Banach space $E$ is the direct sum of its closed subspaces $L$ and $M$. $M$ is finite dimensional. $T:E \rightarrow E$ is a bounded linear operator. I'm asked to prove that $T(E)$ is closed iff $T(L)$ is closed.
I have only learned the open mapping theorem and closed graph theorem, and have no idea how to prove that the image is closed. Any hint will help. Thank you.