Asymptotically isometric copies of in Banach spaces and a theorem of Bessaga and Peczynski

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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

We introduce the notion of a Banach space containing an asymptotically isometric copy of . A well known result of Bessaga and Peczynski states a Banach space contains a complemented isomorphic copy of if and only if contains an isomorphic copy of if and only if contains an isomorphic copy of . We prove an asymptotically isometric analogue of this result.

**[1]**C. Bessaga and A. Peczynski,*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**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**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**and*, 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**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**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
46B20,
46B25

Retrieve articles in all journals with MSC (2000): 46B20, 46B25

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