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The optimality of James's distortion theorems

Authors: P. N. Dowling, W. B. Johnson, C. J. Lennard and B. Turett
Journal: Proc. Amer. Math. Soc. 125 (1997), 167-174
MSC (1991): Primary 46B03, 46B20
MathSciNet review: 1346969
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Abstract: A renorming of $\ell _{1}$, explored here in detail, shows that the copies of $\ell _{1}$ produced in the proof of the Kadec-Pelczynski theorem inside nonreflexive subspaces of $L_{1}[0,1]$ cannot be produced inside general nonreflexive spaces that contain copies of $\ell _{1}$. Put differently, James's distortion theorem producing one-plus-epsilon-isomorphic copies of $\ell _{1}$ inside any isomorphic copy of $\ell _{1}$ is, in a certain sense, optimal. A similar renorming of $c_{0}$ shows that James's distortion theorem for $c_{0}$ is likewise optimal.

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Additional Information

P. N. Dowling
Affiliation: Department of Mathematics and Statistics, Miami University, Oxford, Ohio 45056

W. B. Johnson
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843

C. J. Lennard
Affiliation: Department of Mathematics and Statistics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260

B. Turett
Affiliation: Department of Mathematical Sciences, Oakland University, Rochester, Michigan 48309

Keywords: $\ell _{1}$, $c_{0}$, renorming, James's distortion theorem, asymptotically isometric copies of $\ell _{1}$, fixed point property
Received by editor(s): May 8, 1995
Received by editor(s) in revised form: July 7, 1995
Additional Notes: The second author was supported by NSF 93-06376.
The third author was partially supported by a University of Pittsburgh FAS grant.
Communicated by: Dale Alspach
Article copyright: © Copyright 1997 American Mathematical Society