![]() ![]() by assuming one-one can I get ontoness and structure preserving properties free) I would like to know, if this is really the case or are there any other examples?ĮDIT: Looking at some answers, I thought it is better if the scope of the question is broadened.ĭoes injection (surjection) imply surjection (injection) and isomorphism/isometry? (i.e. 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. It is very easy to find domains where the result fails. ![]() is one-one iff it is onto.ġ) Finite set case: functions from $\lbrace 1,2,\dots,n\rbrace$ to itself is one-one iff onto.Ģ) Linear operators $T\colon V\rightarrow V,$ where $V$ is a finite-dimensional vector space is also one-one iff onto.ģ) Linear operators of the from (I-K) where K is some compact operator acting on a Banach space satisfies this property. I need some more examples for the following really interesting phenomenon: A function from the class.
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