Isometry of a metric space example

Question

I am looking for an example of metric space isometry where the distance functions are different. I know one example of isometry would be a specific function where we have two metric spaces, both of them sets of real numbers and both having the distance function $f(x,y) = |x - y|$. I can see how some function $\sigma$ would be an isometry on those two metric spaces.

Can anyone provide an example of an isometric function on two distinct metric spaces with distinct distance functions?

Answer 1

On $(0,\infty)$, let $d_1(x,y)=\bigl\lvert\log(x)-\log(y)\bigr\rvert$ and on $\mathbb R$ let $d_2(x,y)=\lvert x-y\rvert$. Then$$\begin{array}{ccc}\bigl((0,\infty),d_1\bigr)&\longrightarrow&(\mathbb R,d_2)\\x&\mapsto&\log x\end{array}$$is an isometry.

Answer 2

You can for instance, starting from a vector space $E$ with norm $\|\mkern2mu.\|$ and a positive number $r$, consider $E$ with norm $\|\mkern2mu.\|_r= r\|\kern2mu.\|$. The linear map: \begin{align} (E,\|\kern2mu.\|)&\longrightarrow (E,\|\kern2mu.\|_r)\\ x& \longmapsto \frac x r, \end{align} is such an isometry.