Consider $V : \mathbb{R}^N \rightarrow \mathbb{R}$ and let
$$X(\mathbb{R}^N) = \left\{ u \in H^1(\mathbb{R}^N) : \int_{\mathbb{R}^N} Vu^2 < +\infty \right\}.$$
Also consider the application
$$
||u|| = \left(\int_{\mathbb{R}^N} |\nabla u|^2 + \int_{\mathbb{R}^N} Vu^2\right)^{\frac{1}{2}}.
$$
Under what conditions on $V$, the space $(X(\mathbb{R}^N), ||\cdot||)$ is a Banach space?
Any reference with this subject would be very useful for me.