A metric space $(X,d)$ is said to be *bounded* if there is $r\in\mathbb{R}$ such that for all $x,y\in X$ we have $d(x,y) \leq r$.


A *self-isometry* is a map $\iota:X\to X$ such that for all $x,y\in X$ we have $d(x,y) = d(\iota(x), \iota(y))$.


Does there exist a bounded metric space with a non-surjective self-isometry?



Bounded metric spaces with non-surjective self-isometry
Let $B$ be the unit ball in the sequence space $\ell^2$. Send $(x\_0, x\_1,\dots)$ to $(0,x\_0,x\_1,\dots)$.


