Banach space has uncountable Hamel dimension
= Banach space has uncountable Hamel dimension
No infinite-dimensional <Banach space> has a countable <Hamel basis>. If $(e_n)$ were such a basis, the space would be the countable union of the finite-dimensional subspaces $\operatorname{span}(e_1,\ldots,e_n)$. Each is closed and has empty interior, contradicting the <Baire category theorem>.