From the relation it follows that if Tx=0 then . In other words, .

The same argument with T and T* interchanged shows that . Therefore by the rank plus nullity theorem, where n = dim(V). Interchanging T and T* again, you see that , and therefore . That shows (by the rank plus nullity theorem again) that .

Finally, it follows that . But we know that , and if these spaces have the same dimension then they must be equal.