A proportional Dvoretzky-Rogers

factorization result

Author:
A. A. Giannopoulos

Journal:
Proc. Amer. Math. Soc. **124** (1996), 233-241

MSC (1991):
Primary 46B07

MathSciNet review:
1301496

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: If is an -dimensional normed space and , there exists , such that the formal identity can be written as , with . This is proved as a consequence of a Sauer-Shelah type theorem for ellipsoids.

**[B-S]**J. Bourgain and S. J. Szarek,*The Banach-Mazur distance to the cube and the Dvoretzky-Rogers factorization*, Israel J. Math.**62**(1988), no. 2, 169–180. MR**947820**, 10.1007/BF02787120**[B-T]**J. Bourgain and L. Tzafriri,*Invertibility of “large” submatrices with applications to the geometry of Banach spaces and harmonic analysis*, Israel J. Math.**57**(1987), no. 2, 137–224. MR**890420**, 10.1007/BF02772174**[D-R]**A. Dvoretzky and C. A. Rogers,*Absolute and unconditional convergence in normed linear spaces*, Proc. Nat. Acad. Sci. U. S. A.**36**(1950), 192–197. MR**0033975****[G]**A. A. Giannopoulos,*A note on the Banach-Mazur distance to the cube*, GAFA Seminar (to appear).**[J]**Fritz John,*Extremum problems with inequalities as subsidiary conditions*, Studies and Essays Presented to R. Courant on his 60th Birthday, January 8, 1948, Interscience Publishers, Inc., New York, N. Y., 1948, pp. 187–204. MR**0030135****[L-T]**Joram Lindenstrauss and Lior Tzafriri,*Classical Banach spaces. I*, Springer-Verlag, Berlin-New York, 1977. Sequence spaces; Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 92. MR**0500056****[M-Sc]**Vitali D. Milman and Gideon Schechtman,*Asymptotic theory of finite-dimensional normed spaces*, Lecture Notes in Mathematics, vol. 1200, Springer-Verlag, Berlin, 1986. With an appendix by M. Gromov. MR**856576****[Pi]**Albrecht Pietsch,*Operator ideals*, Mathematische Monographien [Mathematical Monographs], vol. 16, VEB Deutscher Verlag der Wissenschaften, Berlin, 1978. MR**519680****[S-T]**S. J. Szarek and M. Talagrand,*An “isomorphic” version of the Sauer-Shelah lemma and the Banach-Mazur distance to the cube*, Geometric aspects of functional analysis (1987–88), Lecture Notes in Math., vol. 1376, Springer, Berlin, 1989, pp. 105–112. MR**1008718**, 10.1007/BFb0090050**[Sa]**N. Sauer,*On the density of families of sets*, J. Combinatorial Theory Ser. A**13**(1972), 145–147. MR**0307902****[Sh]**Saharon Shelah,*A combinatorial problem; stability and order for models and theories in infinitary languages*, Pacific J. Math.**41**(1972), 247–261. MR**0307903****[Sz.1]**Stanisław J. Szarek,*Spaces with large distance to 𝑙ⁿ_{∞} and random matrices*, Amer. J. Math.**112**(1990), no. 6, 899–942. MR**1081810**, 10.2307/2374731**[Sz.2]**S. J. Szarek,*On the geometry of the Banach-Mazur compactum*, Functional analysis (Austin, TX, 1987/1989) Lecture Notes in Math., vol. 1470, Springer, Berlin, 1991, pp. 48–59. MR**1126736**, 10.1007/BFb0090211**[T-J]**Nicole Tomczak-Jaegermann,*Banach-Mazur distances and finite-dimensional operator ideals*, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 38, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1989. MR**993774**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
46B07

Retrieve articles in all journals with MSC (1991): 46B07

Additional Information

**A. A. Giannopoulos**

Affiliation:
Department of Mathematics, Case Western Reserve University, Cleveland, Ohio 44106

Address at time of publication:
Department of Mathematics, University of Crete, Iraklion, Crete, Greece

Email:
deligia@talos.cc.uch.gr

DOI:
http://dx.doi.org/10.1090/S0002-9939-96-03071-7

Received by editor(s):
February 21, 1994

Received by editor(s) in revised form:
August 15, 1994

Communicated by:
Dale Alspach

Article copyright:
© Copyright 1996
American Mathematical Society