Remote Access Journal of the American Mathematical Society
Green Open Access

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (1991): 46B03, 46B45

Retrieve articles in all journals with MSC (1991): 46B03, 46B45

Additional Information

E. Odell
Affiliation: Department of Mathematics, The University of Texas at Austin, Austin, Texas 78712-1082

Th. Schlumprecht
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368
MR Author ID: 260001

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