For a strict positive potential function $V(x)$, we define the following function space corresponding with $V$:
- $H_{V}^1(R^n):= \{u\in H^1(R^n): \int _{R^n}V(x)u^2dx<\infty\}$
Where $H^1(R^n)$ is the usual Sobolev space. I know that if $V$ is coercive, then the embedding from $H_{V}^1(R^n)$ to $L^2(R^n)$ is compact, but I fail to prove that the space $H_{V}^1(R^n)$ is a closed space in $H^1(R^n)$, hence a complete space. Could anyone give me some tips?