Timeline for Banach spaces locally having a basis
Current License: CC BY-SA 4.0
16 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 13 at 11:00 | vote | accept | M.González | ||
| Mar 11 at 5:14 | comment | added | Onur Oktay | Professor @M.González , I hope the answer below serves the purpose & is somewhat beneficial. | |
| Mar 11 at 5:05 | answer | added | Onur Oktay | timeline score: 4 | |
| Mar 10 at 17:03 | comment | added | M.González | It would be nice if you write an answer indicating how Szarek's results answer both questions. | |
| Mar 10 at 17:02 | comment | added | M.González | @Onur Oktay: I missunderstood your first comment. I thought that only the first question was answered in Szarek's paper. | |
| Mar 9 at 16:31 | comment | added | Bill Johnson | Szarek's negative answer to question 2 is a hard result, but it is not so difficult to give a negative answer to question 1 once you know that there is a subspace of $c_0$ that fails the approximation property: If $c_0$ is finitely representable in $X$, then $X$ has local basis structure. This is a consequence of the fact that every finite dimensional space is $2$-complemented in a finite dimensional space that has a basis with basis constant at most $2$. | |
| Mar 9 at 10:18 | comment | added | M.González | The key is the last phrase of the review. | |
| Mar 9 at 10:17 | comment | added | M.González | Some elementary properties of these concepts are established, and the behaviour of B_λ-condition under duality is studied. It does not seem to be known whether every Banach space has the λ-f.d.s.b. for some λ≥1. Reviewed by N. J. Kalton | |
| Mar 9 at 10:17 | comment | added | M.González | The author defines a notion of local basic structure for a Banach space, by analogy with the concept of local unconditional structure. A Banach space X has the "λ-finite dimensional subspaces basis property'' (λ-f.d.s.b.) if every finite-dimensional subspace can be embedded in a finite-dimensional subspace with a basis of basis constant at most λ; X is a B_λ-space if, in addition, this subspace can be chosen to be complemented with projection constant at most λ. | |
| Mar 9 at 10:16 | comment | added | M.González | MR0399819 (53 #3661) Reviewed Pujara, Lakhpat R. Some local structures in Banach spaces. Math. Japon. 20 (1975), no. 1, 49–54. 46B15 | |
| Mar 9 at 1:28 | comment | added | Onur Oktay | Szarek's result aside, perhaps Enflo's space is another example of a separable space with local basis structure (by Proposition 1.3 in doi.org/10.1007/BF02392555) & without basis? | |
| Mar 9 at 1:20 | comment | added | Onur Oktay | I'm afraid I don't have an access to MathSciNet, and I don't have a copy of Kalton's review in another form. Presently I can record here the exact reference to Szarek's article doi.org/10.1007/BF02392555 Corollary 1.6 within nicely sums up the answers to both questions -- for future visitors to this MO post. | |
| Mar 7 at 9:13 | comment | added | M.González | I have looked at Szarek's paper. I think you should write an answer, trying to say something about the second question using Kalton's review in MathScinet to one of Pujara's papers. | |
| Mar 7 at 9:12 | comment | added | M.González | Thank you for your comment. | |
| Mar 6 at 23:50 | comment | added | Onur Oktay | Professor Gonzalez, as far as I can remember, both questions have a negative answer in Szarek's paper where he provided a space with BAP & without a basis. | |
| Mar 6 at 11:08 | history | asked | M.González | CC BY-SA 4.0 |