An extension of Elton's theorem to complex Banach spaces
Authors:
S. J. Dilworth and Joseph P. Patterson
Journal:
Proc. Amer. Math. Soc. 131 (2003), 14891500
MSC (2000):
Primary 46B07; Secondary 46B04, 46B09
Published electronically:
September 5, 2002
MathSciNet review:
1949879
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Let be sufficiently small. Then, for , there exists such that if are vectors in the unit ball of a complex Banach space which satisfy
(where are independent complex Steinhaus random variables), then there exists a set , with , such that for all (). The dependence on of the threshold proportion is sharp.
 1.
Noga
Alon, On the density of sets of vectors, Discrete Math.
46 (1983), no. 2, 199–202. MR 710891
(85b:05004), http://dx.doi.org/10.1016/0012365X(83)902534
 2.
Béla
Bollobás, Combinatorics, Cambridge University Press,
Cambridge, 1986. Set systems, hypergraphs, families of vectors and
combinatorial probability. MR 866142
(88g:05001)
 3.
William
J. Davis, D.
J. H. Garling, and Nicole
TomczakJaegermann, The complex convexity of quasinormed linear
spaces, J. Funct. Anal. 55 (1984), no. 1,
110–150. MR
733036 (86b:46032), http://dx.doi.org/10.1016/00221236(84)900211
 4.
S. J. Dilworth and Denka Kutzarova, On the optimality of a theorem of Elton on Subsystems, Israel J. Math. 124 (2001), 215220.
 5.
John
Elton, Signembeddings of
𝑙ⁿ₁, Trans. Amer. Math.
Soc. 279 (1983), no. 1, 113–124. MR 704605
(84g:46023), http://dx.doi.org/10.1090/S00029947198307046054
 6.
M.
G. Karpovsky and V.
D. Milman, Coordinate density of sets of vectors, Discrete
Math. 24 (1978), no. 2, 177–184. MR 522926
(80m:05004), http://dx.doi.org/10.1016/0012365X(78)901978
 7.
Alain
Pajor, Plongement de 𝑙ⁿ₁ dans les espaces de
Banach complexes, C. R. Acad. Sci. Paris Sér. I Math.
296 (1983), no. 17, 741–743 (French, with
English summary). MR 707332
(84g:46024)
 8.
Alain
Pajor, Sousespaces 𝑙ⁿ₁ des espaces de
Banach, Travaux en Cours [Works in Progress], vol. 16, Hermann,
Paris, 1985 (French). With an introduction by Gilles Pisier. MR 903247
(88h:46028)
 9.
N.
Sauer, On the density of families of sets, J. Combinatorial
Theory Ser. A 13 (1972), 145–147. MR 0307902
(46 #7017)
 10.
Saharon
Shelah, Stability, the f.c.p., and superstability; model theoretic
properties of formulas in first order theory, Ann. Math. Logic
3 (1971), no. 3, 271–362. MR 0317926
(47 #6475)
 11.
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)
 12.
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
 13.
Michel
Talagrand, Type, infratype and the EltonPajor theorem,
Invent. Math. 107 (1992), no. 1, 41–59. MR 1135463
(92m:46028), http://dx.doi.org/10.1007/BF01231880
 1.
 N. Alon, On the density of sets of vectors, Discrete Math. (1983), 199202. MR 85b:05004
 2.
 B. Bollobàs, Combinatorics: set systems, hypergraphs, families of vectors, and combinatorial probability, Cambridge University Press, Cambridge, 1986. MR 88g:05001
 3.
 W. J. Davis, D. J. H. Garling and N. TomczakJaegermann, The complex convexity of quasinormed linear spaces, J. Funct. Anal. 55 (1984), 110150. MR 86b:46032
 4.
 S. J. Dilworth and Denka Kutzarova, On the optimality of a theorem of Elton on Subsystems, Israel J. Math. 124 (2001), 215220.
 5.
 John Elton, Signembeddings of , Trans. Amer. Math. Soc. (1983), 113124. MR 84g:46023
 6.
 M. G. Karpovsky and V. D. Milman Coordinate density of sets of vectors, Discrete Math. (1978), 177184. MR 80m:05004
 7.
 A. Pajor, Plongement de dans les espaces de Banach complexes, C. R. Acad. Sci. Paris Sér. I Math (1983), 741743. MR 84g:46024
 8.
 A. Pajor, Sousespaces des espaces de Banach, Travaux en Cours, Herman, Paris, 1985. MR 88h:46028
 9.
 N. Sauer, On the density of families of sets, J. Combinatorial Theory Ser. A (1972), pp. 145147. MR 46:7017
 10.
 Saharon Shelah, Stability, the f.c.p., and the superstability; model theoretic properties of formulas in first order theory, Annals of Math. Logic, pp. 271362, 1971. MR 47:6475
 11.
 Saharon Shelah, A combinatorial problem; stability and order for models and theories in infinitary languages, Pacific J. Math. 41 (1972), 247261. MR 46:7018
 12.
 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 (19871988), 105112; Lecture Notes in Math., 1376, Springer, Berlin, 1989. MR 90h:46034
 13.
 M. Talagrand, Type, infratype and the EltonPajor theorem, Invent. Math. 107 (1992), 4159. MR 92m:46028
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (2000):
46B07,
46B04,
46B09
Retrieve articles in all journals
with MSC (2000):
46B07,
46B04,
46B09
Additional Information
S. J. Dilworth
Affiliation:
Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208
Address at time of publication:
Department of Mathematics, University of Texas at Austin, Austin, Texas 78712
Email:
dilworth@math.sc.edu
Joseph P. Patterson
Affiliation:
Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208
Address at time of publication:
2110 Arrowcreek Dr., Apt. 101, Charlotte, North Carolina 28273
Email:
joe_p_chess@yahoo.com
DOI:
http://dx.doi.org/10.1090/S0002993902066510
PII:
S 00029939(02)066510
Received by editor(s):
October 17, 2001
Received by editor(s) in revised form:
December 11, 2001
Published electronically:
September 5, 2002
Additional Notes:
The research of the first author was completed while on sabbatical as a Visiting Scholar at The University of Texas at Austin.
This paper is based on the second author’s thesis for his MS degree at the University of South Carolina.
Communicated by:
N. TomczakJaegermann
Article copyright:
© Copyright 2002
American Mathematical Society
