Subspaces of small codimension of finite-dimensional Banach spaces

Pajor, Alain; Tomczak-Jaegermann, Nicole

doi:10.1090/S0002-9939-1986-0845980-8

Subspaces of small codimension of finite-dimensional Banach spaces
HTML articles powered by AMS MathViewer

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

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

Nombres de Gelfand et sections euclidiennes de grande dimension

On Mahler’s conjecture on the volume of a convex symmetric body and its polar

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