normed space which posses a countable algebraic basis that a Banach space cannot posses such a basis

Question

I have have read some papers related to normed space , it came to mind to Find a normed space wchich posses a countable algebraic basis that a Banach space cannot posses such a basis , But i failed to get answer for that question , I guess about $\ell^2$ space but the problem about the existence of its Basis .

Answer 1

The space $\ell_0$ of finitely nonzero sequences has the standard vectors $e_1,e_2,...$ as a basis. To prove that an infinite dimensional Banach space cannot have a countable basis $e_1,e_2,...$ write $M_n=span \{e_1,e_2,...e_n\}$ and note that $X= \cup_n M_n$. Each $M_n$ is a closed because it is finite dimensional. If $M_n$ has an interior point then it is easy to show that $M_n=X$, a contradiction. Now you have to conclude the proof using Baire Category Theorem.