Some questions about isometries between completions

Question

I have some doubts about completion. The definition that was shown in the book used in my course of Real Analysis is the next one:

We say that a metric space $(X^{*}, d)$ is a completion of $X$ if $X^{*}$ is complete and exists an isometry $\iota: X \rightarrow X^{*}$ such that $\overline{\iota(X)}= X^{*}$, where $\overline{\iota(X)}$ is the closure of $\iota(X)$ inside $X^{*}$.

And then it follows the next theorem:

Theorem Any metric space allows a completion and is unique of to isometry.

To prove this theorem, the book gives you some steps, I haven't prove this one:

Show that if there are completions ($X^{*}, d^{*})$ and $(X^{\#}, d^{\#})$ of X, there exists a bijective isometry $\beta : X^{*} \rightarrow X^{\#}$.

My questions are:

1. Is there a way to prove that $\beta$ exists without using the density of spaces ?
2. Would it be o.k if I just show that some specific function between $X^{*}$ and $X^{\#}$ is an isometry?
3. So far I have only thought in the identity between the completions, but I`m not so sure where to start, since I ' don't know how $d^{*}$ $d^{\#}$ are defined? (Later on the steps it defines $d^{*}$)

Thanks in advance.

Answer 1

1. You will need to use density of $X$ (after all, that is pretty much all you know about $X^*$ and $X^\#$).
2. What do you mean by "specific function"? The whole point is to construct this bijective isometry $\beta$.
3. Use density and isometric embedding of $X$ into these spaces. You don't need to know how $d^*$ or $d^\#$ is defined except they satisfies the definition of completion (you are only proving the uniqueness part assuming existence). So you have isometric embeddings $\iota^*\colon X\to X^*$ and $\iota^\#\colon X\to X^\#$ allowing you to define an isometry $\iota^\#\circ(\iota^*)^{-1}\colon\iota^*(X)\to\iota^\#(X)$. Show that you can extend this (uniquely) to $\overline{\iota^*(X)}\to\overline{\iota^\#(X)}$. Similarly for the other way round, so ...