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
10 events
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| Jun 17, 2012 at 15:18 | comment | added | Kevin Beanland | Valerio: Looks like my laziness got the better of me. I knew it was Lindenstrauss and was only 50/50 on Pelczynski. Thanks. | |
| Jun 17, 2012 at 5:07 | comment | added | Valerio Capraro | @Kevin: The theorem you cite about complemented subspace is not by Lindenstrauss and Pelczynski, but by Lindenstrauss and Tzafriri. Precisely in Lindenstrauss, J. and Tzafriri, L. On the complemented subspaces problem, Israel J. Math. 9 (2) (1971) 263-269. | |
| Jun 17, 2012 at 3:21 | comment | added | Qingping Zeng | Thank Beanland and Brooker for eliminating the confuse. | |
| Jun 17, 2012 at 0:15 | comment | added | Yemon Choi | @Kevin, as someone who still gets upvotes for an answer to one question which consisted solely of pointing to work of Knutson and Tao ... so it goes | |
| Jun 16, 2012 at 23:54 | history | edited | Kevin Beanland | CC BY-SA 3.0 |
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| Jun 16, 2012 at 23:31 | comment | added | Kevin Beanland | On another note. I'm a bit surprised at how popular this question (and answer) is given that the solution is simply to quote a major theorem in Banach spaces that (as it happens) was part of the work that won Gowers the Fields. I suppose Banach Theory needs better PR. | |
| Jun 16, 2012 at 23:28 | history | edited | Kevin Beanland | CC BY-SA 3.0 |
added 153 characters in body
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| Jun 16, 2012 at 23:16 | comment | added | Kevin Beanland | Thanks for clarifying. I should have assumed others would have corrected! | |
| Jun 16, 2012 at 23:12 | comment | added | Philip Brooker | @Kevin: The OP's terminology is confusing and, I think, differs from standard Banach space terminology; what he calls an embedding is what I think you or I would call a complemented embedding. In particular, I presume that, in the OP's terminology, a left inverse for $J$ would be a map $K: Y\longrightarrow X$ such that $KJ$ is the identity on $X$, hence $JK$ is an idempotent operator on $Y$ with range isomorphic to $X$. | |
| Jun 16, 2012 at 22:09 | history | answered | Kevin Beanland | CC BY-SA 3.0 |