Let $F$ be a finite dimensional subspace of an infinite dimensional Banach space $X$, we know that $F$ is always topologically complemented in $X$, that is, there is always a closed subspace $W$ such that $X=F\oplus W$.
I am thinking about the converse. Suppose $W$ is a subspace of $X$ such that $X=F\oplus W$ for some finite dimensional subspace $F$. Is $W$ necessarily closed?
I guess the answer should be negative but I cannot find such an example. Can somebody give a hint?