| Title:
|
On the classes of hereditarily $\ell_p$ Banach spaces (English) |
| Author:
|
Azimi, P. |
| Author:
|
Ledari, A. A. |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
56 |
| Issue:
|
3 |
| Year:
|
2006 |
| Pages:
|
1001-1009 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $X$ denote a specific space of the class of $X_{\alpha ,p}$ Banach sequence spaces which were constructed by Hagler and the first named author as classes of hereditarily $\ell _p$ Banach spaces. We show that for $p>1$ the Banach space $X$ contains asymptotically isometric copies of $\ell _{p}$. It is known that any member of the class is a dual space. We show that the predual of $X$ contains isometric copies of $\ell _q$ where $\frac{1}{p}+\frac{1}{q}=1$. For $p=1$ it is known that the predual of the Banach space $X$ contains asymptotically isometric copies of $c_0$. Here we give a direct proof of the known result that $X$ contains asymptotically isometric copies of $\ell _1$. (English) |
| Keyword:
|
Banach spaces |
| Keyword:
|
asymptotically isometric copy of $\ell _p$ |
| Keyword:
|
hereditarily $\ell _p$ Banach spaces |
| MSC:
|
46B04 |
| MSC:
|
46B20 |
| MSC:
|
46B25 |
| idZBL:
|
Zbl 1164.46304 |
| idMR:
|
MR2261672 |
| . |
| Date available:
|
2009-09-24T11:40:40Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/128125 |
| . |
| Reference:
|
[1] P. Azimi: A new class of Banach sequence spaces.Bull. Iranian Math. Soc. 28 (2002), 57–68. Zbl 1035.46006, MR 1992259 |
| Reference:
|
[2] P. Azimi, J. Hagler: Examples of hereditarily $\ell _{1}$ Banach spaces failing the Schur property.Pacific J. Math. 122 (1986), 287–297. MR 0831114, 10.2140/pjm.1986.122.287 |
| Reference:
|
[3] S. Chen, B.-L. Lin: Dual action of asymptotically isometric copies of $\ell _{p}$ ($1 \le p < \infty $) and $c_{0}$.Collect. Math. 48 (1997), 449–458. MR 1602639 |
| Reference:
|
[4] J. Dilworth, M. Girardi, and J. Hagler: Dual Banach spaces which contains an isometric copy of $L_{1}$.Bull. Polish Acad. Sci. 48 (2000), 1–12. MR 1751149 |
| Reference:
|
[5] P. N. Dowling, C. J. Lennard: Every nonreflexive subspace of $L_1$ fails the fixed point property.Proc. Amer. Math. Soc. 125 (1997), 443–446. MR 1350940, 10.1090/S0002-9939-97-03577-6 |
| Reference:
|
[6] J. Lindenstrauss, L. Tzafriri: Classical Banach Spaces I. Sequence Spaces.Springer Verlag, Berlin, 1977. MR 0500056 |
| . |
