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Random Talk on Random Thoughts

No Schauder Base in $\ell^\infty$

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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.

  1. $c_0$ is closed.
  2. A result deduced from Hahn–Banach Theorem about a non-zero linear functional defined on a closed subspace of a normed space.
  3. $x^*$ is continous.