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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  \(\ell _1\).

MSC:

46B04 Isometric theory of Banach spaces
46B20 Geometry and structure of normed linear spaces
46B25 Classical Banach spaces in the general theory
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References:

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