To show Maps are closed .Closed Graph theorem

Question

Let $V$ and $W$ be two Banach spaces and let $T \in L(V,W)$ be surjective.Let $M$ be any subset of $V$.Prove that $T(M)$ is closed in$W$ iff $M+N(T)$ is closed in V.

I tried to prove it by using closed graph theorem.But I am not able to conclude.

Answer 1

Define $S:V/\ker(T) \to W$ by $S(x+\ker(T))=Tx$. Then $S$ is an isomorphism (by open mapping theorem). Hence $M+\ker(T)$ is closed iff $S(M+\ker(T))$ is closed. But $S(M+\ker(T))=T(M)$. Henec the result.