Does every Hamel basis for an infinite dimensional topological vector space have a maximal countable spanning subset?

Question

‎‎‎‎Let ‎$ X $ ‎be an infinite dimensional topological vector space and ‎$\{ e‎_{i}: i ‎\in I ‎\}$‎ be a Hamel basis for $X$. Does there exist a maximal countable subset ‎$J‎\subset ‎I$ ‎such ‎that for all ‎$a‎\in ‎X$ ‎there ‎are‎ ‎$j‎_{1}, \ldots, j‎_{k}‎\in J‎$ such that $a=‎\sum^{k}_{i=1}\lambda_{i}e_{j_i}‎$‎?

Answer 1

No, there isn't in general.

If $J$ were such a set then $J$ would span the whole vector space. So we would have $J=I$. So your question can be reduced to the question if there exists vector spaces with an uncountable basis. Each infinite dimensional Banach space is such a space. For example take $C_0([0,1])$ with supremum norm.