General Summability Theory and Steinhaus Type Theorems

  • P. N. Natarajan


In this chapter, we recall well-known definitions and concepts. We state and prove Silverman–Toeplitz theorem and Schur’s theorem and then deduce Steinhaus theorem. A sequence space \(\Lambda _r\), \(r \ge 1\) being a fixed integer, is introduced, and we make a detailed study of the space \(\Lambda _r\), especially from the point of view of sequences of zeros and ones. We prove a Steinhaus type result involving the space \(\Lambda _r\), which improves Steinhaus theorem. Some more Steinhaus type theorems are also proved.


Infinite matrix Banach space Convergence preserving or conservative matrix Regular matrix Silverman–Toeplitz theorem Schur’s theorem Steinhaus theorem Steinhaus type theorem Sequence space \(\Lambda _r\) Eventually periodic sequence Non-periodic sequence Sequence of zeros and ones Closed linear span Generalized semiperiodic sequence 


  1. 1.
    Hardy, G.H.: Divergent Series. Oxford University Press, Oxford (1949)MATHGoogle Scholar
  2. 2.
    Maddox, I.J.: Elements of Functional Analysis. Cambridge University Press, Cambridge (1970)MATHGoogle Scholar
  3. 3.
    Peyerimhoff, A.: Lectures on Summability. Lecture Notes in Mathematics, vol. 107. Springer, Berlin (1969)MATHGoogle Scholar
  4. 4.
    Natarajan, P.N.: A Steinhaus type theorem. Proc. Amer. Math. Soc. 99, 559–562 (1987)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Schur, I.: Uber lineare Transformationen in der Theorie der unendlichen Reihen. J. Reine Angew. Math. 151, 79–111 (1921)MathSciNetMATHGoogle Scholar
  6. 6.
    Natarajan, P.N.: On certain spaces containing the space of Cauchy sequences. J. Orissa Math. Soc. 9, 1–9 (1990)MathSciNetMATHGoogle Scholar
  7. 7.
    Lorentz, G.G.: A contribution to the theory of divergent sequences. Acta Math. 80, 167–190 (1948)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Hill, J.D., Hamilton, H.J.: Operation theory and multiple sequence transformations. Duke Math. J. 8, 154–162 (1941)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Zeller, K., Beekmann, W.: Theorie der Limitierungverfahren. Springer, Berlin (1970)CrossRefMATHGoogle Scholar
  10. 10.
    Berg, I.D., Wilansky, A.: Periodic, almost periodic and semiperiodic sequences. Michigan Math. J. 9, 363–368 (1962)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Stieglitz, M., Tietz, H.: Matrixtransformationen von Folgenraümen Eine Ergebnisübersicht. Math. Z. 154, 1–16 (1977)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Natarajan, P.N.: Some Steinhaus type theorems over valued fields. Ann. Math. Blaise Pascal 3, 183–188 (1996)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Maddox, I.J.: On theorems of Steinhaus type. J. London Math. Soc. 42, 239–244 (1967)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Fridy, J.A.: Properties of absolute summability matrices. Proc. Amer. Math. Soc. 24, 583–585 (1970)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Natarajan, P.N.: A theorem of Steinhaus type. J. Anal. 5, 139–143 (1997)MathSciNetMATHGoogle Scholar
  16. 16.
    Natarajan, P.N.: Some more Steinhaus type theorems over valued fields. Ann. Math. Blaise Pascal 6, 47–54 (1999)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Natarajan, P.N.: Some more Steinhaus type theorems over valued fields II. Commun. Math. Anal. 5, 1–7 (2008)MathSciNetMATHGoogle Scholar
  18. 18.
    Natarajan, P.N.: Steinhaus type theorems for \((C, 1)\) summable sequences. Comment Math. Prace Mat. 54, 21–27 (2014)MathSciNetMATHGoogle Scholar
  19. 19.
    Natarajan, P.N.: Steinhaus type theorems for summability matrices. Adv. Dev. Math. Sci. 6, 1–8 (2014)CrossRefGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2017

Authors and Affiliations

  1. 1.Formerly of the Department of MathematicsRamakrishna Mission Vivekananda CollegeChennaiIndia

Personalised recommendations