Timeline for Is $(\ell^1(\mathbb N_0),\sigma(\ell^1,\ell^\infty))$ not quasi-complete?
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| Jun 26, 2017 at 17:46 | history | rollback | Todd Trimble |
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| Jun 25, 2017 at 15:58 | comment | added | traun | We seem to be talking at cross purposes so let me try again (I do want to help). It seems to me that the difficulty lies in the fact that you are talking about quasi-completeness, me about compactness of unit balls. But in the context of your question, these coincide. This is because the unit ball of a Banach space (resp. of a dual Banach space) is precompact for the weak (resp. for the weak star uniformity) so that they are compact for these topologies if and only they are complete for the associated uniformities. Please accept this in the spirit it is intended and not as "teaching". | |
| Jun 25, 2017 at 3:44 | history | edited | traun | CC BY-SA 3.0 |
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| Jun 25, 2017 at 3:37 | comment | added | traun | This is the third edition (3.a edição), São Paulo, 1964. | |
| Jun 25, 2017 at 3:25 | comment | added | traun | It is indeed László. This article probably hasn't been published---it is more a memo. But you should be able to find it simply by googling author plus the title I gave. Anyway it worked for me. | |
| Jun 24, 2017 at 19:49 | comment | added | TaQ | Who is "L. Erdös"? MathSciNet knows László Erdős but I didn't find there any publication titled Banach-Alaoglu theorems. | |
| Jun 24, 2017 at 19:49 | comment | added | TaQ | I just need a precise and simple answer to my question(s) above, most importantly, a precise reference to a proof of Jarchow's assertion "not being quasi-complete" for which he didn't give any. I looked up Grothendieck in MathSciNet, and there is Kelley's review of his text Espaces vectoriels topologiques from 1954 containing 240 pages. So where is the page numbered 277? | |
| Jun 24, 2017 at 8:53 | comment | added | traun | found an article "Banach-Alaoglu theorems" by L. Erdös easily available online which would seem to have everything your heart might desire. | |
| Jun 24, 2017 at 6:04 | comment | added | traun | Have managed to dig up Grotndieck's text "Espaces vectoriels topologiques". The result mentioned is on p. 277. By the way, if $E$ is a Banach space, then it bidual is the linear span of the bipolar of its unit ball in the algebraic dual of its dual. The latter is, by the bipolar theorem, its closure in the weak topology induced by $E'$. From this, everything follows. | |
| Jun 24, 2017 at 5:54 | comment | added | traun | I don't have access to the texts now so can't give you page and theorem numbers. The theorem of Alaoglu (sometimes Banach-Alaoglu) states that the unit ball of the dual of a Banach space is weak star compact. In the case of reflexivity, this gives what you require. | |
| Jun 23, 2017 at 23:09 | comment | added | TaQ | Could you be more specific? You mention 3 texts but no theorem or page numbers. How are you supposing Alaoglu's theorem to be applied here? | |
| Jun 23, 2017 at 17:15 | history | edited | traun | CC BY-SA 3.0 |
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| Jun 23, 2017 at 17:07 | history | answered | traun | CC BY-SA 3.0 |