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?



Basic sequences
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.


