Subspaces of small codimension of finite-dimensional Banach spaces
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- by Alain Pajor and Nicole Tomczak-Jaegermann PDF
- Proc. Amer. Math. Soc. 97 (1986), 637-642 Request permission
Abstract:
Given a finite-dimensional Banach space $E$ and a Euclidean norm on $E$, we study relations between the norm and the Euclidean norm on subspaces of $E$ of small codimension. Then for an operator taking values in a Hilbert space, we deduce an inequality for entropy numbers of the operator and its dual.References
- Jean Bourgain and Vitali D. Milman, Sections euclidiennes et volume des corps symétriques convexes dans $\textbf {R}^n$, C. R. Acad. Sci. Paris Sér. I Math. 300 (1985), no. 13, 435–438 (French, with English summary). MR 794017
- Bernd Carl, Entropy numbers, $s$-numbers, and eigenvalue problems, J. Functional Analysis 41 (1981), no. 3, 290–306. MR 619953, DOI 10.1016/0022-1236(81)90076-8 —, On Gelfand, Kolmogonov and entropy numbers of operators acting between special Banach spaces (to appear).
- 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. MR 815604, DOI 10.1007/BF02776082
- R. M. Dudley, The sizes of compact subsets of Hilbert space and continuity of Gaussian processes, J. Functional Analysis 1 (1967), 290–330. MR 0220340, DOI 10.1016/0022-1236(67)90017-1
- 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. MR 445274, DOI 10.1007/BF02392234
- 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). MR 687610
- 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. MR 879670, DOI 10.1016/0021-9045(87)90100-6
- Thomas Kühn, $\gamma$-Radonifying operators and entropy ideals, Math. Nachr. 107 (1982), 53–58. MR 695735, DOI 10.1002/mana.19821070104
- 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. MR 941807
- Vitali D. Milman and Gilles Pisier, Banach spaces with a weak cotype $2$ property, Israel J. Math. 54 (1986), no. 2, 139–158. MR 852475, DOI 10.1007/BF02764939
- Albrecht Pietsch, Operator ideals, North-Holland Mathematical Library, vol. 20, North-Holland Publishing Co., Amsterdam-New York, 1980. Translated from German by the author. MR 582655
- V. N. Sudakov, Gaussian random processes, and measures of solid angles in Hilbert space, Dokl. Akad. Nauk SSSR 197 (1971), 43–45 (Russian). MR 0288832 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. 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.
- W. J. Davis, V. D. Milman, and N. Tomczak-Jaegermann, The distance between certain $n$-dimensional Banach spaces, Israel J. Math. 39 (1981), no. 1-2, 1–15. MR 617286, DOI 10.1007/BF02762849
- T. Figiel and Nicole Tomczak-Jaegermann, Projections onto Hilbertian subspaces of Banach spaces, Israel J. Math. 33 (1979), no. 2, 155–171. MR 571251, DOI 10.1007/BF02760556
- 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. MR 827765, DOI 10.1007/BFb0074699
- V. D. Milman and G. Pisier, Gaussian processes and mixed volumes, Ann. Probab. 15 (1987), no. 1, 292–304. MR 877605
- A. Yu. Garnaev and E. D. Gluskin, The widths of a Euclidean ball, Dokl. Akad. Nauk SSSR 277 (1984), no. 5, 1048–1052 (Russian). MR 759962
Additional Information
- © Copyright 1986 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 97 (1986), 637-642
- MSC: Primary 46B20; Secondary 41A46, 47B10
- DOI: https://doi.org/10.1090/S0002-9939-1986-0845980-8
- MathSciNet review: 845980