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?
Thanks!
Best Answer
If $f:X\to\mathbb C$ is a discontinuous linear functional, then $\ker f$ is not closed. If $v$ is in $X\setminus \ker f$, then $F=\mathbb C v$ and $W=\ker f$ gives a counterexample. ($X$ is the internal direct sum of $F$ and $W$ as vector spaces, but it is not a topological direct sum.)