In a recent post about $\ell^\infty$, I’ve said that there’s a more direct way to show that a Schauder base doesn’t exist in $\ell^\infty$. This proof can be found on Math Stack Exchange. It uses three facts.
- $c_0$ is closed.
- A result deduced from Hahn–Banach Theorem about a non-zero linear functional defined on a closed subspace of a normed space.
- $x^*$ is continous.