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.

Solution 1:

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

Solution 2:

I think the question you are trying to get at is about the relation between algebraic complements and topological complements ie if

1) M and N are complemented algebraically (complements defined without any topology involved).

2) M is closed.

Does this mean N is closed?

The answer is no, See this answer on the same site for a counterexample.

See this survey for more relations between algebraic and topological complements. In the Banach space setting, two closed subspaces are algebraic complemented if and only if they are topologically complemented.