The statement of the problem is pretty much the title. Given a banach space $X$, I want to show that $X$ cannot be written as a countable $\bigcup_{n\in\mathbb{N}} U_n$, where each $U_n\subseteq X$ is a proper subspace.
Now if we know that each $U_n$ is meager ( for example, if $U_n$ is closed, proper supspace ), then it's just an application of Baire Category Theorem. So the only problem is when $\overline{U_n} = X$, i.e., when $U_n$ is dense in $X$.
I couldn't find any resources for this problem online, so I'm not even sure if this is true. Can anyone give me any counterexamples in that case?
It is not true. Take any infinite-dimensional Banach space $X$. It follows from the Baire category theorem that $X$ has an uncountable (Hamel) basis $B$. Let $C=\{c_1,c_2,c_3,\ldots\}$ be a countably infinite subset of $B$. For each positive natural number $n$, let $X_n$ be the linear span of $(B\setminus C)\cup\{c_1,\ldots,c_n\}$. Then each $X_n$ is a proper subspace of $X$ and $X=\bigcup_n X_n$.