Timeline for Does $X$ embed in $Y$, and $Y$ embed in $X$, always imply that $X$ isomorphic onto $Y$?
Current License: CC BY-SA 3.0
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| Jun 18, 2012 at 12:48 | comment | added | Philip Brooker | for any $1\leq j \leq n-1$; if such $n$ and $X$ do exist, then $X$ and $X^{\ast\ast}$ are nonisomorphic Banach spaces that are isomorphic to complemented subspaces of one another. | |
| Jun 18, 2012 at 12:47 | comment | added | Philip Brooker | @Kevin: I first came across the question in a paper of Plichko and Wojtowicz, Note on a Banach space having equal linear dimension with its second dual, Extracta Mathematica 18(3) (2003), p.311--314 (in particular, see the final remark of the paper). I wrote to one of the authors and also to Galego a couple of years ago enquiring as to the status of the problem, and at that time it was still open as far as they knew. More generally, I think it is open whether there exists an integer $n\geq 3$ and a Banach space $X$ such that $X$ is isomorphic to its $n$th dual but not to its $j$th dual | |
| Jun 18, 2012 at 0:17 | comment | added | Kevin Beanland | @Philip: I did not know this was open and I'm surprised it is not obviously true. I suppose one might construct a counterexample using a non-reflexive (so-called Jamesification, coined by Spiros) version of Gowers' space. Also, it's at least possible that Spiros has already constructed a counterexample, perhaps without knowing, in one of his papers. Where did you see this question? | |
| Jun 17, 2012 at 3:11 | comment | added | Qingping Zeng | Thanks Douglas for you reminding me the Schroeder-Bernstein problem. | |
| Jun 17, 2012 at 1:46 | vote | accept | Qingping Zeng | ||
| Jun 17, 2012 at 1:46 | vote | accept | Qingping Zeng | ||
| Jun 17, 2012 at 1:46 | |||||
| Jun 16, 2012 at 13:31 | comment | added | Philip Brooker | There is a noteworthy special case of this problem that is still open: if $X$ and $X^{\ast \ast}$ are isomorphic to complemented subspaces of one another, are they in fact isomorphic? | |
| Jun 16, 2012 at 10:20 | history | answered | Douglas Zare | CC BY-SA 3.0 |