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Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)

 

Asymptotic properties of Banach spaces
under renormings


Authors: E. Odell and Th. Schlumprecht
Journal: J. Amer. Math. Soc. 11 (1998), 175-188
MSC (1991): Primary 46B03, 46B45
MathSciNet review: 1469118
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Abstract: It is shown that a separable Banach space $X$ can be given an equivalent norm $|\!|\!|\, \cdot |\!|\!|\, $ with the following properties: If $(x_{n})\subseteq X$ is relatively weakly compact and $\lim _{m\to \infty } \lim _{n\to \infty } |\!|\!|\, x_{m}+x_{n}|\!|\!|\, = 2\lim _{m\to \infty } |\!|\!|\, x_{m}|\!|\!|\,$, then $(x_{n})$ converges in norm. This yields a characterization of reflexivity once proposed by V.D. Milman. In addition it is shown that some spreading model of a sequence in $(X,|\!|\!|\, \cdot |\!|\!|\, )$ is 1-equivalent to the unit vector basis of $\ell _{1}$ (respectively, $c_{0}$) implies that $X$ contains an isomorph of $\ell _{1}$ (respectively, $c_{0}$).


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Additional Information

E. Odell
Affiliation: Department of Mathematics, The University of Texas at Austin, Austin, Texas 78712-1082
Email: odell@math.utexas.edu

Th. Schlumprecht
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368
Email: schlump@math.tamu.edu

DOI: http://dx.doi.org/10.1090/S0894-0347-98-00251-3
PII: S 0894-0347(98)00251-3
Keywords: Spreading model, Ramsey theory, $\ell _{1}$, $c_{0}$, reflexive Banach space
Received by editor(s): May 12, 1997
Received by editor(s) in revised form: September 15, 1997
Additional Notes: Research of both authors was supported by NSF and TARP
Article copyright: © Copyright 1998 American Mathematical Society