Probabilistic Normed Groups

Document Type: Research Paper

Authors

1 Faculty of Mathematics, K.N.Toosi University of Technology, P.O.Box 16315-1618, Tehran, Iran.

2 Department of Mathematics, Payame Noor University, P.O.BOX 19395-3697, Tehran, Iran.

Abstract

In this paper, we introduce the  probabilistic normed groups. Among other results, we investigate the continuity
of inner automorphisms of a group and the continuity of left and right shifts in probabilistic group-norm. We also
 study midconvex functions defined  on probabilistic normed groups and  give  some results about locally boundedness of such  functions.

Keywords


[1] C. Alsina, B. Schweizer and A. Sklar, On the de nition of a probabilistic normed space,
Aequationes Math, 46(2) (1993), 91{98.
[2] N. H. Bingham and A. J. Ostaszewski, Normed versus topological groups: dichotomy and
duality, Dissertationes Math, 472 (2010), 138p.
[3] G. Birkho , A note on topological groups, Compositio Math, 3 (1936), 427{430.
[4] D. R. Farkas, The algebra of norms and expanding maps on groups, J. Algebra, 133(2)
(1990), 386{403.

[5] M. Frechet, Sur quelques points du calcul fonctionnel, Rendiconti de Circolo Matematico di
Palermo, 22 (1906), 1{74.
[6] S. Kakutani,  Uber die Metrisation der topologischen Gruppen, (German) Proc. Imp. Acad,
12(4) (1936), 82{84. (also in Selected Papers, Vol. 1, ed. R. Robert Kallman, Birkhuser,
(1986), 60{62.)
[7] V. L. Klee, Invariant metrics in groups (solution of a problem of Banach), Proc. Amer.
Math. Soc, 3 (1952), 484{487.
[8] E. Klement, R. Mesiar and E. Pap, Triangular norms, Trends in Logica{Studia Logica Library,
Kluwer Academic Publishers, Dordrecht, 8 (2000).
[9] M. Kuczma, An introduction to the theory of functional equations and inequalities, Cauchy's
equation and Jensen's inequality, Second edition, Birkhauser Verlag, Basel, 2009.
[10] K. Menger, Statistical metrics, Proc. Nat. Acad. Sci. U. S. A, 28 (1942), 535{537.
[11] A. A. Pavlov, Normed groups and their application to noncommutative di erential geometry,
J. Math. Sci., 113(5) (2003), 675{682.
[12] B. J. Pettis, On continuity and openness of homomorphisms in topological groups, Ann. of
Math, 52(2) (1950), 293{308.
[13] B. Schweizer and A. Sklar, Probabilistic metric spaces, North-Holland Series in Probability
and Applied Mathematics, North-Holland Publishing Co., New York, 1983.
[14] A. N. Serstnev, On the concept of a stochastic normalized space, (Russian), Dokl. Akad.
Nauk SSSR, 149 (1963), 280{283.
[15] D. A. Sibley, A metric for weak convergence of distribution functions, Rocky Mountain J.
Math, 1(3) (1971), 427{430.