Completion of a normed space

Remarks

Every normed space has a completion: take the Cauchy sequences in $X$ modulo the sequences that tend to $0$, with $\norm{[(x_n)]} = \lim_n \norm{x_n}$, and $\iota(x)$ the constant sequence. Any two completions $(\widehat{X}, \iota)$ and $(\widetilde{X}, \kappa)$ are related by a unique isometric isomorphism $\phi$ with $\phi \circ \iota = \kappa$. Example: let $P$ be the space of real polynomial functions on $[0, 1]$ with the sup norm, and let $\iota$ be the inclusion of $P$ in $C[0, 1]$. Clauses 1 and 2 hold, and clause 3 holds by the Weierstrass approximation theorem, so $C[0, 1]$ is the completion of $P$. Non-example: the inclusion of the finitely supported sequences $c_{00}$, with the sup norm, into $\ell^\infty$ satisfies clauses 1 and 2 but not 3. The reason is that the closure of $c_{00}$ in $\ell^\infty$ is the space $c_0$ of null sequences. That closure, $c_0$, is the completion of $c_{00}$. As a metric space, $\widehat{X}$ is the completion of the metric space $X$, with the vector operations extended by continuity.