Asymptotic properties of Banach spaces under renormings
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- by E. Odell and Th. Schlumprecht PDF
- J. Amer. Math. Soc. 11 (1998), 175-188 Request permission
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, $c_{0}$).References
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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
- MR Author ID: 260001
- Email: schlump@math.tamu.edu
- 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
- © Copyright 1998 American Mathematical Society
- Journal: J. Amer. Math. Soc. 11 (1998), 175-188
- MSC (1991): Primary 46B03, 46B45
- DOI: https://doi.org/10.1090/S0894-0347-98-00251-3
- MathSciNet review: 1469118