Let $X$ be a vector space over any field $F$, and let $\mathcal{B}$ and $\mathcal{C}$ Hamel basis for $X$. My question is

Is there a bijection $\phi : \mathcal{B}\to \mathcal{C}\ ?$ i.e. do $\mathcal{B}$ and $\mathcal{C}$ have the same cardinality?

We know this fact for finite dimensional vector spaces but, does this hold in general?

(Please ignore any concept of norm: I know that if $X$ is a Bannach space, then its dimension is at least $2^{\aleph_0}$, however, this doesn't prove my question, even in this case)


marked as duplicate by Nate Eldredge, Brian M. Scott, Community Apr 4 '15 at 15:50

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  • $\begingroup$ Yes, this assertion is true. However, I cannot provide you with a reference. $\endgroup$ – TZakrevskiy Apr 4 '15 at 15:43
  • $\begingroup$ Yes; you’ll find a proof of a slightly more general result here. $\endgroup$ – Brian M. Scott Apr 4 '15 at 15:44
  • $\begingroup$ I knew it!! I was thinking in using Zorn's lemma in order to obtain a proof, but unfortunally, I couldn't. I'm really sorry if this is a duplicate, I searched for it in MSE and couldn't find anything, that's why I've posted the question. $\endgroup$ – Daniel Apr 4 '15 at 15:44
  • $\begingroup$ Also, this question might be useful. $\endgroup$ – TZakrevskiy Apr 4 '15 at 15:46
  • $\begingroup$ @TZakrevskiy That last one was the only one I could find! That's why I've said: "Please ignore any concept of norm". I really appreciate all your comments. $\endgroup$ – Daniel Apr 4 '15 at 15:47

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