Let $(V,\|\cdot\|_0)$ be a normed vector space with $\overline{V}$ being its completion. Show that $\overline{V}$ is an $\mathbb{R}$-vector space and that the norm $\|\cdot\|_0$ extends to a norm $\|\cdot\|$ on $\overline{V}$.
I don't quite understand this problem. What does it mean for a normed space to be an $\mathbb{R}$-vector space?
Also, does a norm on $V$ have extension on the completion of $V$ because it has to operate on the additional (w.r.t. $V$) elements in the completion of $V$? But then, how can a norm not have an extension on a completion, even if it is the same as the norm on $V$?