Geodesic
On the classes of hereditarily $\ell_p$ Banach spaces
Czechoslovak Mathematical Journal, Tome 56 (2006) no. 3, pp. 1001-1009
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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 $\ell _1$.
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 $\ell _1$.
Classification : 46B04, 46B20, 46B25
Keywords: Banach spaces; asymptotically isometric copy of $\ell _p$; hereditarily $\ell _p$ Banach spaces 
@article{CMJ_2006_56_3_a18,
     author = {Azimi, P. and Ledari, A. A.},
     title = {On the classes of hereditarily $\ell_p$ {Banach} spaces},
     journal = {Czechoslovak Mathematical Journal},
     pages = {1001--1009},
     year = {2006},
     volume = {56},
     number = {3},
     mrnumber = {2261672},
     zbl = {1164.46304},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2006_56_3_a18/}
}
TY  - JOUR
AU  - Azimi, P.
AU  - Ledari, A. A.
TI  - On the classes of hereditarily $\ell_p$ Banach spaces
JO  - Czechoslovak Mathematical Journal
PY  - 2006
SP  - 1001
EP  - 1009
VL  - 56
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/CMJ_2006_56_3_a18/
LA  - en
ID  - CMJ_2006_56_3_a18
ER  - 
%0 Journal Article
%A Azimi, P.
%A Ledari, A. A.
%T On the classes of hereditarily $\ell_p$ Banach spaces
%J Czechoslovak Mathematical Journal
%D 2006
%P 1001-1009
%V 56
%N 3
%U http://geodesic.mathdoc.fr/item/CMJ_2006_56_3_a18/
%G en
%F CMJ_2006_56_3_a18
Azimi, P.; Ledari, A. A. On the classes of hereditarily $\ell_p$ Banach spaces. Czechoslovak Mathematical Journal, Tome 56 (2006) no. 3, pp. 1001-1009. http://geodesic.mathdoc.fr/item/CMJ_2006_56_3_a18/

[1] P.  Azimi: A new class of Banach sequence spaces. Bull. Iranian Math. Soc. 28 (2002), 57–68. | MR | Zbl

[2] P.  Azimi, J.  Hagler: Examples of hereditarily $\ell _{1}$ Banach spaces failing the Schur property. Pacific J.  Math. 122 (1986), 287–297. | DOI | MR

[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

[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

[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. | DOI | MR

[6] J. Lindenstrauss, L. Tzafriri: Classical Banach Spaces  I. Sequence Spaces. Springer Verlag, Berlin, 1977. | MR