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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Asymptotically isometric copies of $\ell ^{\infty }$ in Banach spaces and a theorem of Bessaga and Pe\lczynski


Authors: Patrick N. Dowling and Narcisse Randrianantoanina
Journal: Proc. Amer. Math. Soc. 128 (2000), 3391-3397
MSC (2000): Primary 46B20, 46B25
Published electronically: May 18, 2000
MathSciNet review: 1690984
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Abstract:

We introduce the notion of a Banach space containing an asymptotically isometric copy of $\ell^\infty$. A well known result of Bessaga and Pe\lczynski states a Banach space $X$ contains a complemented isomorphic copy of $\ell^1$ if and only if $X^*$ contains an isomorphic copy of $c_0$ if and only if $X^*$ contains an isomorphic copy of $\ell^\infty$. We prove an asymptotically isometric analogue of this result.


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Additional Information

Patrick N. Dowling
Affiliation: Department of Mathematics and Statistics, Miami University, Oxford, Ohio 45056
Email: pndowling@miavx1.muohio.edu

Narcisse Randrianantoanina
Affiliation: Department of Mathematics and Statistics, Miami University, Oxford, Ohio 45056
Email: randrin@muohio.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-00-05447-2
PII: S 0002-9939(00)05447-2
Received by editor(s): June 26, 1998
Received by editor(s) in revised form: January 22, 1999
Published electronically: May 18, 2000
Additional Notes: The second author was supported in part by a Miami University Summer Research Appointment and by NSF grant DMS-9703789.
Communicated by: Dale Alspach
Article copyright: © Copyright 2000 American Mathematical Society