×

Subspaces of small codimension of finite-dimensional Banach spaces. (English) Zbl 0623.46008

Let \(E=(R^ n,\| \cdot \|)\) be an n-dimensional Banach space. Let (.,.) be an inner product on \(R^ n\) and let \(\|| \cdot \||\) be the associated Euclidean norm on \(R^ n\) defined by \(\|| x\|| =\sqrt{(x,x)}\) for \(x\in R^ n\). Set \(\| x\|_*=\sup \{| (x,y)|\); \(\| y\| \leq 1\}\) for \(x\in R^ n\). Clearly \((R^ n,\| \cdot \|_*)\) can be identified with the dual space \(E^*\). Let \(S=\{x\in R^ n\); \(\|| x\|| =1\}\) and let \(\mu\) be the normalized rotation invariant measure on S. Define the Levy mean \(M_*\) by \(M_*=(\int_{S}\| x\|^ 2_* d\mu (x))^{1/2}.\)
In this paper the following problem is studied: Given an n-dimensional Banach space E, an euclidean norm \(\|| \cdot \||\) on E and \(0<\lambda <1\), find a subspace F of E with dim(F)\(\geq \lambda n\) such that \(\|| x\|| \leq M_*f(1-\lambda)\| x\|\) for \(x\in F\). The main result of this note is a theorem proving the above inequality with the function \(f(1-\lambda)\leq K(1-\lambda)^{-}\), a result which is in a certain sense the best possible. Weaker estimates had been previously obtained by V. Milman. The present result seems to be a useful tool for various applications in Banach space geometry, as well as in operator theory (for what concerns s-numbers, entropy numbers, Gel’fand numbers): see e.g. V. D. Milman and G. Pisier, Isr. J. Math. 54, 139- 158 (1986; Zbl 0611.46022).
Reviewer: P.L.Papini

MSC:

46B20 Geometry and structure of normed linear spaces
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
46B03 Isomorphic theory (including renorming) of Banach spaces
47B06 Riesz operators; eigenvalue distributions; approximation numbers, \(s\)-numbers, Kolmogorov numbers, entropy numbers, etc. of operators

Citations:

Zbl 0611.46022
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Jean Bourgain and Vitali D. Milman, Sections euclidiennes et volume des corps symétriques convexes dans \?\(^{n}\), C. R. Acad. Sci. Paris Sér. I Math. 300 (1985), no. 13, 435 – 438 (French, with English summary). · Zbl 0576.52002
[2] Bernd Carl, Entropy numbers, \?-numbers, and eigenvalue problems, J. Funct. Anal. 41 (1981), no. 3, 290 – 306. · Zbl 0466.41008 · doi:10.1016/0022-1236(81)90076-8
[3] -, On Gelfand, Kolmogonov and entropy numbers of operators acting between special Banach spaces (to appear).
[4] Stephen Dilworth and Stanisław Szarek, The cotype constant and an almost Euclidean decomposition for finite-dimensional normed spaces, Israel J. Math. 52 (1985), no. 1-2, 82 – 96. · Zbl 0657.46010 · doi:10.1007/BF02776082
[5] R. M. Dudley, The sizes of compact subsets of Hilbert space and continuity of Gaussian processes, J. Functional Analysis 1 (1967), 290 – 330. · Zbl 0188.20502
[6] T. Figiel, J. Lindenstrauss, and V. D. Milman, The dimension of almost spherical sections of convex bodies, Acta Math. 139 (1977), no. 1-2, 53 – 94. · Zbl 0375.52002 · doi:10.1007/BF02392234
[7] E. D. Gluskin, Norms of random matrices and diameters of finite-dimensional sets, Mat. Sb. (N.S.) 120(162) (1983), no. 2, 180 – 189, 286 (Russian). · Zbl 0528.46015
[8] Y. Gordon, H. König, and C. Schütt, Geometric and probabilistic estimates for entropy and approximation numbers of operators, J. Approx. Theory 49 (1987), no. 3, 219 – 239. · Zbl 0647.47035 · doi:10.1016/0021-9045(87)90100-6
[9] Thomas Kühn, \?-Radonifying operators and entropy ideals, Math. Nachr. 107 (1982), 53 – 58. · Zbl 0533.47042 · doi:10.1002/mana.19821070104
[10] V. D. Milman, Random subspaces of proportional dimension of finite-dimensional normed spaces; approach through the isoperimetric inequality, Séminaire d’Analyse Fonctionelle 1984/1985, Publ. Math. Univ. Paris VII, vol. 26, Univ. Paris VII, Paris, 1986, pp. 19 – 29.
[11] Vitali D. Milman and Gilles Pisier, Banach spaces with a weak cotype 2 property, Israel J. Math. 54 (1986), no. 2, 139 – 158. · Zbl 0611.46022 · doi:10.1007/BF02764939
[12] Albrecht Pietsch, Operator ideals, North-Holland Mathematical Library, vol. 20, North-Holland Publishing Co., Amsterdam-New York, 1980. Translated from German by the author. · Zbl 0434.47030
[13] V. N. Sudakov, Gaussian random processes, and measures of solid angles in Hilbert space, Dokl. Akad. Nauk SSSR 197 (1971), 43 – 45 (Russian). · Zbl 0231.60025
[14] A. Pajor and N. Tomczak-Jaegermann, Nombres de Gelfand et sections euclidiennes de grande dimension, Séminaire d’Analyse Fonctionelle 84/85, Université Paris VI et VII, Paris. · Zbl 0659.46017
[15] J. Bourgain and V. D. Milman, On Mahler’s conjecture on the volume of a convex symmetric body and its polar, Preprint, I.H.E.S.
[16] W. J. Davis, V. D. Milman, and N. Tomczak-Jaegermann, The distance between certain \?-dimensional Banach spaces, Israel J. Math. 39 (1981), no. 1-2, 1 – 15. · Zbl 0475.46010 · doi:10.1007/BF02762849
[17] T. Figiel and Nicole Tomczak-Jaegermann, Projections onto Hilbertian subspaces of Banach spaces, Israel J. Math. 33 (1979), no. 2, 155 – 171. · Zbl 0427.46010 · doi:10.1007/BF02760556
[18] V. D. Milman, Volume approach and iteration procedures in local theory of normed spaces, Banach spaces (Columbia, Mo., 1984) Lecture Notes in Math., vol. 1166, Springer, Berlin, 1985, pp. 99 – 105. , https://doi.org/10.1007/BFb0074699 V. D. Milman, Random subspaces of proportional dimension of finite-dimensional normed spaces: approach through the isoperimetric inequality, Banach spaces (Columbia, Mo., 1984) Lecture Notes in Math., vol. 1166, Springer, Berlin, 1985, pp. 106 – 115. · doi:10.1007/BFb0074700
[19] V. D. Milman and G. Pisier, Gaussian processes and mixed volumes, Ann. Probab. 15 (1987), no. 1, 292 – 304. · Zbl 0636.60036
[20] A. Yu. Garnaev and E. D. Gluskin, The widths of a Euclidean ball, Dokl. Akad. Nauk SSSR 277 (1984), no. 5, 1048 – 1052 (Russian). · Zbl 0588.41022
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.