New Classes of Generalized PN Spaces and Their Normability

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.

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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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Correspondence to P. K. Harikrishnan.

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Dedicated to Professor Carlo Sempi Dipartimento di Matematica “Ennio De Giorgi”, Universita del Salento, Leece, Italy

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Harikrishnan, P.K., Guillén, B., Cho, Y. et al. New Classes of Generalized PN Spaces and Their Normability. Acta Math Vietnam 42, 727–746 (2017). https://doi.org/10.1007/s40306-017-0218-z

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Keywords

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

Mathematics Subject Classification (2010)

  • 54E70