Timeline for $\ell_\infty^*$ has Dunford--Pettis property
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Oct 23, 2012 at 16:26 | comment | added | user27480 | Yes, I'm following Harald's advice and trying both foruns. | |
| Oct 23, 2012 at 15:15 | comment | added | Martin Sleziak | The question was posted at MSE: math.stackexchange.com/q/219454/8297 | |
| Oct 23, 2012 at 15:07 | comment | added | user27480 | Suppose $T\colon \ell_\infty \to \ell_1$ is an isomorphism onto it's image. Then $T^{-1}\colon \mathrm{Im} \; T \to \ell_1$ is onto, bounded and linear. Therefore $\mathrm{Im} \; T$ contains a complemented copy of $\ell_1$. So I can see that $\ell_1$ is a complemented subspace of a (not complemented) subspace of $\ell_\infty$. I can't seem to close the gap. Is this the right approach ? | |
| Oct 23, 2012 at 14:53 | comment | added | user27480 | right. Here's my approach. I found this result that says that $X^*$ has DPP iff $X$ has $DPP$ and doesn't contais a copy of $\ell_1$. So I'm trying to show that $\ell_\infty$ does not contain a copy of $\ell_1$ . I found a result that says that if $T\colon X \to \ell_1$ is bounded linear and onto then $X$ contains a complemented copy of $\ell_1$. It looked like I could use this to prove the result by contradiction, showing that $\ell_\infty$ has a complemented copy of $\ell_1$, which is absurd, since $\ell_\infty$ is prime. But I got stuck, I'm afraid I got the wrong road. | |
| Oct 23, 2012 at 13:57 | comment | added | Bill Johnson |
Here is an addition to your homework assignment: Prove that $\ell_\infty$ has a subspace isometrically isomorphic to $\ell_1(2^{\aleph_0})$.
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| Oct 23, 2012 at 7:39 | comment | added | Harald Hanche-Olsen | Homework? Maybe you should take this question over to Math.SE instead. It is more likely to be positively received there. Also, when you do ask, say a bit about what approaches you have tried. | |
| Oct 23, 2012 at 7:36 | history | edited | Harald Hanche-Olsen | CC BY-SA 3.0 |
Minor fixes.
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| Oct 23, 2012 at 7:29 | history | asked | user27480 | CC BY-SA 3.0 |