I know that the completion of $C^k$ under the sobolev norm is the sobolev space $W^{k,p}$, and I was trying to find an example to show that $C^k$ is not complete with respect to the sobolev norm itsself. Does anybody have a nice example?
Thank you in advance!
If $k=0$ then this reduces to showing that continuous functions are not complete w.r.t. the $L^p$ norm.
To see this, you can use the sequence $u_n(x):=\operatorname{sgn}(x)|x|^n\in L^p(-1,1)$ so that $u_n\to \bar{u}:=\operatorname{sgn}(x)$ in $L^p(-1,1)$ for any $p\in [1,\infty)$. So each $u_n$ is continuous, but $\bar{u}$ is not.
Now from this we can construct a counterexample for general $k$.
Integrate each $u_n$ $k$ times to obtain a sequence $\left\{v_n\right\}$ (we can do this while prescribing the first $k-1$ derivatives of $v_n$ in $x=-1$ , so choose them to be $0$ for all $n$), and $\bar{u}$ to obtain an element $\bar{v}$.
Then $v_n^{(k)}\to \bar{v}^{(k)}$ in $L^p(-1,1)$. Since $\bar{u}$ is not continuous, then $\bar{v}$ cannot be $C^k$. However, each $v_n\in C^k([-1,1])$. It remains to show that $v_n\to \bar{v}$ in $W^{k,p}$. We already know that $v_n^{(k)}\to \bar{v}^{(k)}$ in $L^p$, so it remains to show $L^p$ convergence for lower order derivatives, which can be done by (reverse) induction using \begin{align*}\|v_n^{(h)}-\bar{v}^{(h)}\|_{L^p}^p&=\int_{-1}^{1}\left|\int_{-1}^{t}\left(v_n^{(h+1)}(s)-\bar{v}^{(h+1)}(s)\right)\,\mathrm{d}s\right|^p\,\mathrm{d}t\\ &\leq \int_{-1}^1(t+1)^p\int_{-1}^t \left|v_n^{(h+1)}(s)-\bar{v}^{(h+1)}(s)\right|^p\,\mathrm{d}s\,\mathrm{d}t\\ &\leq 2^{p+1}\|v_n^{(h+1)}-\bar{v}^{(h+1)}\|_{L^p}^p\to 0, \end{align*} for $h=k-1,\dots, 0$.