1
$\begingroup$

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?

$\endgroup$
2
  • $\begingroup$ Something is missing, for if $t_n=||x_n||=1$ the series can't converge... $\endgroup$ Oct 14, 2011 at 20:54
  • $\begingroup$ Right, the crucial assumption was missing :) Thanks. $\endgroup$ Oct 14, 2011 at 20:55

1 Answer 1

2
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.