## The $\ell ^{1}$-indices of Tsirelson type spaces

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- by Denny H. Leung and Wee-Kee Tang
- Proc. Amer. Math. Soc.
**131**(2003), 511-521 - DOI: https://doi.org/10.1090/S0002-9939-02-06586-3
- Published electronically: June 3, 2002
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## Abstract:

If $\alpha$ and $\beta$ are countable ordinals such that $\beta \neq 0$, denote by $tilde{T}_{\alpha ,\beta }$ the completion of $c_{00}$ with respect to the implicitly defined norm \[ \Vert x\Vert = \max \{\Vert x\Vert _{\mathcal {S}_{\alpha }},\frac {1}{2}\sup \sum _{i=1}^{j}\Vert E_{i}x\Vert \}, \] where the supremum is taken over all finite subsets $E_{1}$, โฆ, $E_{j}$ of $\mathbb {N}$ such that $E_{1}<\dots <E_{j}$ and $\{\min E_{1}, \dots , \min E_{j} \} \in \mathcal {S}$_{๐ฝ}$. It is shown that the Bourgain$โ^{1}$-index of$Tฬ_{๐ผ,๐ฝ}$is$๐^{๐ผ+๐ฝโ ๐}$. In particular, if$๐_{1}>๐ผ=๐^{๐ผ_{1}}โ m_{1}+โฆ+๐^{๐ผ_{n}}โ m_{n}$in Cantor normal form and$๐ผ_{n}$is not a limit ordinal, then there exists a Banach space whose$โ^{1}$-index is$๐^{๐ผ}$.$## References

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## Bibliographic Information

**Denny H. Leung**- Affiliation: Department of Mathematics, National University of Singapore, Singapore 117543
- MR Author ID: 113100
- Email: matlhh@nus.edu.sg
**Wee-Kee Tang**- Affiliation: Mathematics and Mathematics Education, National Institute of Education, Nanyang Technological University, 1 Nanyang Walk, Singapore 637616
- Email: wktang@nie.edu.sg
- Received by editor(s): March 7, 2001
- Received by editor(s) in revised form: July 10, 2001, and September 20, 2001
- Published electronically: June 3, 2002
- Communicated by: N. Tomczak-Jaegermann
- © Copyright 2002 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**131**(2003), 511-521 - MSC (2000): Primary 46B20; Secondary 05C05
- DOI: https://doi.org/10.1090/S0002-9939-02-06586-3
- MathSciNet review: 1933342