Advertisement

Entropy Numbers and Duality for Operators with Values in a Hilbert Space

  • H. König
  • V. D. Milman
  • N. Tomczak-Jaegermann
Part of the Progress in Probability book series (PRPR, volume 20)

Abstract

Let Y be a Banach space and let TY be a compact body. Let KY be a compact set. Recall that the covering number N(K, T) is defined by
$$N(K,T)=inf\left\{ N:\exists y1,...,yNinYsuchthatK\subset \underset{1}{\overset{N}{\mathop{\bigcup }}}\,(yi+T) \right\}$$
.

Keywords

Hilbert Space Convex Body Isoperimetric Inequality Absolute Constant Random Projection 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [B-M.1]
    J. Bourgain and V.D. Milman, Sections euclidiennes et volume des corps symétriques convexes dans R n, C.R. Acad. Sci. Paris 300 (1985), 435–438.MathSciNetMATHGoogle Scholar
  2. [B-M.2]
    J. Bourgain and V.D. Milman, New volume ratio properties for convex symmetric bodies in R n, Inventiones Math. 88 (1987), 319–340.MathSciNetMATHCrossRefGoogle Scholar
  3. [B-Z]
    Y.D. Burago and V,A. Zalgaler, Geometric inequalities. in “Nauka”, Leningrad, 1980 (Russian) and Springer Verlag 1987.Google Scholar
  4. [C]
    B. Carl, On Gelfand, Kolmogorov and entropy numbers of operators acting between special Banach spaces, Journal of Approx. Theory.Google Scholar
  5. [F-L-M]
    T. Figiel, J. Lindenstrauss and V.D. Milman, The dimension of almost spherical sections of convex bodies, Acta Math. 139 (1977), 53–94.MathSciNetMATHCrossRefGoogle Scholar
  6. [G-K-S]
    Y. Gordon, H. König and C. Schútt, Geometric and probabilistic estimates for entropy and approximation numbers of operators, J. of Approx. Theory 49 (1987), 219–239.MATHCrossRefGoogle Scholar
  7. [J-L]
    W.B. Johnson and J. Lindenstrauss, Extensions of Lipschitz mappings into a Hilbert space, Contemporary Mathematics 26 (1984), 189–206.MathSciNetMATHCrossRefGoogle Scholar
  8. [K-M]
    H. König and V.D. Milman, On the covering number of convex bodies, pp. 82–95 in “Geometric Aspects of Functional Analysis”, Israel Seminar 1985/86, Lecture Notes in Math. 1267. Springer Verlag 1987.CrossRefGoogle Scholar
  9. [M]
    V.D. Milman, A new proof of the theorem of A. Dvoretzky on sections of convex bodies, Funct. Anal. Appl. 5 (1971), 28–38. (translated from Russian).MathSciNetGoogle Scholar
  10. [M-Sch]
    V.D. Milman and G. Schechtman, “Asymptotic theory of finite dimensional normed spaces,” Lecture Notes in Math. 1200, Springer Verlag, 1986.MATHGoogle Scholar
  11. [M-T]
    V.D. Milman and N. Tomczak-Jaegermann, Sudakov type inequalities for convex bodies in R n, pp. 113–121 in “Geometric Aspects of functional Analysis”, Israel Seminar 1985/86, Lecture Notes in Math. 1267. Springer Verlag, 1987.CrossRefGoogle Scholar
  12. [P]
    G. Pisier. private letter.Google Scholar
  13. [Pie]
    A. Pietsch, “Operator ideals,” VEB Deutscher Verlag, Berlin, North Holland, 1980.MATHGoogle Scholar
  14. [P-T]
    A. Pajor and N. Tomczak-Jaegermann, Remarques sur les nombres d’entropie d’un operateur et son transposé, C.R. Acad. Sci. Paris 301 (1985), 743–746.MathSciNetMATHGoogle Scholar
  15. [S]
    C. Schutt, Entropy numbers of diagonal operators between symmetric spaces, J. of Appr. Theory 40 (1983), 121–128.MathSciNetCrossRefGoogle Scholar
  16. [Sa]
    L.A. Santalo, Un inveriant afin pasa los cuerpos convexos del espacio de n-dimensiones, Portugal Math. 8 (1949), 155–161.MathSciNetMATHGoogle Scholar

Copyright information

© Birkhäuser Boston 1990

Authors and Affiliations

  • H. König
    • 1
  • V. D. Milman
    • 2
  • N. Tomczak-Jaegermann
    • 3
  1. 1.Mathematisches SeminarUniversität KielKiel 1West Germany
  2. 2.School of Mathematical Sciences, Raymond and Beverley Sackler, Faculty of Exact SciencesTel Aviv UniversityTel AvivIsrael
  3. 3.Department of MathematicsUniversity of AlbertaEdmontonCanada

Personalised recommendations