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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?

edited Oct 14 2011 at 20:55

asked Oct 14 2011 at 20:44

Sellapan Nathan
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	Something is missing, for if $t_n=\|\|x_n\|\|=1$ the series can't converge... – Pietro Majer Oct 14 2011 at 20:54
	Right, the crucial assumption was missing :) Thanks. – Sellapan Nathan Oct 14 2011 at 20:55

Bill Johnson · Accepted Answer · 2011-10-14 21:03:38Z UTC

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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.

answered Oct 14 2011 at 21:03

Bill Johnson
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Basic sequences

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