Let $X$ be an infinite dimensional vector space with an algebraic basis $\{ e_1 , ... , e_n, ... \}$. I need to prove that $X$ cannot be a Banach space with respect to any norm.
By the definition of Banach space, I should show that any convergent sequence in $X$ converges in $X$ but I am not getting any starting approach.
Finite dimensional spaces are nowhere dense in infinite dimensional spaces.
X is a countable union of finite dimensional spaces, each of which are nowhere dense in $X$, hence $X$ is of the first category. Therefore, since complete spaces are of the second category, $X$ is not complete, and hence not Banach.