I'm having trouble in exercise 2.10 in Brezis' book in Functional Analysis: let $E$ and $F$ be two Banach spaces and let $T \in \cal{L}$$(E,F)$ be surjective. Let $M$ be any subset of $E$. Prove that $T(M)$ is closed in $F$ iff $M+N(T)$ is closed in $E$.
I was able to prove that $M+N(T)$ is closed in $F$ if $T(M)$ is closed in $E$ by noting that $$M+N(T) = T^{-1}(T(M))$$ and using the fact that $T$ is continuous. For the other part, my idea was to prove that $$(T(M))^{c} = T((M+N(T))^{c})$$ and then use the open mapping theorem. However, I can only prove that $$(T(M))^{c} \subset T((M+N(T))^{c}),$$ for the other inclusion it seems that I need $T$ to also be injetive, so I guess this method does not work. Am I missing something here, or is there a better way to do it? Thanks!
Since $M+N(T)$ is closed, its complementary subspace $U$ is open, since $T$ is surjective, the open map theorem implies that $T(U)$ is open, but $T(U)$ is the complementary of $T(M)$, thus $T(M)$ is closed.