Answer 1

Any continuous piecewise linear function $f$ with $f(0)=f(1)=0$ is in $H_0^1(0,1)$ but not in $D(0,1)$. One explicit example would be $f(x)=\min(x,1-x)$.