Let $X$ be an infinite-dimensional Banach space. Suppose that $ M$ is an infinite- dimensional subspace of $X$ that has a countable Hamel basis. I want to show that $ M $ is a meager subset of $ X$.
My attempt: Since $M$ has countable Hamel basis, let $\{x_n\}_{n \in \mathbb{N}}$ be a countable Hamel basis of $M$ and let $F_n=\text{span}\{x_1,…,x_n\}$. Then $F_n$ is nowhere dense. Then I want to use Baire Category Theorem however it requires the completeness of $M$. But if $M$ is complete then it's a infinite-dimensional Banach space, and we know there won't be countable Hamel basis in $M$. Which part was I wrong? Any help or hint is appreicated.
Best Answer
Well, $M$ is by definition a countable union of nowhere dense sets, so meagre. This is without completeness, just by definitions. There is a bit of work to see that the $F_n$ are closed and have empty interior (as $X$ is infinite-dimensional), but that should be standard.
Completeness of $X$ then tells us that $M \neq X$, even has empty interior. So $X$ cannot have a countable Hamel base, which is usually the point.