Deviation of elements of a Banach space from a system of subspaces

  • S. V. Konyagin


We prove that if X is a real Banach space, Y 1Y 2 ⊂ ... is a sequence of strictly embedded closed linear subspaces of X, and d 1d 2 ≥ ... is a nonincreasing sequence converging to zero, then there exists an element xX such that the distance ρ(x, Y n ) from x to Y n satisfies the inequalities d n ρ(x, Y n ) ≤ 8d n for n = 1, 2, ....


Hilbert Space Banach Space Positive Integer STEKLOV Institute Number Sequence 
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  1. 1.
    P. A. Borodin, “On the existence of an element with given deviations from an expanding system of subspaces,” Mat. Zametki 80(5), 657–667 (2006) [Math. Notes 80, 621–630 (2006)].CrossRefMathSciNetGoogle Scholar
  2. 2.
    S. N. Bernstein, “On the inverse problem in the theory of best approximation of continuous functions,” in Collected Works (Akad. Nauk SSSR, Moscow, 1954), Vol. 2, pp. 292–294 [in Russian].Google Scholar
  3. 3.
    A. F. Timan, Theory of Approximation of Functions of a Real Variable (Fizmatgiz, Moscow, 1960; Pergamon, Oxford, 1963).Google Scholar
  4. 4.
    I. S. Tyuremskikh, “(B) property of Hilbert spaces,” Uch. Zap. Kalinin. Gos. Pedagog. Inst. 39, 53–64 (1964).Google Scholar
  5. 5.
    V. N. Nikol’skii, “On some properties of reflexive spaces,” Uch. Zap. Kalinin. Gos. Pedagog. Inst. 29, 121–125 (1963).Google Scholar
  6. 6.
    I. S. Tyuremskikh, “On a problem of S.N. Bernstein,” Uch. Zap. Kalinin. Gos. Pedagog. Inst. 52, 123–129 (1967).Google Scholar
  7. 7.
    J. M. Almiro and N. Del Toro, “Some remarks on negative results in approximation theory,” in Proc. Fourth Int. Conf. on Functional Analysis and Approximation Theory, Potenza, Italy, 2000 (Circ. Mat. Palermo, Palermo, 2002), Vol. 1, Rend. Circ. Mat. Palermo, Ser. 2, Suppl. 68 (1), pp. 245–256.Google Scholar
  8. 8.
    P. A. Borodin, “Selected approximative properties of sets in Banach spaces,” Doctoral (Phys.-Math.) Dissertation (Moscow State Univ., Moscow, 2012).Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2014

Authors and Affiliations

  1. 1.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia

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