A mapping $f:X\to Y$ between metric spaces $(X,d_X)$ and $(Y,d_Y)$ is called an isometry if it preserves distances, i.e.
$$d_Y(f(a), f(b))=d_X(a,b)\text{ }\forall\text{ } a,b\in X$$
My question is:
Is an isometry necessarily surjective? If yes, how can we prove it? If no, is there any counterexample?
My guess is yes. To show that a function is surjective we need to show for all $y\in Y$, there exist some $x\in X$ such that $y=f(x)$.
Let's suppose $f$ is not surjective, then there exist some $y_1\in Y$ such that for all $x\in X$, $y_1\neq f(x)$. Then $d_Y(y, y_1)$ is well defined but $d_X(x, x_1)$ is not defined because there is no $x_1$ such that $f(x_1)=y_1$. Hence $f$ is not an isometry, a contradiction. Therefore $f$ must be surjective.
I am not sure whether my argument is correct. I would really appreciate any help that shows whether it must be surjective or not and suggestion to improve my proof or counterexample.
Thanks very much!
Best Answer
Consider the following linear map on the Hilbert space $l^2(\mathbb{N})$. Define $T : l^2 \to l^2$ by $ T((x_1,x_2,....)) = (0,x_1,x_2,...)$.
Clearly $T$ is an isometry, but not onto.