# Is it true that every two Hamel Basis for a Vector Space have the same cardinality? [duplicate]

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

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

• Yes, this assertion is true. However, I cannot provide you with a reference. – TZakrevskiy Apr 4 '15 at 15:43
• Yes; you’ll find a proof of a slightly more general result here. – Brian M. Scott Apr 4 '15 at 15:44
• 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. – Daniel Apr 4 '15 at 15:44
• Also, this question might be useful. – TZakrevskiy Apr 4 '15 at 15:46
• @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. – Daniel Apr 4 '15 at 15:47