Arkiv för Matematik

  • Ark. Mat.
  • Volume 12, Number 1-2 (1974), 41-49.

Every sequence converging to O weakly in L2 contains an unconditional convergence sequence

János Komlós

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Throughout the paper all functions are measurable functions on some measure space {X, S, μ}. It is clear that it is sufficient to prove our Theorem in case of finite measure, thus we can take μ(X)=1.

As a rule, we do not indicate the arguments of functions: writing ϕ, f etc. instead of ϕ(x), f(x) etc., and μ(f>λ) instead of μ({x; f(x)>λ}), and the measure: writing ∫ ϕ, ∫ ϕ1ϕ2 etc. instead of Xϕ1(x)ϕ2(x)μ(dx) etc.; we also say «almost everywhere» instead of «μ-almost everywhere». $\alpha _n \mathop \rightharpoonup \limits_{L_p } \alpha $ will stand for weak convergence in LP.

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Ark. Mat., Volume 12, Number 1-2 (1974), 41-49.

Received: 7 December 1972
First available in Project Euclid: 31 January 2017

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1974 © Institut Mittag-Leffler


Komlós, János. Every sequence converging to O weakly in L 2 contains an unconditional convergence sequence. Ark. Mat. 12 (1974), no. 1-2, 41--49. doi:10.1007/BF02384745.

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  • Alexits, G., Convergence problems of orthogonal series. Publishing House of the Hungarian Academy of Sciences, Budapest, 1961.
  • Menchov, D., Sur la convergence et la sommation des séries de fonctions orthogonales. Bull. Soc. Math. France 64 (1936), 147–170.
  • Révész, P.: On a problem of Steinhaus. Acta Math. Acad. Sci. Hungar 16 (1965), 310–318.
  • Gaposhkin, V. F., Lacunary series and independent functions. Uspehi Mat. Nauk 6/21 (1966), 3–82. (In Russian).
  • Chatterji, S. D., A general strong law. Invent. Math. 9 (1970), 235–245.
  • Ziza, O. A.: On a property of orthonormal sequences (in Russian). Mat. Sbornik, 55 (100) (1962), 3–16.
  • Uljanov, P. L., Solved an unsolved problems in the theory of trigonometric and orthonormal series. Uspehi Mat. Nauk 19/1 (1964), 3–69.. (In Russian).
  • Billingsley, P., Convergence of probability measures. John Wiley and Sons Inc., 1968.
  • Komlós, J. & Révész, P., Remark to a paper of Gaposhkin. Acta Sci. Math. Szeged 33 (1972), 237–241.
  • , On the series Σckϕk. Studia Sci. Math. Hungar 7 (1972), 451–458.
  • Garsisa, A., Topics in almost everywhere convergence. Chicago, 1970.
  • Komlós, J., A generalization of a problem of Steinhaus. Acta Math. Acad. Sci. Hungar. 18 (1967), 217–229.