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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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The $\ell ^{1}$-indices of Tsirelson type spaces
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by Denny H. Leung and Wee-Kee Tang PDF
Proc. Amer. Math. Soc. 131 (2003), 511-521 Request permission

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 $E_{1}$, โ€ฆ, $E_{j}$ of $\mathbb {N}$ such that $E_{1}<\dots <E_{j}$ and $\{\min E_{1}, \dots , \min E_{j} \} \in \mathcal {S}$_{๐›ฝ}$. It is shown that the Bourgain$โ„“^{1}$-index of$Tฬƒ_{๐›ผ,๐›ฝ}$is$๐œ”^{๐›ผ+๐›ฝโ‹…๐œ”}$. In particular, if$๐œ”_{1}>๐›ผ=๐œ”^{๐›ผ_{1}}โ‹…m_{1}+โ€ฆ+๐œ”^{๐›ผ_{n}}โ‹…m_{n}$in Cantor normal form and$๐›ผ_{n}$is not a limit ordinal, then there exists a Banach space whose$โ„“^{1}$-index is$๐œ”^{๐›ผ}$.$
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Additional 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