Timeline for Ekeland's standardness-property inheritable?
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 21, 2022 at 5:10 | comment | added | TaQ | @JochenWengenroth I have added as an answer a short discussion. | |
| Sep 21, 2022 at 5:09 | answer | added | TaQ | timeline score: 1 | |
| Sep 16, 2022 at 14:15 | comment | added | Jochen Wengenroth | @TaQ Is this adversity of Ekeland's version discussed somewhere in more detail? | |
| Apr 6, 2021 at 10:20 | comment | added | TaQ | The question is a bit inadequate for the following reason: The conditions (3) and especially (5) in Ekeland's results seem to be in such a way restrictive (hence not at all "weak") for the maps to be considered that it is questionable whether these results of Ekeland posses any nontrivial applications. In particular, they do not generalize Nash−Moser. The joint paper with Séré from about 2015 (see arXiv) seems more promising but there the spaces are more restricted sinse they need to allow smoothing operators. | |
| Apr 5, 2021 at 8:35 | comment | added | Jan Bohr | Aha, thanks! Just for the record: A Fréchet spaces called countably normed if its topology can be generated by a sequence of norms and if additionally the canonical inclusion maps $i_k$ are injective (p.168 in Dubinsky's The structure of nuclear Fréchet-spaces). Then Vogt's paper, together with reference [3], states: If $E$ admits a continuous norm, then: BAP $\Leftrightarrow$ $E$ countably normed. So the point is: Ekeland's paper only considers countably normed Fréchet-spaces and Vogt's paper gives an example of a Fréchet-space which fails to satisfy this property. | |
| Apr 4, 2021 at 18:12 | comment | added | Jochen Wengenroth | You're right. The maps $i_k$ need not be injective as Vogt's example (and others) show. The relation to the bounded approximation property goes back to Pelczynski. This is probably explained in Vogt's article (or in the references cited by Vogt.) | |
| Apr 3, 2021 at 13:49 | comment | added | Jan Bohr | Thanks for pointing this out! Where exactly is the fallacy in Ekeland's paper: Are the maps $i_k$ between the local Banach spaces not necessarily injective? Also, can you please elaborate on the connection with the bounded approximation property? (Are you suggesting that if $F$ is the intersection of Banach spaces, then it satisfies the BAP and hence Vogt's paper gives a counterexample? I don't wee why this would be true though.) | |
| Apr 3, 2021 at 7:41 | comment | added | Jochen Wengenroth | By the way, the article of Ekeland you refer to has a very bad start: The very first claim that, if the topology of a Fréchet space is given by a sequence of norms (not only semi-norms) then it is an intersection of Banach spaces, is wrong. This is known since a long time and is quite relevant for a problem of Grothendieck about the bounded approximation property. Perhaps the simplest example is in an article of Dietmar Vogt from 2010 in the Proc. Amer. Math. Soc. Hopefully, this is the only mistake in Ekeland's article. | |
| Apr 2, 2021 at 11:53 | comment | added | Jan Bohr | @JochenWengenroth Well, that would at least justify any extra effort in proving that a certain (sub-)space is standard. Do you know of any reference where this property is discussed in more detail? I skimmed some papers that cite Ekeland, but none gave me any more insight in how people typically verify standardness (beyond the two remarks above). | |
| Apr 2, 2021 at 11:42 | comment | added | Jochen Wengenroth | This does, of course, not answer your question, but my feeling is that a positive answer would be much too good to be true. | |
| Apr 2, 2021 at 10:53 | history | asked | Jan Bohr | CC BY-SA 4.0 |