Timeline for Finding closed subspaces whose sum isn't closed
Current License: CC BY-SA 4.0
13 events
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| Jan 8, 2023 at 10:23 | comment | added | S Argyros | I also add a Remark in my last answer concerning your Theorem. | |
| Jan 8, 2023 at 9:42 | comment | added | S Argyros | Yes I mean" strongly suggests" | |
| Jan 8, 2023 at 0:44 | comment | added | Bill Johnson | Right, if by "indicates" you mean "suggests". In the example I was looking at $X/Y=c_0(\Gamma)$, which of course has a separable quotient. | |
| Jan 7, 2023 at 21:51 | comment | added | S Argyros | Bill, your theorem is strong and indicates that the problem that every separable subspace of a Banach space is quasi complemented has a positive solution. In the example you mention $c_0$ is indeed quasi complemented. | |
| Jan 7, 2023 at 16:40 | comment | added | Bill Johnson | (Continued) Thinking along these lines reminded me of a Pacific J. Math. paper I wrote in 1972 titled "Quasi-complements", where I improved some results of James and do some other things. In Theorem 2, I showed that if $Y^*$ is weak$^*$ separable and $X/Y$ has a separable quotient, then $Y$ is quasi-complemented in $X$. I mentioned in the paper that the result was due to Lindenstrauss and Rosenthal [unpublished]. (After I told Haskell I had proved this, he informed me that both he and Joram had observed the same thing. I guess neither thought it worth publishing.) | |
| Jan 7, 2023 at 16:18 | comment | added | Bill Johnson | Spyros, I thought there might be example by taking $X$ to be the closure in $\ell_\infty$ of $c_0$ and the characteristic functions of an uncountable almost disjoint family of infinite subsets of the natural numbers (which can be given by branches of a tree, as in the examples you mentioned). Here also $Y=c_0$, but $Y^\perp$ is $\ell_1$ saturated. Nevertheless, Rosenthal's construction yields that there is a non separable closed subspace of $X$ that intersects $Y$ trivially (and maybe is even a quasi-complement of $Y$, but I did not verify that). (TBC) | |
| Jan 6, 2023 at 11:45 | comment | added | S Argyros | Bill, thank you very much for pointing out to me the $ 𝑙_1$ case. I believe that it is correct but it needs much more explanation. The 𝑤∗ closure of $𝑙_1 $ is one point but also how we adapt Haskell' s proof is another one. I will try to write an answer discussing this matter | |
| Jan 6, 2023 at 11:38 | history | edited | S Argyros | CC BY-SA 4.0 |
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| Jan 5, 2023 at 23:04 | comment | added | Bill Johnson | Spyros, by "I think that the result remains valid if $Y_\perp$ contains $\ell_1$" do you mean "$Y_\perp$ contains a weak$^*$ closed subspace that is isomorphic to $\ell_1$?" | |
| Jan 5, 2023 at 13:04 | history | edited | S Argyros | CC BY-SA 4.0 |
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| Jan 5, 2023 at 12:46 | history | edited | S Argyros | CC BY-SA 4.0 |
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| Jan 5, 2023 at 11:24 | history | edited | S Argyros | CC BY-SA 4.0 |
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| Jan 5, 2023 at 10:50 | history | answered | S Argyros | CC BY-SA 4.0 |