A proportional DvoretzkyRogers factorization result
Author:
A. A. Giannopoulos
Journal:
Proc. Amer. Math. Soc. 124 (1996), 233241
MSC (1991):
Primary 46B07
MathSciNet review:
1301496
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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 SauerShelah type theorem for ellipsoids.
 [BS]
J.
Bourgain and S.
J. Szarek, The BanachMazur distance to the cube and the
DvoretzkyRogers factorization, Israel J. Math. 62
(1988), no. 2, 169–180. MR 947820
(89g:46026), http://dx.doi.org/10.1007/BF02787120
 [BT]
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
(89a:46035), http://dx.doi.org/10.1007/BF02772174
 [DR]
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
(11,525a)
 [G]
A. A. Giannopoulos, A note on the BanachMazur 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
(10,719b)
 [LT]
Joram
Lindenstrauss and Lior
Tzafriri, Classical Banach spaces. I, SpringerVerlag,
BerlinNew York, 1977. Sequence spaces; Ergebnisse der Mathematik und ihrer
Grenzgebiete, Vol. 92. MR 0500056
(58 #17766)
 [MSc]
Vitali
D. Milman and Gideon
Schechtman, Asymptotic theory of finitedimensional normed
spaces, Lecture Notes in Mathematics, vol. 1200, SpringerVerlag,
Berlin, 1986. With an appendix by M. Gromov. MR 856576
(87m:46038)
 [Pi]
Albrecht
Pietsch, Operator ideals, Mathematische Monographien
[Mathematical Monographs], vol. 16, VEB Deutscher Verlag der
Wissenschaften, Berlin, 1978. MR 519680
(81a:47002)
 [ST]
S.
J. Szarek and M.
Talagrand, An “isomorphic” version of the SauerShelah
lemma and the BanachMazur 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
(90h:46034), http://dx.doi.org/10.1007/BFb0090050
 [Sa]
N.
Sauer, On the density of families of sets, J. Combinatorial
Theory Ser. A 13 (1972), 145–147. MR 0307902
(46 #7017)
 [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
(46 #7018)
 [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
(91j:46023), http://dx.doi.org/10.2307/2374731
 [Sz.2]
S.
J. Szarek, On the geometry of the BanachMazur compactum,
Functional analysis (Austin, TX, 1987/1989) Lecture Notes in Math.,
vol. 1470, Springer, Berlin, 1991, pp. 48–59. MR 1126736
(93b:46019), http://dx.doi.org/10.1007/BFb0090211
 [TJ]
Nicole
TomczakJaegermann, BanachMazur distances and finitedimensional
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 (90k:46039)
 [BS]
 J. Bourgain and S. J. Szarek, The BanachMazur distance to the cube and the DvoretzkyRogers factorization, Israel J. Math. 62 1988, 169180. MR 89g:46026
 [BT]
 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), 137224. MR 89a:46035
 [DR]
 A. Dvoretzky and C. A. Rogers, Absolute and unconditional convergence in normed linear spaces, Proc. Nat. Acad. Sci. U.S.A. 36 (1950), 192197. MR 11:525a
 [G]
 A. A. Giannopoulos, A note on the BanachMazur distance to the cube, GAFA Seminar (to appear).
 [J]
 F. John, Extremum problems with inequalities as subsidiary conditions, Courant Anniversary Volume, Interscience, New York, 1948. MR 10:719b
 [LT]
 J. Lindenstrauss and L. Tzafriri, Classical Banach spaces I: Sequence spaces, SpringerVerlag, Berlin and New York, 1977. MR 58:17766
 [MSc]
 V. D. Milman and G. Schechtman, Asymptotic theory of finite dimensional normed spaces, Lecture Notes in Math., vol. 1200, SpringerVerlag, Berlin and New York, 1986. MR 87m:46038
 [Pi]
 A. Pietsch, Operator ideals, NorthHolland, Amsterdam, 1978. MR 81a:47002
 [ST]
 S. J. Szarek and M. Talagrand, An ``isomorphic'' version of the SauerShelah lemma and the BanachMazur distance to the cube, GAFA Seminar 8788, Lecture Notes in Math., vol. 1376, SpringerVerlag, Berlin and New York, 1989, 105112. MR 90h:46034
 [Sa]
 N. Sauer, On the density of families of sets, J. Combin. Theory Ser. A 13 (1972), 145147. MR 46:7017
 [Sh]
 S. Shelah, A combinatorial problem: stability and order for models and theories in infinitary languages, Pacific J. Math. 41 (1972), 247261. MR 46:7018
 [Sz.1]
 S. J. Szarek, Spaces with large distance to and random matrices, Amer. J. Math. 112 (1990), 899942. MR 91j:46023
 [Sz.2]
 , On the geometry of the BanachMazur compactum, Lecture Notes in Math., vol. 1470, SpringerVerlag, Berlin and New York, 1991, pp. 4859. MR 93b:46019
 [TJ]
 N. TomczakJaegermann, BanachMazur distances and finite dimensional operator ideals, Longman Sci. Tech., Harlow, 1988. MR 90k:46039
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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/S0002993996030717
PII:
S 00029939(96)030717
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
