I need some more examples for the following really interesting phenomenon:
A function from the class ... is one-one iff it is onto.
Some examples I know:
1) Finite set case: functions from $\lbrace 1,2,\dots,n\rbrace$ to itself is one-one iff onto.
2) Linear operators $T\colon V\rightarrow V,$ where $V$ is a finite-dimensional vector space is also one-one iff onto.
3) Linear operators of the from (I-K) where K is some compact operator acting on a Banach space satisfies this property. This is the famous result of Fredholm.
It is very easy to find domains where the result fails.
I remember my teacher telling me that 'compactness is the next best thing to finiteness', hence this result which trivially holds in the finite case can happen only in the compact setting. I would like to know, if this is really the case or are there any other examples?
Thank you in advance.
EDIT: Looking at some answers, I thought it is better if the scope of the question is broadened.
Does injection (surjection) imply surjection (injection) and isomorphism/isometry? (i.e. by assuming one-one can I get ontoness and structure preserving properties free)