Uncountable Hamel basis of Banach space

Question

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.

Answer 1

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.