I haven't had much experience with infinite dimensional vector spaces, and I was working on a problem that asks to prove that for a finite dimensional vector space $V$, and linear transformation $T:V\to V$, $V=imT + ker T \implies V=imT \bigoplus ker T$ . I think I've done this correctly by using the rank-nullity theorem to show $dim(imT \cap kerT)=0$. Next I'm asked to find a counterexample to the assertion for an infinite dimensional $V$ and $T$.
I'm not exactly sure what I'm looking for here. It seems like it should be a linear transformation that maps one or more non-zero elements of $V$ to its own kernel (since their intersection is has to be non-trivial), but its image and kernel still span $V$ somehow.