Highly tempering infinite matrices

  • Luis Bernal-González
  • J. Alberto Conejero
  • Marina Murillo-Arcila
  • Juan B. Seoane-Sepúlveda
Original Paper


In this short note, it is proved the existence of infinite matrices that not only preserve convergence and limits of sequences but also convert every member of some dense vector space consisting, except for zero, of divergent sequences, into a convergent sequence.


Summation method Infinite matrix Divergent sequence Toeplitz–Silverman theorem Dense linear subspace 

Mathematics Subject Classification

15B05 15A04 40C05 



The first author has been supported by the Plan Andaluz de Investigación de la Junta de Andalucía FQM-127 Grant P08-FQM-03543 and by MEC Grant MTM2015-65242-C2-1-P. The second and third authors have been supported by MEC, Grant MTM2016-75963-P. The fourth author has been supported by Grant MTM2015-65825-P.


  1. 1.
    Aizpuru, A., Pérez-Eslava, C., Seoane-Sepúlveda, J.B.: Linear structure of sets of divergent sequences and series. Linear Algebra Appl. 418(2–3), 595–598 (2006)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Aron, R.M., Bernal-González, L., Pellegrino, D., Seoane-Sepúlveda, J.B.: Lineability: The Search for Linearity in Mathematics. Monographs and Research Notes in Mathematics. Chapman & Hall/CRC, Boca Raton, FL (2016)MATHGoogle Scholar
  3. 3.
    Aron, R.M., Gurariy, V.I., Seoane-Sepúlveda, J.B.: Lineability and spaceability of sets of functions on \({\mathbb{R}}\). Proc. Am. Math. Soc. 133(3), 795–803 (2005)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bernal-González, L., Pellegrino, D., Seoane-Sepúlveda, J.B.: Linear subsets of nonlinear sets in topological vector spaces. Bull. Am. Math. Soc. (N.S.) 51(1), 71–130 (2014). doi: 10.1090/S0273-0979-2013-01421-6 MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Gámez-Merino, J.L., Seoane-Sepúlveda, J.B.: An undecidable case of lineability in \({\mathbb{R}}^{\mathbb{R}}\). J. Math. Anal. Appl. 401(2), 959–962 (2013)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Gelbaum, B.R., Olmsted, J.M.H.: Counterexamples in Analysis. Dover Publications Inc., Mineola, NY (2003)MATHGoogle Scholar
  7. 7.
    Horváth, J.: Topological Vector Spaces and Distributions, vol. I. Addison-Wesley, Reading, MA (1966)MATHGoogle Scholar
  8. 8.
    Peyerimhoff, A.: Lectures on Summability. Lecture Notes in Mathematics. Springer, Berlin (1970)MATHGoogle Scholar
  9. 9.
    Kuratowski, K., Mostowski, A.: Set Theory. North Holland, Amsterdam (1976)MATHGoogle Scholar
  10. 10.
    Wilansky, A.: Summability Through Functional Analysis. North-Holland, Amsterdam (1984)MATHGoogle Scholar

Copyright information

© Springer-Verlag Italia 2017

Authors and Affiliations

  1. 1.Departamento de Análisis Matemático, Facultad de MatemáticasUniversidad de SevillaSevilleSpain
  2. 2.Instituto Universitario de Matemática Pura y AplicadaUniversitat Politècnica de ValènciaValenciaSpain
  3. 3.Institut Universitari de Matemàtiques i Aplicacions de Castelló (IMAC), Escuela Superior de Tecnología y Ciencias ExperimentalesUniversitat Jaume ICastelló de la PlanaSpain
  4. 4.Instituto de Matemática Interdisciplinar (IMI), Departamento de Análisis Matemático, Facultad de Ciencias MatemáticasUniversidad Complutense de MadridMadridSpain

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