Difference between revisions of "Two-sided inverse"

Definition: Two-sided inverse
If is a function, then is a two-sided inverse of if and .

If has a two-sided inverse, it must be unique, so we are justified in writing the two-sided inverse of . We also write to denote the inverse of if it exists.

If and then is a left inverse of , because

For the same reason, is a right inverse of .

However, is not a right inverse of (nor is a left inverse of ) because

Finally, if and , then and are two-sided inverses of each other, because

and