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Acta Mathematica Vietnamica

, Volume 42, Issue 4, pp 727–746 | Cite as

New Classes of Generalized PN Spaces and Their Normability

  • P. K. HarikrishnanEmail author
  • Bernardo Lafuerza Guillén
  • Yeol Je Cho
  • K. T. Ravindran
Article

Abstract

In this paper, we obtain some properties of invertible operators; convex, balanced, absorbing sets; and \(\mathcal {D}\)-boundedness in Šerstnev spaces. We prove that some PN spaces (V,ν,τ,τ ), which are not Šerstnev spaces, in which the triangle function τ is not Archimedean can be endowed with a structure of a topological vector space, and we give suitable example to illustrate this result. Also, we show that the topological spaces obtained in such a manner are normable under certain given conditions: some examples are given.

Keywords

Probabilistic normed spaces Šerstnev spaces Normability Topological vector spaces (TVS) Local convexity Archimedean 

Mathematics Subject Classification (2010)

54E70 

Notes

Acknowledgements

The authors are grateful to the anonymous referees for careful reading of the manuscript. Referees’ valuable comments and suggestions have improved the paper. The first author acknowledges MIT, Manipal University, India; the second author acknowledges Universidad de Almeria, Spain; the third author acknowledges Gyeongsang National University, South Korea; and the fourth author acknowledges Payyanur College, Kannur University, India, for their kind encouragement.

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Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2017

Authors and Affiliations

  • P. K. Harikrishnan
    • 1
    Email author
  • Bernardo Lafuerza Guillén
    • 2
  • Yeol Je Cho
    • 3
  • K. T. Ravindran
    • 4
  1. 1.Department of Mathematics, Manipal Institute of TechnologyManipal UniversityManipalIndia
  2. 2.Departamento de Matemática Aplicada y EstadísticaUniversidad de AlmeríaAlmeríaSpain
  3. 3.Department of Mathematics Education and the RINSGyeongsang National UniversityJinjuSouth Korea
  4. 4.Department of Mathematics, Payyanur CollegeKannur UniversityKannurIndia

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