On the classes of hereditarily \(\ell _p\) Banach spaces. (English) Zbl 1164.46304

Summary: Let \(X\)  denote a specific space of the class of  \(X_{\alpha ,p}\) Banach sequence spaces which were constructed by J. 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 \(1/p+1/q=1\). 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  \(\ell _1\).


46B04 Isometric theory of Banach spaces
46B20 Geometry and structure of normed linear spaces
46B25 Classical Banach spaces in the general theory
Full Text: DOI EuDML


[1] P. Azimi: A new class of Banach sequence spaces. Bull. Iranian Math. Soc. 28 (2002), 57–68. · Zbl 1035.46006
[2] P. Azimi, J. Hagler: Examples of hereditarily 1 Banach spaces failing the Schur property. Pacific J. Math. 122 (1986), 287–297. · Zbl 0609.46012
[3] S. Chen, B.-L. Lin: Dual action of asymptotically isometric copies of p (1 p < and c 0. Collect. Math. 48 (1997), 449–458.
[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. · Zbl 0956.46006
[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. · Zbl 0861.47032
[6] J. Lindenstrauss, L. Tzafriri: Classical Banach Spaces I. Sequence Spaces. Springer Verlag, Berlin, 1977. · Zbl 0362.46013
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.