On the classes of hereditarily ℓ p Banach spaces

Parviz Azimi; A. A. Ledari

Czechoslovak Mathematical Journal (2006)

  • Volume: 56, Issue: 3, page 1001-1009
  • ISSN: 0011-4642

Abstract

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Let X denote a specific space of the class of X α , p Banach sequence spaces which were constructed by Hagler and the first named author as classes of hereditarily ℓ p Banach spaces. We show that for p > 1 the Banach space X contains asymptotically isometric copies of ℓ 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 ℓ q where 1 p + 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 ℓ 1 .

How to cite

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Azimi, Parviz, and Ledari, A. A.. "On the classes of hereditarily $\ell _p$ Banach spaces." Czechoslovak Mathematical Journal 56.3 (2006): 1001-1009. <http://eudml.org/doc/31086>.

@article{Azimi2006,
abstract = {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 $\ell _1$.},
author = {Azimi, Parviz, Ledari, A. A.},
journal = {Czechoslovak Mathematical Journal},
keywords = {Banach spaces; asymptotically isometric copy of $\ell _p$; hereditarily $\ell _p$ Banach spaces; asymptotically isometric copy of },
language = {eng},
number = {3},
pages = {1001-1009},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the classes of hereditarily $\ell _p$ Banach spaces},
url = {http://eudml.org/doc/31086},
volume = {56},
year = {2006},
}

TY - JOUR
AU - Azimi, Parviz
AU - Ledari, A. A.
TI - On the classes of hereditarily $\ell _p$ Banach spaces
JO - Czechoslovak Mathematical Journal
PY - 2006
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 56
IS - 3
SP - 1001
EP - 1009
AB - 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 $\ell _1$.
LA - eng
KW - Banach spaces; asymptotically isometric copy of $\ell _p$; hereditarily $\ell _p$ Banach spaces; asymptotically isometric copy of 
UR - http://eudml.org/doc/31086
ER -

References

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  1. A new class of Banach sequence spaces, Bull. Iranian Math. Soc. 28 (2002), 57–68. (2002) Zbl1035.46006MR1992259
  2. 10.2140/pjm.1986.122.287, Pacific J.  Math. 122 (1986), 287–297. (1986) MR0831114DOI10.2140/pjm.1986.122.287
  3. Dual action of asymptotically isometric copies of  ℓ p ( 1 ≤ p < ∞ ) and c 0 , Collect. Math. 48 (1997), 449–458. (1997) MR1602639
  4. Dual Banach spaces which contains an isometric copy of  L 1 , Bull. Polish Acad. Sci. 48 (2000), 1–12. (2000) MR1751149
  5. 10.1090/S0002-9939-97-03577-6, Proc. Amer. Math. Soc. 125 (1997), 443–446. (1997) MR1350940DOI10.1090/S0002-9939-97-03577-6
  6. Classical Banach Spaces  I. Sequence Spaces, Springer Verlag, Berlin, 1977. (1977) MR0500056

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