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Suppose $E$ is a Banach space with an unconditional shrinking basis. Must the dual basis of $E^*$ be shrinking as well?

edit: Of course not, $E^{**}$ might be non-separable.

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May be you should write a detailed explanation or delete this question –  userNaN Oct 3 '12 at 13:05
Take $c_0$. Then the usual basis has the desired properties, yet the dual basis of $\ell_1$ is not shrinking because of the obvious reason that $\ell_\infty$ cannot have a basis. Could you please delete my post? –  TrzyTrypy Oct 3 '12 at 14:14
Sorry, I have no such privileges. –  userNaN Oct 3 '12 at 14:16
@TrzyTrypy You can delete the question yourself and that would be the best method to do so. –  Jayesh Badwaik Oct 4 '12 at 16:20

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