I am wondering if I have understood the consept of completion of a metric/normed space correctly. As I have understood the completion theorem, it is: $$\textbf{Completion theorem for metric spaces}$$
Let $(X,d)$ be a metric space. Then there exists a metric space $(M,D)$ and a linear injective function $\phi: X\rightarrow M$ such that $\overline{\phi(X)}=M$, and such that $d(x,y)=D(\phi(x),\phi(y)) \quad \forall x,y \in X $.
Moreover, this completion $(M,D)$ of $(X,d)$ is unique up to isomorphism. This means that if $(M',D')$ is another metric space with $\gamma:X\rightarrow M'$ a injective linear function satisfying $\overline{\gamma(X)}=M'$ with $d(x,y)=D'(\gamma(x),\gamma(y)) \quad \forall x,y\in X$,
then there also exists a unique bijection $F:M\rightarrow M'$ satisfying:
$1)\quad (F\circ\phi)(x)=\gamma(x) \quad \forall x\in X$
$2)\quad D(u,v)=D'(\phi(u),\phi(v)) \quad \forall u,v \in M$
$$\textbf{What i am wondering is:}$$
$1) \quad $Have i stated this theorem correctly?
$2)\quad $In the lecture notes i was handed they required that $X\subset M$. But if this was the case then the field $\mathbb{Q}_p$ would per definition not be a completion of $\mathbb{Q}$ with respect to the $p-adic$ norm, since the rationals in $\mathbb{Q_p}$ are isomorphic but not equal to the rationals in $\mathbb{Q}$ (they are defined in terms of equvialence classes of cauchy sequences. ) Am I mistaken? Must a completion contain the actual set?
$3) \quad $ Is property 2) of the isomorphism $F$ superfluos/wrong?
$4) \quad $ Is $\mathbb{C}$ not per definiton a completion of $\mathbb{Q}$ with respect to the absolute value? (But $\mathbb{R}$ is)
