Timeline for $\ell_1$ and $\ell_\infty$ as complementary subspaces of Banach space
Current License: CC BY-SA 3.0
20 events
| when toggle format | what | by | license | comment | |
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| Jun 25, 2022 at 12:48 | comment | added | Ali Taghavi | I guess you mean "$X'$ is CLOSED subspace of $X$ | |
| Nov 7, 2017 at 20:22 | review | Close votes | |||
| Nov 8, 2017 at 9:23 | |||||
| Nov 7, 2017 at 20:11 | vote | accept | Evgeny | ||
| Nov 7, 2017 at 20:05 | comment | added | Pietro Majer | the proof is indeed short, but needs some facts. I post a sketch below (although the question could be closed as it is not of research level) | |
| Nov 7, 2017 at 20:03 | answer | added | Pietro Majer | timeline score: 3 | |
| Nov 7, 2017 at 19:35 | history | edited | Evgeny | CC BY-SA 3.0 |
more details
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| Nov 7, 2017 at 19:25 | history | edited | Evgeny | CC BY-SA 3.0 |
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| Nov 7, 2017 at 19:21 | comment | added | Evgeny | @PietroMajer how to prove it with not a big cost of words. I saw a proof for the first prop, but it is not trivial, well | |
| Nov 7, 2017 at 19:13 | comment | added | Pietro Majer | btw, what is the question? | |
| Nov 7, 2017 at 18:01 | comment | added | Evgeny | @LSpice sure. I found a book with answers to my questions, after reading I will post the answer, if nobody else will do it. | |
| Nov 7, 2017 at 17:26 | comment | added | LSpice | 'Complementary' means 'complemented', right? | |
| Nov 7, 2017 at 16:12 | comment | added | Bill Johnson | I give both as exercises when I teach the second semester of real analysis. | |
| Nov 7, 2017 at 15:34 | comment | added | Mateusz Wasilewski | The first statement is true, due to the lifting property of $\ell_1$, but it does not hold for other $\ell_p$. | |
| Nov 7, 2017 at 15:02 | history | edited | Evgeny | CC BY-SA 3.0 |
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| Nov 7, 2017 at 14:16 | comment | added | Evgeny | @FedorPetrov well thanks, I was in doubts that it should be necessary the direct summand, and it is not in general, then at least this is not trivial :) | |
| Nov 7, 2017 at 14:15 | comment | added | Fedor Petrov | But it does not prove that $X'$ is complemented! The factor space always exists, but it does not provide a complement. | |
| Nov 7, 2017 at 14:08 | comment | added | Evgeny | @FedorPetrov I meant that $X'$ complement should be $X/X'$, but $X/X'\cong \ell_1$, that is | |
| Nov 7, 2017 at 13:01 | comment | added | Fedor Petrov | How $\ell^1$ is a complement of $X'$? You do not have a priori $\ell^1$ in $X$. | |
| Nov 7, 2017 at 12:49 | history | edited | Martin Sleziak | CC BY-SA 3.0 |
minor typo
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| Nov 7, 2017 at 12:46 | history | asked | Evgeny | CC BY-SA 3.0 |