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Equicontinuity, affine mean ergodic theorem and linear equations in random normed spaces


Author: V. Radu
Journal: Proc. Amer. Math. Soc. 57 (1976), 299-303
MSC: Primary 47A50; Secondary 46A15, 47A35
DOI: https://doi.org/10.1090/S0002-9939-1976-0473883-3
MathSciNet review: 0473883
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Abstract: The aim of this paper is to give a characterization of equicontinuous families of linear operators in random normed spaces which generalizes the normed spaces case.

This characterization is used to generalize some results on iterative solutions of linear equations by using the affine mean ergodic theorem for locally convex spaces (note that every Fr
'echet space is a random normed space with the $ t$-norm $ T = {\text{Min}}$).

References [Enhancements On Off] (What's this?)

  • [1] W. G. Dotson Jr., Mean ergodic theorems and iterative solution of linear functional equations, J. Math. Anal. Appl. 34 (1971), 141–150. MR 0301529, https://doi.org/10.1016/0022-247X(71)90164-8
  • [2] Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, With the assistance of W. G. Bade and R. G. Bartle. Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers, Ltd., London, 1958. MR 0117523
  • [3] V. I. Istraţescu, Introducere în teoria spaţiilor metrice probabiliste şi aplicaţii, Editura Tehnica, Bucureşti, 1974.
  • [4] Karl Menger, Statistical metrics, Proc. Nat. Acad. Sci. U. S. A. 28 (1942), 535–537. MR 0007576
  • [5] D. H. Muštari, The linearity of isometric mappings of random normed spaces, Kazan. Gos. Univ. Učen. Zap. 128 (1968), no. 2, 86–90 (Russian). MR 0487377
  • [6] Octav Onicescu, Nombres et systèmes aléatoires, Éditions de l’ Académie de la R. P. Roumaine, Bucharest; Éditions Eyrolles, Paris, 1964 (French). MR 0172314
  • [7] V. Radu, Linear operators in random normed spaces, Bull. Math. Soc. Sci. Math. R. S. Roumanie (N.S.) 17(65) (1973), 217–220 (1975). MR 0365650
  • [8] -, On a random norm and continuity of linear operators on random normed spaces, C. R. Acad. Sci. Paris 280 (1975), 1303-1305.
  • [9] B. Schweizer, Lectures on $ P$. $ M$-spaces, Lectures Notes, the University of Arizona, 1965 (unpublished).
  • [10] B. Schweizer and A. Sklar, Statistical metric spaces, Pacific J. Math. 10 (1960), 313–334. MR 0115153
  • [11] A. N. Šerstnev, The notion of random normed space, Dokl. Akad. Nauk SSSR 149 (1963), 280-283 = Soviet Math. Dokl. 4 (1963), 388-391. MR 27 #4268.
  • [12] A. N. Šerstnev, Random normed spaces. Questions of completeness, Kazan. Gos. Univ. Učen. Zap. 122 (1962), no. kn. 4, 3–20 (Russian). MR 0174032

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1976-0473883-3
Keywords: Random normed space, equicontinuity, affine mean ergodic theorem, random asymptotically $ a$-bounded, random asymptotically $ a$-regular
Article copyright: © Copyright 1976 American Mathematical Society