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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