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.
2025-01-13 05:35:12.1736746512
Hamel Bases: Cardinality?
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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.