Let $X$ be a (real or complex) infinite dimensional vector space. (Not Normed or Banach one).
Is every Hamel Basis for $X$ necessarily uncountable ?
2025-01-13 05:43:16.1736746996
Is Hamel Basis necessarily uncountable?
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Take any infinite dimensional vector space. Take a set $S$ of countably infinitely many linearly independent vectors from that vector space. Be $V$ the subspace spanned by $S$ (using finite linear combinations). Then the vectors in $S$ form a Hamel basis of $V$.
Thus $V$ has a countably infinite Hamel basis.
Note however, that in a vector space that has an uncountable Hamel basis, all other Hamel bases are also uncountable, as all Hamel bases of the same vector space have the same cardinality.