Let $X$ be a infinite dimensional Banach space. If $\mathcal{B}=\{e_i:i\in I\}$ is a Hamel base for $X$. How to show that only a finite number of the coordinate functionals $e^{*}_i$ will be continuous.
My approach: Suppose that there is a $I_0\subseteq I$ infinite such that, for each $i\in I_0$, $e^*_i$ is continuous. Then, there exists a countable infinite number of $i\in I_0$ such that $e^*_i$ is continuous. But having trouble to derive an absurd from this fact.
