# On the classes of hereditarily ${\ell}_{p}$ Banach spaces

Czechoslovak Mathematical Journal (2006)

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

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topAzimi, 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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- Dual action of asymptotically isometric copies of ${\ell}_{p}$ ($1\le p<\infty $) and ${c}_{0}$, Collect. Math. 48 (1997), 449–458. (1997) MR1602639
- Dual Banach spaces which contains an isometric copy of ${L}_{1}$, Bull. Polish Acad. Sci. 48 (2000), 1–12. (2000) MR1751149
- 10.1090/S0002-9939-97-03577-6, Proc. Amer. Math. Soc. 125 (1997), 443–446. (1997) MR1350940DOI10.1090/S0002-9939-97-03577-6
- Classical Banach Spaces I. Sequence Spaces, Springer Verlag, Berlin, 1977. (1977) MR0500056

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