Document Type : Research Paper

Authors

Department of Mathematics, Sahand University of Technology, P.O.Box 53318-17634, Tabriz, Iran.

Abstract

In this paper, we first show that the  induced topologies by Felbin and Bag-Samanta type fuzzy norms on a linear space $X$ are equivalent. So all results in Felbin-fuzzy normed linear spaces are valid in Bag-Samanta fuzzy normed linear spaces and vice versa. Using this, we will be able to define a fuzzy norm on $FB(X,Y)$, the space of all fuzzy bounded linear operators from $X$ into $Y$, where $X$ and $Y$ are fuzzy normed linear spaces.

Keywords

###### ##### References
[1] O. Kaleva and S. Seikala, On fuzzy metric spaces, Fuzzy Sets Syst., 12 (1984), pp. 143-154.
[2] C. Felbin, Finite dimensional fuzzy normed linear spaces, Fuzzy Sets Syst., 48 (1992), pp. 239-248.
[3] J. Xiao and X. Zhu, On linearly topological structure and property of fuzzy normed linear space, Fuzzy Sets Syst., 125 (2002), 153-161.
[4] J. Xiao, and X. Zhu, Fuzzy normed space of operators and its completeness, Fuzzy Sets Syst., 133 (2003), pp. 389-399.
[5] M. Saheli, S.A.M.Mohsenialhosseini and H. Saedi, The Operators’ Theorems on Fuzzy Topological Spaces, Sahand Commun. Math. Anal., 19 (2022), pp. 57-76.

[6] T. Bag and and S.K. Samanta, Fuzzy bounded linear operators, Fuzzy Sets Syst., 151 (2005), pp. 513-547.
[7] T. Bag and S.K. Samanta, A comparative study of fuzzy norms on a linear space, Fuzzy Sets Syst., 159(2008), pp. 670-684.
[8] T. Bag and S.K. Samanta, Fuzzy bounded linear operators in Felbin's type fuzzy normed linear spaces, Fuzzy Sets Syst., 14 (2007), pp. 234-251.
[9] I. Sadeqi and F.Y. Azari, Gradual normed linear space, Iran. J. Fuzzy Syst., 8 (2011), pp. 131-139.
[10] M. Ettefagh, F. Y. Azari and S. Etemad, S, On some topological properties in gradual normed    spaces, Facta Univ. Ser. Math. Inform., 35 (2020), pp. 549-559.
[11] M. Ettefagh, S. Etemad and F.Y. Azari, (2020), Some properties of sequences in gradual normed spaces, Asian-Eur. J. Math., 13 (2020), pp. 2050085.
[12] C. Choudhury and S. Debnath, On I−convergence of sequences in gradual normed linear spaces, Facta Univ., Ser. Math. Inf., (2021) pp. 595-251.
[13] I. Sadeqi and F.Y. Azari, The Comparative Study of Gradual and Fuzzy Normed Linear Spaces, Journal of Intelligent and Fuzzy Systems,30 (2016) pp. 1195–1198.
[14] B. Daraby, Z. Solimani and A. Rahimi, A note on fuzzy Hilbert spaces, J. Intell. Fuzzy Syst., 159 (2016), pp. 313-319.
[15] B. Daraby, Z. Solimani and A. Rahimi, Some properties of fuzzy Hilbert spaces, Complex Anal. Oper. Theory, 11 (2017), pp. 119-138.
[16] B.C. Tripathy and M. Sen, On fuzzy I-convergent difference sequence space, J. Intell. Fuzzy Syst., 25(3) (2013), pp. 643-647.
[17] B.C. Tripathy and A. Baruah, New type of difference sequence spaces of fuzzy real numbers, Math. Modell. Analysis, 14(3)(2009), pp. 391-397.

[18] D. Dubois and H. Prade, Gradual elements in a fuzzy set, Soft Comput., 12 (2008), pp. 165-175.
[19] T. Bag and S.K. Samanta, Finite dimensional fuzzy normed linear spaces, J. Fuzzy Math, 11 (2003), pp. 687-705.
[20] A. K. Katsaras, Fuzzy topological vector space II, Fuzzy Sets Syst., 12 (1984), pp. 143-154.
[21] F. Lael and K. Nourouzi, Fuzzy compact linear operators, Chaos, Solitons and Fractals, 34 (2007), 1584-1589.
[22] I. Sadeqi and M. Salehi, Fuzzy compact operators and topological degree theory,
Fuzzy Sets Syst., 160 (2009), pp. 1277-1285.
[23] I. Sadeqi and M. Salehi, Relating fuzzy normed linear spaces to topological vector spaces using fuzzy intervals, Fuzzy Sets Syst. (2012), pp. 134-157.

[24] I. Sadeqi and F. Solaty kia, Fuzzy normed linear space and it's topological structure, Chaos, Solitons and Fractals, 40 (2009), pp. 2576-2589.
[25] B.C. Tripathy and A. Baruah Nörlund and Riesz mean of sequences of fuzzy real numbers, Applied Math. Letters, 23(2010), pp. 651-655.
[26] M. Et, B. C. Tripathy and A.J. Dutta, On pointwise statistical convergence of order alpha of sequences of fuzzy mappings, Kuwait J. Sci., 41(3) (2014), pp. 17-30.
[27] J. Xiao and X. Zhu, Topological degree theory and fixed point theorems in fuzzy normed     space, Fuzzy Sets Syst., 147, (2004), pp. 437-452.