A proportional Dvoretzky-Rogers factorization result

Giannopoulos, A.

doi:10.1090/S0002-9939-96-03071-7

A proportional Dvoretzky-Rogers factorization result
HTML articles powered by AMS MathViewer

by A. A. Giannopoulos

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

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

PDF | Request permission

Abstract:

If $X$ is an $n$-dimensional normed space and $\varepsilon \in (0,1)$, there exists $m\geq (1-\varepsilon )n$, such that the formal identity $i_{2,\infty }\colon l^m_2\to l^m_\infty$ can be written as $i_{2,\infty }=\alpha \circ \beta ,\beta \colon l^m_2\to X,\alpha \colon X\to l^m_\infty$, with $\|\alpha \|\cdot \|\beta \|\leq c/\varepsilon$. This is proved as a consequence of a Sauer-Shelah type theorem for ellipsoids.

References

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, DOI 10.1007/BF02787120
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, DOI 10.1007/BF02772174
Morgan Ward and R. P. Dilworth, The lattice theory of ova, Ann. of Math. (2) 40 (1939), 600–608. MR 11, DOI 10.2307/1968944
A. A. Giannopoulos, A note on the Banach-Mazur distance to the cube, GAFA Seminar (to appear).
Morgan Ward, Ring homomorphisms which are also lattice homomorphisms, Amer. J. Math. 61 (1939), 783–787. MR 10, DOI 10.2307/2371336
Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. I, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 92, Springer-Verlag, Berlin-New York, 1977. Sequence spaces. MR 0500056, DOI 10.1007/978-3-642-66557-8
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
Albrecht Pietsch, Operator ideals, Mathematische Monographien [Mathematical Monographs], vol. 16, VEB Deutscher Verlag der Wissenschaften, Berlin, 1978. MR 519680
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, DOI 10.1007/BFb0090050
N. Sauer, On the density of families of sets, J. Combinatorial Theory Ser. A 13 (1972), 145–147. MR 307902, DOI 10.1016/0097-3165(72)90019-2
Saharon Shelah, A combinatorial problem; stability and order for models and theories in infinitary languages, Pacific J. Math. 41 (1972), 247–261. MR 307903, DOI 10.2140/pjm.1972.41.247
Stanisław J. Szarek, Spaces with large distance to $l^n_\infty$ and random matrices, Amer. J. Math. 112 (1990), no. 6, 899–942. MR 1081810, DOI 10.2307/2374731
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, DOI 10.1007/BFb0090211
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