Timeline for A separable Banach space and a non-separable Banach space having the same dual space?
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 6, 2011 at 19:46 | comment | added | Valerio Capraro | don't worry, it's always important to have the opportunity to learn something. Thank you very much. | |
| Oct 6, 2011 at 19:39 | comment | added | Philip Brooker | Sorry Valerio, I now see that it was someone else (Andres Caicedo) who asked about a reference for dual spaces. | |
| Oct 6, 2011 at 19:35 | comment | added | Philip Brooker | That $c_0$ is not complemented in $\ell_\infty$ is shown in the Albiac-Kalton book, but they also give a proof that $c_0$ is not linearly homeomorphic to a subspace of a separable dual space (so therefore isn't a dual itself). They also give a proof that $L_1[0,1]$ does not embed in a separable dual, so $L_1[0,1]$ is not a dual either. Despite this, $L_1[0,1]$ is in fact complemented in its bidual. I strongly recommend the Albiac-Kalton book. | |
| Oct 6, 2011 at 19:32 | comment | added | Philip Brooker | Valerio, the book Topics in Banach Space Theory by Albiac and Kalton contains lots of the information that you are after, including the definition of the James Tree space. You mentioned in a comment to Bill Johnson's answer that you want to know which spaces are duals; obviously reflexive spaces are, and it is classical (I think due to Civin and Yood?) that quasi-reflexive spaces are duals of quasi-reflexive spaces. Note that a dual space is complemented in its bidual, so for example you can show that $c_0$ is not a dual space by showing that it is not complemented in $\ell_\inty = c_0^{**}$ | |
| Oct 6, 2011 at 19:11 | comment | added | Valerio Capraro | Thannk you! Can you please tell me what is the James tree space? | |
| Oct 6, 2011 at 18:59 | history | answered | Philip Brooker | CC BY-SA 3.0 |