A lemma is that:
Let $V$ be a finite dimensional vector space over a field $F$ and let $T\in L(V,V)$. Then there exists a nonzero linear transformation $S\in L(V,V)$ such that $TS=0$ if and only if there exists a nonzero vector $v\in V$ such that $T(v)=0$.
This proof is not difficult. I want to know that whether it is true when $V$ is infinite dimensional? When the such $S$ exists, we can easy to find the such $v$. But I can not do the converse statement. In fact, I need to find a $S\neq 0$ such that $\operatorname{Image}(S)\subseteq \ker(T)$. We know that $\ker(T)$ is a subspace of $V$ and so $S\in L(V,\ker(T))$. Thus if $L(V,\ker(T))\neq \{0\}$ when $\ker(T)\neq 0$?