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Every vector space admits a Hamel basis by AC.
That is there are maximally linear independent sets.
But how to prove their cardinalities necessarily agree?

..I couldn't really find any reference.

José Carlos Santos
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freishahiri
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  • The easy answer is: this is just like the finite dimensional case, only with transfinite recursion to construct the bijection between the two bases, rather than just an induction up to $n$ or whatever. – Asaf Karagila Jun 15 '16 at 13:59

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If there is a finite Hamel base, the vector space is finite dimensional and we can assume to be known that any basis has the same number of elements. Suppose that $\mathcal B$ is an infinite Hamel basis for the $F$-vector space $V$. Here $F$ is the field of the scalars and I assume in this answer that $F$ has the cardinality of $\mathbb R$. Any vector is a finite linear combination of element of $\mathcal B$ in a unique way. Note that this imply that $V$ has the cardinality of $F$, because there is an injection. $$ V\to \bigcup_{n=1}^\infty\left(F\cup F^2\ldots\cup F^n\right) $$ So, there are two possibilities: $\mathcal B$ is infinite but countable or $\mathcal B$ has the continuum cardinality of $F$. For the case of topological VS The first possibility is ruled out here; No infinite-dimensional $F$-space has a countable Hamel basis.

Community
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guestDiego
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    The result you cite is for topological vector spaces. The space of sequences of reals (or whatever field) with finite support is decidedly a space with a countable basis. – Milo Brandt Jun 15 '16 at 13:43
  • How do you construct your injection? I think you miss substantially lot of expressions.. – freishahiri Jun 15 '16 at 13:46
  • Many thanks to Milo Brandt for having highlighted my oversight! The remark is absolutely correct. I will have to re-edit or erase the "answer". – guestDiego Jun 15 '16 at 13:52
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    I posted some results along these lines in this 23 July 2000 sci.math post (see also my follow-up here), which you're welcome to make use of to rewrite your answer, if you wish. – Dave L. Renfro Jun 15 '16 at 14:07
  • @DaveL.Renfro sadly that link is not working at the moment. – theHigherGeometer Jun 24 '24 at 12:06
  • @theHigherGeometer: Fortunately I learned many years ago (early 2000s, when Math Forum and google both kept changing URLs for their Usenet post archives) to give specific dates when citing these posts, allowing me to more easily search and identify them when needed. The first of those posts is this 24 July 2000 sci.math post (date difference probably due to archiving site time zone differences) and the 2nd (a correction/addition with same date) is here. – Dave L. Renfro Jun 24 '24 at 12:17
  • @theHigherGeometer: I notice that one of your Ph.D. advisers is James Stasheff. I had a reading course out of Vick's Homology Theory with him Spring of my 2nd undergraduate year at UNC (Spring 1979; I was terribly bad in both motivation and achievement in that course -- see my comments here), and later that semester (April?) Milnor was nearby NC State Univ. to give several days of talks, and Stasheff invited me along the last day (continued) – Dave L. Renfro Jun 24 '24 at 12:31
  • when Milnor gave a "general math/physics audience" colloquium talk titled (I think) "Is the Universe Simply Connected?" Afterwards, when meeting up with Stasheff to ride back to UNC, he walked us (there was also a post-doc who rode there with Stasheff and me) to the crowd around Milnor, pushing through to the front, and at that point I learned that Stasheff was driving Milnor to the airport, although we first had dinner at (what was for me at the time) a fairly nice restaurant. (continued) – Dave L. Renfro Jun 24 '24 at 12:50
  • At one point during the dinner I asked Milnor something about dyads (a mathematical device that showed up in the 2nd semester of an upper level classical mechanics course I had previously taken; text was Symon's Mechanics) and I was astounded when Milnor said he didn't know what a dyad is (but I tried not to show my surprise). At that time I suppose I assumed someone with Milnor's stature in math knew pretty much everything under the sun. Incidentally, Stasheff knew what they were and explained it in mathematical terminology. – Dave L. Renfro Jun 24 '24 at 12:50
  • @Dave thanks for the updated links (why not copy the content of those emails into an answer below? More copies helps preserve these things in case Google kills yet another service) and also the reminiscences of Jim! We have never in fact met in person, and only ever corresponded by email. – theHigherGeometer Jun 24 '24 at 23:46
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    @theHigherGeometer: why not copy the content of those emails into an answer below? --- I'd have to stick it in another place, probably one of the cited questions it's a duplicate of. I'll probably do this at some point, but not anytime soon as it is a nontrivial task to Mathjax things and make other wording changes I'd want to do, as well as review all the math. I've done this for many of my old sci.math posts, such as this 3 July 2005 sci.mat post to these mathoverflow answers. – Dave L. Renfro Jun 25 '24 at 00:40
  • @theHigherGeometer: Regarding reminiscences of Stasheff, I was originally enrolled in his Fall 1978 graduate algebraic topology class, but despite a decent background in algebra and general topology (2 semesters Herstein's Topics in Algebra, Kasriel's Undergraduate Topology and Munkres Topology and about 1/3 of Willard's General Topology), I felt lost right away (used Vick's Homology Theory), and wound up auditing the class (still have the notes). But early that semester, maybe within the first week, Stasheff broke his leg falling off a ladder while painting his house (continued) – Dave L. Renfro Jun 25 '24 at 00:48
  • (I think this is how it happened), and Robert Heyneman taught more than half of the course, with the result that it had much more algebraic flavor than what Stasheff would have done (e.g. exact sequences of chain complexes vs. CW complexes). The reason for my taking the reading course the next semester was to do the course for credit, but it didn't work out very well, as algebraic topology did not mesh very well with me. – Dave L. Renfro Jun 25 '24 at 01:02
  • @theHigherGeometer: Thinking back about Stasheff's injury a bit more, I think he hurt his back rather than broke a leg. Also, I wouldn't think a broken leg would keep him from teaching for 2-3 months. – Dave L. Renfro Jun 25 '24 at 01:17