Does this result for inverses of linear transformation holds for infinite dimensional case.

Question

Q.If V is an arbitrary vector space over $F$ and if $T$ belongs $Hom(V,V)$ is right-invertible with a unique right inverse, prove that its invertible. This was question asked in Topics in Algebra by I.N. Herstein. I was able to prove this result by using the argument by contradiction, let us assume $T$ is not invertible and So there exist a $S\not=0$ such that $ST=TS=0$ and let $S_1$ is unique right inverse, now $ S_1TS=IS=S=0$, which is a contradiction. I had question if the vector space is not finite dimensional does the statement holds true? If yes, does my argument above holds.

Answer 1

In general the statement is not true. The problem is that $S_1T$ is the identity on the image of $T$ which could be smaller than $V$. Think for example in the space of polynomial ring the multiplication by $x$ which is linear and has an inverse in the image (i.e. divide by $x$ by shifting all the coefficient) but clearly is not invertible on the all space (a polynomial of degree zero has not pre image)