0
$\begingroup$

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.

$\endgroup$
  • 1
    $\begingroup$ The version of the Baire Category Theorem they're using requires the completeness of the space. $\endgroup$ – Shivering Soldier Apr 26 at 13:43
  • $\begingroup$ You're right. I forgot that $\endgroup$ – Maskoff Apr 26 at 13:50
1
$\begingroup$

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.

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.