Let $X$ be a Banach space and $M$ be a closed subspace of $X$. Suppose that there exists a subspace $N$ of $X$ such that $X=M\oplus N$. Does it imply that $N$ is closed ?
I know that not every closed subspace of a Banach space is complemented (see here). But my question is slightly different from that question. I think the answer is no. But I do not able to construct a counter example.