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Nowadays, we know that there exist Banach spaces without unconditional basic sequences. Do we know if something a bit milder holds? Namely, is that true each non-reflexive Banach space contains a normalised basic sequence $(x_n)_{n=1}^\infty$ such that for each $(t_n)_{n=1}^\infty\in c_0$ the series

$$\sum_{n=1}^\infty t_n x_n$$

is convergent in this space? Has anyone studied this sort of basic sequences?

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Something is missing, for if $t_n=||x_n||=1$ the series can't converge... – Pietro Majer Oct 14 '11 at 20:54
Right, the crucial assumption was missing :) Thanks. – Sellapan Nathan Oct 14 '11 at 20:55
up vote 1 down vote accepted

The Banach spaces that admit such a sequence are the Banach spaces that contain a subspace isomorphic to $c_0$. Look at, e.g., the beginning part of the book of Albiac-Kalton.

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