# Uncountable Hamel basis of Banach space

I came across this problem: Let $X$ be an infinite dimensional Banach space. Prove that every Hamel basis of X is uncountable.. The proof is absolutely correct however I was wondering if this is true for any normed space since I didn't see the use of completeness in the answer.

• The version of the Baire Category Theorem they're using requires the completeness of the space. – Shivering Soldier Apr 26 at 13:43
• You're right. I forgot that – Maskoff Apr 26 at 13:50

No, it is not true. Take, say, the space of all sequences $$(a_n)_{n\in\Bbb N}$$ of real numbers which are $$0$$ if $$n\gg0$$, with$$\lVert(a_n)_{n\in\Bbb N}\rVert=\sup_n|a_n|.$$Then a Hamel basis of space is $$(e_n)_{n\in\Bbb N}$$, where $$e_n$$ is the sequence which takes the value $$1$$ when $$n=1$$ and which is $$0$$ otherwise.