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Subspaces of \(\ell^ N_ p\) of small codimension. (English) Zbl 0792.46010

Let \(1\leq p\leq\infty\) and \(m\) and \(N\) integers. For what numbers \(k\) do there exist good copies of \(\ell^ k_ p\) in \(X\) for arbitrary \(m\)- dimensional subspaces \(X\) of \(\ell^ k_ p\)? In the introduction one finds a thorough summary of results concerning this question.
Whereas previous work mainly has concentrated on the case of small \(m\) here the case \(m= N- n\) with small \(n\) is of interest. A typical result reads as follows: Let \(E\) be a subspace of \(\ell^ N_ 1\) of dimension \(N- n\). Then \(E\) contains a \(k\)-dimensional subspace \(F\) such that \(d(F,\ell^ k_ 1)\leq 6\), where \[ k\geq (1/24)^ 2\min\bigl\{(N/4n)\log(N/2N),N\bigr\}. \] The estimates in the case \(p=\infty\) are in terms of the Gelfand numbers of the formal identity between \(\ell^ N_ 1\) and \(\ell^ N_ \infty\).

MSC:

46B25 Classical Banach spaces in the general theory
46B07 Local theory of Banach spaces

Keywords:

Gelfand numbers
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References:

[1] Bennett, G.; Dor, L. E.; Goodman, V.; Johnson, W. B.; Newman, C. M., On uncomplemented subspaces of L_p, 1<p<2, Israel J. Math., 26, 178-187 (1971) · Zbl 0339.46022 · doi:10.1007/BF03007667
[2] Bourgain, J., Subspaces ofL_N^∞, arithmetical diameter and Sidon sets, Probability in Banach Spaces V, Proc. Medford 1984, 96-127 (1985), Berlin: Springer-Verlag, Berlin · Zbl 0583.43008
[3] J. Bourgain,A remark on the Behaviour of L_p-multipliers and the range of operators acting on L_p-spaces, Israel J. Math., this issue. · Zbl 0808.46037
[4] Bourgain, J.; Kalton, N.; Tzafriri, L.; Lindenstrauss, J.; Milman, V., Geometry of finite-dimensional subspaces and quotients of L_p, Geometric Aspects of Functional Analysis, Israel Seminar 1987-88, 138-176 (1986), Berlin: Springer-Verlag, Berlin · Zbl 0676.46008 · doi:10.1007/BFb0090053
[5] Bourgain, J.; Lindenstrauss, J.; Milman, V. D., Approximation of zonoids by zonotopes, Acta Math., 162, 73-141 (1989) · Zbl 0682.46008 · doi:10.1007/BF02392835
[6] Bourgain, J.; Tzafriri, L., Invertibility of “large” submatrices with applications to the geometry of Banach spaces and harmonic analysis, Israel J. Math., 57, 137-224 (1987) · Zbl 0631.46017 · doi:10.1007/BF02772174
[7] J. Bourgain and L. Tzafriri,Embedding l_p^k in subspaces of L_p for p > 2, Israel J. Math., to appear. · Zbl 0753.46012
[8] Figiel, T.; Johnson, W. B., Large subspaces ofl_∞^N and estimates of the Gordon-Lewis constant, Israel J. Math., 37, 92-112 (1980) · Zbl 0445.46012 · doi:10.1007/BF02762871
[9] Figiel, T.; Lindenstrauss, J.; Milman, V. D., The dimension of almost spherical sections of convex bodies, Acta Math., 139, 53-94 (1977) · Zbl 0375.52002 · doi:10.1007/BF02392234
[10] Garnaev, A. Yu.; Gluskin, E. D., On diameters of the Euclidean ball, Dokl. Akad. Nauk SSSR, 277, 1048-1052 (1984)
[11] Gluskin, E. D., On some finite-dimensional problems from the theory of widths, Vestnik Lenin. Univ., 13, 5-10 (1981) · Zbl 0482.41018
[12] Gluskin, E. D., Norms of random matrices and diameters of finite-dimensional sets, Mat. Sb., 120, 180-189 (1983) · Zbl 0528.46015
[13] Gluskin, E. D., The octahedron is badly approximated by random subspaces, Funk. Analiz i Ego Priloz., 20, 14-20 (1986) · Zbl 0619.46014
[14] Höllig, K., Approximationzahlen von Sobolev-Einbettungen, Math. Ann., 242, 273-281 (1979) · Zbl 0394.41011 · doi:10.1007/BF01420731
[15] W. B. Johnson and G. Schechtman,On subspaces of L_1with maximal distance to Euclidean spaces, Proc. Res. Workshop on Banach Spaces Theory (Bor-Luh-Lin, ed.), 1981, 83-96. · Zbl 0522.46014
[16] Kashin, B. S., On the Diameter of octahedra, Usp. Mat. Nauk, 30, 251-252 (1975) · Zbl 0311.46014
[17] Kashin, B. S., On the Kolmogorov diameters of octahedra, Dokl. Akad. Nauk SSSR, 214, 1024-1026 (1974)
[18] Kashin, B. S., Sections of some finite-dimensional sets and classes of smooth functions, Izv. Akad. Nauk SSSR, 41, 334-351 (1977) · Zbl 0354.46021
[19] Lewis, D. R., Finite-dimensional subspaces of L_p, Studia Mathematica, 63, 207-212 (1978) · Zbl 0406.46023
[20] Lindenstrauss, J.; Tzafriri, L., Classical Banach Spaces. I. Sequence Spaces (1977), Berlin: Springer-Verlag, Berlin · Zbl 0362.46013
[21] Milman, V. D., A new proof of a theorem of A. Dvoretzky on sections of convex bodies, Funct. Anal. Appl., 5, 288-295 (1971) · Zbl 0239.46018 · doi:10.1007/BF01086740
[22] Milman, V. D.; Schechtman, G., Asymptotic theory of finite-dimensional normed spaces (1986), Berlin: Springer-Verlag, Berlin · Zbl 0606.46013
[23] Tomczak-Jaegermann, N., Banach-Mazur Distances and Finite-dimensional Operator Ideals (1989), Harlow: Longman Scientific & Technical, Harlow · Zbl 0721.46004
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