Journal of Applied and Pure Mathematics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
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A CLASS OF ULTRAMETRIC QUASI 2-NORMED SPACES

  • Received : 2025.09.21
  • Accepted : 2026.02.08
  • Published : 2026.03.30
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Abstract

In the present paper, we introduce and study ultrametric quasi 2-normed spaces and ultrametric quasi 2-Banach spaces. In particular, we prove several results on Cauchy sequences in ultrametric quasi 2-normed spaces. Moreover, we introduce and study the notion of bounded linear 2-functionals on ultrametric quasi 2-Banach spaces and investigate some of their properties. On the other hand, the new norm on the ultrametric quasi 2-normed space is constructed. The concepts of closed operators between ultrametric 2-normed spaces and b-linear functionals in ultrametric quasi 2-normed spaces are introduced. Finally, a necessary and sufficient condition for a linear operator to be closed in terms of its graph is proved, and some results on bounded b-linear functionals in ultrametric quasi 2-normed spaces are given.

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References

  1. A. Bayoumi, Foundations of complex analysis in non locally convex spaces: function theory without convexity condition, In: Mathematics Studied 193, North Holland, Amsterdam, New York, Tokyo, 2003.
  2. T. Diagana and F. Ramaroson, Non-archimedean Operators Theory, Springer, 2016.
  3. J. Ettayb, Spectral Theory of Linear Operators on Non-Archimedean Quasi-Banach Spaces, p-Adic Num. Ultra. Anal. 416 (2024), 351-374.
  4. R. Freese and Y.J. Cho, Geometry of Linear 2-normed Spaces, Nova Science Publishers, New York, 2001.
  5. S. Gähler, Lineare 2-Normierte Räume, Math. Nachr. 28 (1964), 1-43. https://doi.org/10.1002/mana.v28:1/2
  6. S. Gähler, 2-metrische Räume und ihre topologische Struktur, Math. Nachr. 26 (1963), 115-122. https://doi.org/10.1002/mana.v26:1/4
  7. S. Gähler, A.H. Siddique and S.C. Gupta, Contribution to non Archimedean functional analysis, Math. Nachr. 69 (1975), 162-175.
  8. K. Iséki, On finite dimensional quasinorm spaces, Proc. Japan Acad. 35 (1959), 536-537.
  9. K. Iséki, An approximation problem in quasi-normed spaces, Poc. Japan Acad. 35 (1959), 465-466. https://doi.org/10.3792/pja/1195524253
  10. K. Iséki, A class of quasi-normed spaces, Proc. Japan Acad. 36 (1960), 22-23. https://doi.org/10.2183/pjab1945.36.675
  11. T. Konda, On quasi-normed space. I, Proc. Japan Acad. 35 (1959), 340-342.
  12. T. Konda, On quasi-normed space. II, Proc. Japan Acad. 35 (1959), 584-587.
  13. T. Konda, On quasi-normed space. III, Proc. Japan Acad. 36 (1959), 189-191.
  14. T. Konda, On quasi-normed spaces over fields with non-archimedean valuation, Proc. Japan Acad. 36 (1960), 543-546.
  15. L.A. Lujsternik and W.L. Sobolew, Elemente der Funktionalanalysis, Akademie-Verlag, Berlin, 1955.
  16. T.D. Narang and S.K. Garg, Best approximation in non-Archimedean quasi normed spaces, Journal of the Korea Society of Mathematical Education 21 (1982), 41-44.
  17. M. Pavel, On quasi normed spaces, Bull. Acad. Sci. 5 (1957), 479-487.
  18. S. Rolewicz, On a certain class of linear metric spaces, Bull. Acad. Sci. 5 (1957), 471-473.
  19. A.C.M. van Rooij, Non-Archimedean functional analysis, Monographs and Textbooks in Pure and Applied Math., 51, Marcel Dekker, Inc., New York, 1978.
  20. W.H. Schikhof, Ultrametric Calculus, Cambrige Unersity Press, Cambridge, 1984.
  21. W.H. Schikhof and C. Perez-Garcia, Locally Convex Spaces over Non-archimedean Fields, Cambridge Studies and Advanced Mathematics, 119, Cambridge University Press, Cambridge, New York, 2010.
  22. A. White, 2-Banach spaces, Math. Nachr. 42 (1969), 43-60. https://doi.org/10.1002/mana.v42:1/3