Let $\Omega\subset\mathbb{R}^N$ be a bounded domain. Define $W=W_0^{1,p}(\Omega)\cap L^\infty(\Omega)$. Is $W$ complete with respect to the norm $\|v\|=\|v\|_{1,p}+\|v\|_\infty$?
If $u_n$ is a Cauchy sequence then, by using the definition of $\|\cdot\|$, we can find $u_1$ and $u_2$ such that $u_n\rightarrow u_1$ in $W_0^{1,p}(\Omega)$ and $u_n\rightarrow u_2$ in $L^\infty(\Omega)$. Is there any chance that $u_1=u_2$?
Yes. Since $\Omega$ is bounded, both $W^{1,p}_0$ and $L^\infty$ are contained in $L^1$ with continuous inclusion. If $u_n$ converges to $v_1$ in $W^{1,p}_0$ and to $v_2$ in $L^\infty$, then it converges to $v_1$ and to $v_2$ in $L^1$. It follows that $v_1=v_2$ almost everywhere.