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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
DOI: https://doi.org/10.1090/S0002-9939-00-05447-2
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.


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

  • [1] C. Bessaga and A. Pe\lczynski, On bases and unconditional convergence of series in Banach spaces, Studia Math. 17 (1958), 151-164. MR 22:5872
  • [2] B.J. Cole, T.W. Gamelin and W.B. Johnson, Analytic disks in fibers over the unit ball of a Banach space, Michigan Math. J. 39 (3) (1992), 551-569. MR 93i:46090
  • [3] J. Diestel, Sequences and Series in Banach Spaces, Graduate Texts in Mathematics, vol. 92, Springer-Verlag, New York-Berlin, 1984. MR 85i:46020
  • [4] P.N. Dowling, W.B. Johnson, C.J. Lennard and B. Turett, The optimality of James' distortion theorems, Proc. Amer. Math. Soc. 125 (1) (1997), 167-174. MR 97d:46010
  • [5] P.N. Dowling and C.J. Lennard, Every nonreflexive subspace of $L_{1}[0,1]$ fails the fixed point property, Proc. Amer. Math. Soc. 125 (2) (1997), 443-446. MR 97d:46034
  • [6] P.N. Dowling, C.J. Lennard and B. Turett, Reflexivity and the fixed point property for nonexpansive maps, J. Math. Anal. Appl. 200 (3) (1996), 653-662. MR 97c:47062
  • [7] P.N. Dowling, C.J. Lennard and B. Turett, Asymptotically isometric copies of $c_{0}$ in Banach spaces, J. Math. Anal. Appl. 219 (2) (1998), 377-391. MR 98m:46023
  • [8] P.N. Dowling, C.J. Lennard and B. Turett, Some fixed point results in $\ell ^{1}$ and $c_{0}$, Nonlinear Analysis (to appear).
  • [9] S. Heinrich and P. Mankiewicz, Applications of ultrapowers to the uniform and Lipschitz classification of Banach spaces, Studia Math. 73 (3) (1982), 225-251. MR 84h:46026
  • [10] W.B. Johnson and H.P. Rosenthal, On $\omega ^{*}$ basic sequences and their applications to the study of Banach spaces, Studia Math. 43 (1972), 77-92. MR 46:9696
  • [11] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I. Sequence Spaces, Ergebnisse der Mathematik und Ihrer Grenzgebiete, vol. 92, Springer-Verlag, Berlin-Heidelberg-New York, 1977. MR 58:17766
  • [12] J.R. Partington, Equivalent norms on spaces of bounded functions, Israel J. Math. 35 (3) (1980), 205-209. MR 81h:46013

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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: https://doi.org/10.1090/S0002-9939-00-05447-2
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

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