Advertisement

On geometry of Orlicz spaces

  • Z. G. Gorgadze
  • V. I. Tarieladze
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 828)

Abstract

In terms of the function Φ it is established when Orlicz space LΦ does not contain l n uniformly and when it has some type or cotype.

We study some geometrical properties of Orlicz spaces. Namely using the results of [1] we establish when the given Orlicz space does not contain l n uniformly and when it has some type or cotype.

Let X be a real Banach space, X* the dual space and let (ɛk)k ∈ N be the sequence of independent random variables with P[ɛk=1]=P[ɛk=−1]=1/2 (the Bernoulli, or the Rademacher sequence). A Banach space X is said to be a space of type p, 0<p⩽2, if there exists a constant c>0 such that for each finite collection x1,...,xn of elements of X there holds the inequality
$$E\parallel \sum\limits_{k = 1}^n {x_k \varepsilon _k } \parallel _X^p \leqslant c\sum\limits_{k = 1}^n {\parallel x_k \parallel _X^p }$$
Here and below E denotes the mathematical expectation,

It is clear that every Banach space has type p,0<p⩽1.

A Banach space X is said to be a space of cotype q 2<q<∞, if there exists a constant c′>0 such that for each finite collection x1,...,xn of elements of X there holds the inequality
$$E\parallel \sum\limits_{k = 1}^n {x_k \varepsilon _k } \parallel _X^q \geqslant c'\sum\limits_{k = 1}^n {\parallel x_k \parallel _X^q .}$$

If X is of type p, 1<p⩽2, then X* is of cotype p′=p/p−1. Đenote by l n the Rn with the maximum-norm. We shall say that a Banach space X contains l n uniformly if for each ɛ>0 and any integer n there exists an injective linear operator J: l n → X such that ‖J‖‖J−1‖<1+ɛ. X does not contain l n uniformly if and only if it has certain cotype q ([2]).

Let (T, Σ, ϑ) be a positive σ-finite measure space and Φ : R+ → R+ denotes a convex continious non-decreasing and vanishing at zero function. For measurable function x: T → R define
$$\rho _\Phi (x) = \int\limits_T {\Phi (\left| {x(t)} \right|)d \vartheta (t)}$$
and denote LΦ=LΦ (T, Σ, ϑ) the collection of all measurable functions x with ρΦ(λ x)<∞ for some λ>0. LΦ is a vector space. Moreover, LΦ is Banach space under the norm
$$\left\| x \right\|_\Phi = \inf \{ \lambda > 0:\rho _\Phi (x/\lambda ) \leqslant 1\}$$
and this space is said to be Orlicz space.

By Δ2 we denote the family of functions Φ that satisfy the so-called Δ2 condition (i.e. Φ (2u)⩽c Φ (u) for some c>0 and every u ∈ R+). If Φ ∈ Δ2, then x ∈ LΦ if and only if ρΦ(x)<∞.

Keywords

Banach Space Measurable Function Geometrical Property Measure Space Independent Random Variable 
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. [1]
    S.A. Chobanjan, Z.G. Gorgadze, V.I. Tarieladze, Gaussian covariances in Banach sulattices of Lo. (in Russian) Dokl.AN SSR, 241, 3(1978), 528–531; Soviet Math. Dokl., 19, 4 (1978), 885–888.Google Scholar
  2. [2]
    B. Maurey, G. Pisier, Caracterization d’une classe d’espaces de Banach par des series aleatoires vectorielles, C.R.Acad.Sci.Paris, 277 (1973), 687–690.MathSciNetzbMATHGoogle Scholar
  3. [3]
    S.A. Rakov, On Banach spaces for which Orlicz’s theorem does not hold. (in Russian), Mat.Zamet. 14, 1 (1973), 101–106.MathSciNetGoogle Scholar
  4. [4]
    N.N. Vakhania, Probability distributions in Banach spaces. (in Russian), Metzniereba, Tbilisi 1971.Google Scholar
  5. [5]
    Z.G. Gorgadze, V.I. Tarieladze, Gaussian measuries in Orlicz spaces, Soobsc.AN Gruz.SSR 74, 3 (1974) 557–559.MathSciNetzbMATHGoogle Scholar
  6. [6]
    W. Matuszewska, W. Orlicz, On certain properties of φ-functions, Bull.Acad.Polon.Sci.Ser.Math.Astronom.Phys. 8, 7 (1960), 439–443.MathSciNetzbMATHGoogle Scholar
  7. [7]
    T. Figiel, J. Lidenstrauss, V.D. Milman, The dimension of almost spherical sections of convex bodies, Acta Mathematica, 139, 1–2 (1977), 53–94.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 1980

Authors and Affiliations

  • Z. G. Gorgadze
    • 1
    • 2
  • V. I. Tarieladze
    • 1
    • 2
  1. 1.Tbilisi State UniversityTbilisiRussia
  2. 2.Academy of Sciences of the Georgian SSR, Computing CenterTbilisiRussia

Personalised recommendations