Definition (Complete)

A normed vector space $(V,\|\cdot\|)$ is complete if every Cauchy Sequence in $V$ converges.

Definition (Completion)

A completion of a normed vector space $(V,\|\cdot\|)$ is a normed $\mathbb{F}$-vector space $(\overline V,\overline{\|\cdot\|})$ such that there exists a linear injection $l_{V}:V\mapsto\overline V$ with the following properties:

1. $\overline{\|l_{V}\|}=\|v\|$ for every $v\in V$;
2. if $\overline v \in \overline V$, then there exists a sequence, $(v_{i})_{i\in\mathbb{N}}$, for which the sequence $(l_{V}(v_{i}))_{i\in\mathbb{N}}$ converges to $\overline v$.