Seeking a clarification regarding the bijectivity of the mapping between isometric spaces. If $X$ and $Y$ are isometric spaces, does the mapping between them $f:X\to Y$ have to be bijective? I feel only injectivity is required to satisfy the relation $$d_y(f(a),f(b))=d_x(a,b)$$
and surjectivity is not needed. 
Even this proof of the completion theorem seems to imply $f$ need only be injective. However, my textbook seems to insist $f$ should be bijective. I don't know if surjectivity is a condition that is useful later on.
Thanks in advance!
 A: For some people (I personally don't adhere to this definition), the statement '$X$ and $Y$ are isometric spaces' is not equivalent to 'there is an isometry $f\colon X\rightarrow Y$'. This comes down to whether or not you impose the extra condition that isometries are bijective. I would personally call such a map which wasn't necessarily surjective an isometric embedding and then if the map also happens to be surjective I would call it an isometry, so that isometries became the class of isomorphisms in the category of metric spaces. Many authors adopt this position.
However, many others adopt the definition that an isometry* (I'll use a * to distinguish the two definitions) only needs to preserve distance, and so if $f\colon X\rightarrow Y$ is an isometry* then the space $X$ and $f(X)$ are isometric metric spaces. However, $X$ and $Y$ are only isometric if $f(X)=Y$, that is $f$ is surjective.
The moral of the story is that an author should make clear which definition they are using, and a reader should check to see which definition the author is using.
