## Taiwanese Journal of Mathematics

### ON GEOMETRIC AND TOPOLOGICAL PROPERTIES OF THE CLASSES OF HEREDITARILY $\ell_{p}$ BANACH SPACES

Parviz Azimi

#### Abstract

A class of hereditarily $\ell_{p}$ ($1 \leq p \lt \infty$) Banach sequence spaces is constructed and denoted by $X_{\alpha,p}$. Any constructed space is a dual space. We show that (i) the predual of any member $X$ of the class of $X_{\alpha,1}$ contains asymptotically isometric copies of $c_0$.(ii) Every infinite dimensional subspace of $X$ contains asymptotically isometric complemented copies of $\ell_{1}$, and consequently, the dual X$^*$ of X contains subspaces isometrically isomorphic to $C[0,1]^*$. (iii) Every member of the class of $X_{\alpha,p}$ ($1 \leq p \lt \infty$) fails the Dunford-Pettis property. (iv) We observe that all $X_{\alpha,p}$ spaces are Banach spaces without unconditional basis but all constructed spaces contain a subspace which is weakly sequentially complete with an unconditional basis which is weakly null sequence but not in norm. (v) All spaces have asymptotic-norming and Kadec-Klee property. The predual of any $X_{\alpha,p}$ is an Asplund space.

#### Article information

Source
Taiwanese J. Math., Volume 10, Number 3 (2006), 713-722.

Dates
First available in Project Euclid: 18 July 2017

https://projecteuclid.org/euclid.twjm/1500403857

Digital Object Identifier
doi:10.11650/twjm/1500403857

Mathematical Reviews number (MathSciNet)
MR2206324

Zentralblatt MATH identifier
1108.46009

Subjects
Primary: 46B04: Isometric theory of Banach spaces
Secondary: 46B20: Geometry and structure of normed linear spaces

#### Citation

Azimi, Parviz. ON GEOMETRIC AND TOPOLOGICAL PROPERTIES OF THE CLASSES OF HEREDITARILY $\ell_{p}$ BANACH SPACES. Taiwanese J. Math. 10 (2006), no. 3, 713--722. doi:10.11650/twjm/1500403857. https://projecteuclid.org/euclid.twjm/1500403857

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