Some properties of continuous linear operators in topological vector PN-spaces

Document Type : Research Paper


1 Department of Mathematics, Iran University of Science and Technology, Narmak, Tehran, Iran.

2 Departamento de Estadistica Ymatem'atica Aplicada, Universidad de Almeria, e-04120, Spain

3 Department of Mathematics Iran University of Science and Technology, Narmak, Tehran, Iran.


The notion of a probabilistic metric space corresponds to the situations when we do not know exactly the distance.  Probabilistic Metric space was introduced by Karl Menger. Alsina, Schweizer and Sklar gave a general definition of probabilistic normed space based on the definition of Menger [1]. In this note we study the PN spaces which are topological vector spaces and the open mapping and closed graph   Theorems in this spaces are proved.


  1. C. Alsina, B. Schweizer and A. Sklar, On the definition of a probabilistic normed space, Aequationes Math., 46 (1993) 91-98.
  2. C. Alsina, B. Schweizer and A. Sklar, Continuity properties of probabilistic norms, J. Math. Anal. Appl., 208 (1997) 446-452.
  3. M. B. Ghaemi, B. Lafuerza-Guill´en, Probabilistic Total Paranormed Spaces, International Journal of Mathematics and Statistics, Autumn 2010, Volume 6, Number A10, 69-78.
  4. U. Hohle, Probabilistic metrization of generalized topologies, Bull. Acad. Polon. Sci. Srie des sciences math., astr. et phys. 25, 493-498, (1977).
  5. B. Lafuerza–Guill´en, D–bounded sets in probabilistic normed spaces and their products, Rend. Mat., Serie VII 21 (2001) 17-28.
  6. B. Lafuerza–Guill´en, J.L. Rodr´iguez, Translation–invariant generalized topologies induced by probabilistic norms, Note di Matematica, in press.
  7. B. Lafuerza-Guill´en, J.A. Rodr´ıguez Lallena, probabilistic norms for linear operator,J. Math. Anal. Appl., 220 (1998) 462-476.
  8. B. Lafuerza–Guill´en, J. A. Rodr´iguez Lallena and C. Sempi, A study of boundedness in probabilistic normed spaces, J. Math. Anal. Appl., 232 (1999) 183-196.
  9. B. Lafuerza–Guill´en, J.A. Rodr´ıguez Lallena and C. Sempi, Some classes of Probabilistic Normed Spaces. Rend. Mat., 17 (1997), 237-252.
  10. H. H. Schaefer and M.P. Wolff, Topological vector spaes, Second Edition, Springer (1991).
  11. B. Schweizer, Multiplication on the space of probability distribution functions, Aequationes Math. 12 (1975), 156–183.
  12. B. Schweizer and A. Sklar, B. Schweizer and A. Sklar, Probabilistic Metric Spaces, Elsevier, New York, 1983; 2nd ed., Dover, Mineola, NY, 2005.
  13. D. A. Seibley, A metric for weak convergence of distribution functions, Rocky Mountain J. Math. 1 (1971), 427–430.
  14. C. Sempi, A short and partial history of probabilistic normed spaces, Mediterr. J. Math. 3 (2006), 283-300.
  15. C. Sempi, On the space on distribution functions, Riv. Mat. Univ. Parma 8 (1982), 243-250.
  16. A. N. Serstnev, On the motion of a random normed space, ˇ Dokl. Akad. Nauk SSSR 149 (1963), 280-283 (English translation in Soviet Math. Dokl. 4 (1963), 388-390.
  17. E. O. Thorp, Generalized topologies for statistical metric spaces, Fund. Math. 51, 9–21 (1962).
  18. A. Wilansky, Functional Analysis, Blaisdell Publishing Company, New York, Toronto, London, 1964
Volume 1, Issue 1 - Serial Number 1
January 2010
Pages 58-64
  • Receive Date: 11 March 2009
  • Revise Date: 14 October 2009
  • Accept Date: 23 October 2009