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According to Mathworld, a Hamel basis is a basis for $\mathbb R$ considered as a vector space over $\mathbb Q$.

According to Wikipedia, the term is used in the context of infinite-dimensional vector spaces

over $\mathbb R$ or $\mathbb C$.

According to the description of the Mathematics Stack Exchange tag ,

a Hamel basis of a vector space $V$ over a field $F$ is a linearly independent subset of $V$ that spans it.

(That is often called simply a basis, and there is no mention of infinite dimension.)

I find it difficult to reconcile these three different explanations of the term Hamel basis,

though the first two seem to be different particular examples of the third

(and for finite-dimensional vector spaces different kinds of bases are the same).

J. W. Tanner
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    The Mathworld definition is far too specific and should be ignored.

    The Wikipedia and MSE tag definitions don't disagree. Rather, the Wikipedia definition focuses on the actual situation where you would want to distinguish between different "basis-flavored" notions. That is, the phrase "Hamel basis" is really only going to be worth noting if meaningfully different things also called bases are at risk of being used, and this is a somewhat limited situation. At a purely formal level, "Hamel basis" is synonymous with "basis."

     – Noah Schweber Oct 29 '21 at 03:49
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    I have always thought a Hamel basis is just a vector space basis that ignores topology. A standard example being that it is difficult to write down an explicit basis for example for the Hilbert space of square summable sequence, aka $\ell^2(\Bbb{N})$. You rather work with the complete orthonormal system of sequences with a single component $1$ and a bunch of zeros. Those don't span the whole thing as a vector space, which makes their use in proofs awkward. In more complicated spaces you won't have complete orthonormal systems, and need to cope somehow (think Hahn-Banach). – Jyrki Lahtonen Oct 29 '21 at 03:52
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    (cont'd) Admittedly it was misguided to bring up topology here. But that is chiefly where I have run into the difference. Another description might be: If you need to invoke the axiom of choice to construct it, then call it a Hamel basis. If you can write it down, call it a basis. – Jyrki Lahtonen Oct 29 '21 at 03:59
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    Mathworld is generally a quite unreliable source. – Eric Wofsey Oct 29 '21 at 04:20
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    @JyrkiLahtonen: It's not really about "writteability"... the distinction is between "basis" and "Hilbert basis". When work in Hilbert spaces, you want to work with a Hilbert basis rather than algebraic basis, because the Hilbert basis is "well-behaved" relative to the extra structure (the inner product), whereas an arbitrary basis need not be; and of course turns out to suffice once you throw in the approximation. "Hamel basis" is used in contrast to the Hilbert basis to mean "basis as a vector space in the usual sense". – Arturo Magidin Oct 29 '21 at 06:09

1 Answers1

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(1) The Mathworld definition is what Hamel had in mind, but not the way most of us use today.

(2) In modern mathematics, Hamel basis of a vector space $V$ over an arbitrary field is just a basis (i.e. a maximal linearly independent subset, or equivalently every vector can be uniquely represented as a (finite) linear combination of vectors from the set).

(3) Hamel basis is more often used in functional analysis to emphasize the algebraic nature of such basis, where other types of basis exist (e.g. Schauder basis), and because of this, sometimes it refers to vector spaces over only $\mathbb R$ or $\mathbb C$ upon which traditional functional analysis is established. For example, the orthonormal basis of an infinite dimensional Hilbert space is not a Hamel basis: It is linearly independent but not maximal. The orthonormal basis can represent every vector only if infinite linear combination is allowed (through a limit process, which is not meaningful when we are only given a vector space with no topology).

Just a user
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    What do you mean by "maximal" in this answer? Does that mean spanning? – EE18 Apr 03 '23 at 17:56
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    It means any strictly super set would be linearly dependent. More precisely, if $B$ is linearly independent and $B\subsetneq B'$ implies $B'$ must be lineearly dependent, then $B$ is called a (Hamel) basis. – Just a user Apr 04 '23 at 04:59
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    Ah, I see. And I suppose I can prove that's equivalent to $B$ being l.i. and $span(B) = V$? – EE18 Apr 04 '23 at 12:56
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    That is, do we have the equivalences ($B$ is l.i. and spans $V) \iff (B$ is maximal and l.i.) $\iff$ (every $v \in V$ can be uniquely represented as a (finite) linear combination of vectors from the set), each one of these serving equally well as the definition of a Hamel basis? – EE18 Apr 04 '23 at 14:20
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    Yes, these are all equivalent and all are about linear algebra alone (instead of limit and functional analysis). – Just a user Apr 05 '23 at 07:55
  • Got it, thanks again for all your help! – EE18 Apr 05 '23 at 13:28