A question about complement of a closed subspace of a Banach space 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.

Writing $X = M \oplus N$ implies that the projection $\pi_1: X \to M$ is continuous.  Then $N = \pi^{-1}(\{0\})$ must be closed. 
