A Banach space cannot be countable union of its proper subspaces

Question

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?

Is this really a question about Banach spaces, or is it really a question about vector spaces over an uncountable field? — Gerry Myerson, Mar 30 '20 at 12:40
@GerryMyerson I don't understand what you mean. Will you please elaborate? — Potemkin Metro Card, Mar 30 '20 at 12:47
I mean, is it true more generally that a vector space over an uncountable field can't be the union of a countable set of its vector subspaces? — Gerry Myerson, Mar 30 '20 at 21:50
Doesn't that sound like a much stronger statement? I don't know any proofs of that, I can't think of anything either :/ — Potemkin Metro Card, Apr 03 '20 at 16:43
It certainly implies the statement about Banach spaces, which is why I suggested trying to prove it. See https://math.stackexchange.com/questions/1284301/show-that-mathbbrn-cannot-be-written-as-a-countable-union-of-proper-subspa — Gerry Myerson, Apr 04 '20 at 00:40

score 1 · Answer 1 · answered Apr 13 '22 at 21:21

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 $X=\bigcup_n X_n$.

A Banach space cannot be countable union of its proper subspaces

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