Statistical convergence in function spaces. (English) Zbl 1242.40003

Summary: We study statistical versions of several classical kinds of convergence of sequences of functions between metric spaces (Dini, Arzelà, and Alexandroff) in different function spaces. Also, we discuss a statistical approach to recently introduced notions of strong uniform convergence and exhaustiveness.


40A30 Convergence and divergence of series and sequences of functions
40A35 Ideal and statistical convergence
Full Text: DOI


[1] C. Arzelà, “Intorno alla continuità della somma di infinite funzioni continue,” Rendiconti Accademia Delle Scienze Dell’istituto Di Bologna, pp. 79-84, 1883-1884, Opere, UMI, Vol. I, Ediz. Cremonese, pp. 215-220, 1992.
[2] C. Arzelà, “Sulle serie di funzioni,” Memorie della R. Accademia delle Scienze dell’Istituto di Bologna, vol. 5, no. 8, pp. 131-186, 701-744, 1899-1900, Opere, UMI, Vol. II, Ediz. Cremonese, 1992, pp. 407-466, 467-510.
[3] E. Borel, Lecons sur les Functions de Variables Réelles et les Développements en Séries de Polynômes, Gauthie-Villars, Paris, France, 1905.
[4] R. G. Bartle, “On compactness in functional analysis,” Transactions of the American Mathematical Society, vol. 79, pp. 35-57, 1955. · Zbl 0064.35503 · doi:10.2307/1992835
[5] P. S. Alexandroff, Einführung in die Mengenlehre und die Theorie der reellen Funktionen, Deutscher Verlag der Wissenschaften, Berlin, Germany, 1956, (Translated from the 1948 Russian edition). · Zbl 0071.38304
[6] A. Caserta, G. Di Maio, and L’. Holá, “Arzelà’s theorem and strong uniform convergence on bornologies,” Journal of Mathematical Analysis and Applications, vol. 371, no. 1, pp. 384-392, 2010. · Zbl 1202.54004 · doi:10.1016/j.jmaa.2010.05.042
[7] N. Dunford and J. T. Schwartz, Linear Operators, Part I: General Theory, Interscience Publishers, New York, NY, USA, 3rd edition, 1966. · Zbl 0146.12601
[8] G. Beer and S. Levi, “Strong uniform continuity,” Journal of Mathematical Analysis and Applications, vol. 350, no. 2, pp. 568-589, 2009. · Zbl 1161.54003 · doi:10.1016/j.jmaa.2008.03.058
[9] A. Caserta and L. D. R. Ko\vcinac, “On statistical exhaustiveness,” . Submitted.
[10] J. L. Kelley, General Topology, D. Van Nostrand Company, Princeton, NJ, USA, 1955. · Zbl 0066.16604
[11] V. Gregoriades and N. Papanastassiou, “The notion of exhaustiveness and Ascoli-type theorems,” Topology and its Applications, vol. 155, no. 10, pp. 1111-1128, 2008. · Zbl 1141.26001 · doi:10.1016/j.topol.2008.02.005
[12] H. Hogbe-Nlend, Bornologies and Functional Analysis, North-Holland Publishing Co., Amsterdam, The Netherlands, 1977. · Zbl 0359.46004
[13] S.-T. Hu, “Boundedness in a topological space,” Journal de Mathematiques Pures et Appliquees, vol. 28, pp. 287-320, 1949. · Zbl 0041.31602
[14] G. Beer, “The Alexandroff property and the preservation of strong uniform continuity,” Applied General Topology, vol. 11, no. 2, pp. 117-133, 2010. · Zbl 1252.54004
[15] A. Caserta, G. Di Maio, and L’. Holá, “(Strong) weak exhaustiveness and (strong uniform) continuity,” Filomat, vol. 24, no. 4, pp. 63-75, 2010. · Zbl 1265.54027 · doi:10.2298/FIL1004063C
[16] A. Caserta, G. Di Maio, and L. D. R. Ko\vcinac, “Bornologies, selection principles and function spaces,” Topology and its Applications. In press. · Zbl 1253.54021
[17] A. Zygmund, Trigonometric Series, Cambridge University Press, Cambridge, UK, 2nd edition, 1979.
[18] H. Fast, “Sur la convergence statistique,” Colloquium Mathematicum, vol. 2, pp. 241-244, 1951. · Zbl 0044.33605
[19] H. Steinhaus, “Sur la convergence ordinaire et la convergence asymptotique,” Colloquium Mathematicum, vol. 2, pp. 73-74, 1951.
[20] G. Di Maio and L. D. R. Ko\vcinac, “Statistical convergence in topology,” Topology and its Applications, vol. 156, no. 1, pp. 28-45, 2008. · Zbl 1155.54004 · doi:10.1016/j.topol.2008.01.015
[21] H. \cCakalli, “Slowly oscillating continuity,” Abstract and Applied Analysis, vol. 2008, Article ID 485706, 5 pages, 2008. · Zbl 1153.26002 · doi:10.1155/2008/485706
[22] S. Aytar and S. Pehlivan, “Statistically monotonic and statistically bounded sequences of fuzzy numbers,” Information Sciences, vol. 176, no. 6, pp. 733-744, 2006. · Zbl 1089.40001 · doi:10.1016/j.ins.2005.03.015
[23] J. A. Fridy, “On statistical convergence,” Analysis, vol. 5, no. 4, pp. 301-313, 1985. · Zbl 0588.40001
[24] T. , “On statistically convergent sequences of real numbers,” Mathematica Slovaca, vol. 30, no. 2, pp. 139-150, 1980. · Zbl 0437.40003
[25] A. V. Arhangel’skiĭ, Topological Function Spaces, Kluwer Academic Publishers, Dodrecht, The Netherlands, 1992.
[26] R. Engelking, General Topology, Heldermann Verlag, Berlin, Germany, 1989. · Zbl 0684.54001
[27] R. A. McCoy and I. Ntantu, Topological Properties of Spaces of Continuous Functions, Lecture Notes in Mathematics, vol. 1315, Springer-Verlag, Berlin, Germany, 1988. · Zbl 0647.54001
[28] U. Dini, Fondamenti per la Teorica delle Funzioni di Variabili Reali, T. Nistri, T. Nistri, Italy, 1878, [UMI, 1990].
[29] E. W. Hobson, The Theory of Functions of a Real Variable and the Theory of Fourier’s Series, vol. 2, Cambridge University Press, Cambridge, UK, 2nd edition, 1926.
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