In this paper, firstly, we introduce the concepts of continuity and boundedness of linear operators between two fuzzy quasi-normed spaces with general continuous t-norms, prove the equivalence of them, and point out that the set of all continuous linear operators forms a convex cone. Secondly, we establish the family of star quasi-seminorms on the cone of continuous linear operators, and construct a fuzzy quasi-norm of a continuous linear operator.
Citation: Han Wang, Jianrong Wu. The norm of continuous linear operator between two fuzzy quasi-normed spaces[J]. AIMS Mathematics, 2022, 7(7): 11759-11771. doi: 10.3934/math.2022655
In this paper, firstly, we introduce the concepts of continuity and boundedness of linear operators between two fuzzy quasi-normed spaces with general continuous t-norms, prove the equivalence of them, and point out that the set of all continuous linear operators forms a convex cone. Secondly, we establish the family of star quasi-seminorms on the cone of continuous linear operators, and construct a fuzzy quasi-norm of a continuous linear operator.
| [1] |
C. Alegre, S. Romaguera, On paratopological vector spaces, Acta Math. Hung. , 101 (2003), 237. https://doi.org/10.1023/b:amhu.0000003908.28255.22 doi: 10.1023/b:amhu.0000003908.28255.22
|
| [2] |
C. Alegre, S. Romaguera, Characterizations of metrizable topological vector spaces and their asymmetric generalizations in terms of fuzzy (quasi-) norms, Fuzzy Set. Syst. , 161 (2010), 2181–2192. https://doi.org/10.1016/j.fss.2010.04.002 doi: 10.1016/j.fss.2010.04.002
|
| [3] |
C. Alegre, S. Romaguera, On the uniform boundedness theorem in fuzzy quasi-normed spaces, Fuzzy Set. Syst. , 282 (2016), 143–153. https://doi.org/10.1016/j.fss.2015.02.009 doi: 10.1016/j.fss.2015.02.009
|
| [4] | T. Bag, S. Samanta, Finite dimensional fuzzy normed linear spaces, J. Fuzzy Math. , 11 (2003), 687–705. |
| [5] |
T. Bag, S. Samanta, Fuzzy bounded linear operators, Fuzzy Set. Syst. , 151 (2005), 513–547. https://doi.org/10.1016/j.fss.2004.05.004 doi: 10.1016/j.fss.2004.05.004
|
| [6] |
T. Bag, S. Samanta, Fuzzy bounded linear operators in Febin's type fuzzy normed linear spaces, Fuzzy Set. Syst. , 159 (2008), 685–707. https://doi.org/10.1016/j.fss.2007.09.006 doi: 10.1016/j.fss.2007.09.006
|
| [7] | T. Bag, S. Samanta, Finite dimensional fuzzy normed linear spaces, Ann. Fuzzy Math. Inform. , 6 (2013), 271–283. |
| [8] | S. Cheng, J. Mordeson, Fuzzy linear operators and fuzzy normed linear spaces, Bull. Am. Math. Soc. , 86 (1994), 429–436. |
| [9] | Y. Cho, T. Rassias, R. Saadati, Fuzzy operator theory in mathematical analysis, Cham: Springer, 2018. https://doi.org/10.1007/978-3-319-93501-0 |
| [10] | S. Cobzas, Functional analysis in asymmetric normed spaces, Basel: Birkhäuser, 2013. https://doi.org/10.1007/978-3-0348-0478-3 |
| [11] |
C. Felbin, Finite dimensional fuzzy normed linear spaces, Fuzzy Set. Syst. , 48 (1992), 239–248. https://doi.org/10.1016/0165-0114(92)90338-5 doi: 10.1016/0165-0114(92)90338-5
|
| [12] | C. Felbin, Finite dimensional fuzzy normed linear spaces Ⅱ, Journal of Analysis, 7 (1999), 117–131. |
| [13] |
R. Gao, X. Li, J. Wu, The decomposition theorem for a fuzzy quasi-norm, J. Math. , 2020 (2020), 8845283. https://doi.org/10.1155/2020/8845283 doi: 10.1155/2020/8845283
|
| [14] |
A. George, P. Veeramani, On some results in fuzzy metric space, Fuzzy Set. Syst. , 64 (1994), 395–399. https://doi.org/10.1016/0165-0114(94)90162-7 doi: 10.1016/0165-0114(94)90162-7
|
| [15] |
B. Hussein, F. Al-Basri, On the completion of quasi-fuzzy normed algebra over fuzzy field, J. Interdiscip. Math. , 23 (2020), 1033–1045. https://doi.org/10.1080/09720502.2020.1776942 doi: 10.1080/09720502.2020.1776942
|
| [16] | I. Kramosil, J. Michalek, Fuzzy metrics and statistical metric spaces, Kybernetika, 11 (1975), 326–334. |
| [17] |
O. Kaleva, S. Seikkala, On fuzzy metric spaces, Fuzzy Set. Syst., 12 (1984), 215–229. https://doi.org/10.1016/0165-0114(84)90069-1 doi: 10.1016/0165-0114(84)90069-1
|
| [18] |
A. Katsars, Fuzzy topological vector spaces Ⅱ, Fuzzy Set. Syst., 12 (1984), 143–154. https://doi.org/10.1016/0165-0114(84)90034-4 doi: 10.1016/0165-0114(84)90034-4
|
| [19] |
R. Li, J. Wu, Hahn-Banach type theorems and the separation of convex sets for fuzzy quasi-normed spaces, AIMS Mathematics, 7 (2021), 3290–3302. https://doi.org/10.3934/math.202218320 doi: 10.3934/math.202218320
|
| [20] | V. Radu, Some fixed point theorems in probabilistic metric spaces, In: Lecture notes in mathematics, Berlin: Springer, 2006, 125–133. https://doi.org/10.1007/BFb0072718 |
| [21] |
B. Schweizer, A. Sklar, Statistical metric spaces, Pacific J. Math., 10 (1960), 313–334. https://doi.org/10.2140/pjm.1960.10.673 doi: 10.2140/pjm.1960.10.673
|
| [22] |
I. Sadeqi, F. Kia, Fuzzy normed linear space and its topological structure, Chaos Soliton. Fract., 40 (2009), 2576–2589. https://doi.org/10.1016/j.chaos.2007.10.051 doi: 10.1016/j.chaos.2007.10.051
|
| [23] |
M. Sharma, D. Hazarika, On linear operators in Felbin's fuzzy normed space, Ann. Fuzzy Math. Inform., 13 (2017), 749–758. https://doi.org/10.30948/afmi.2017.13.6.749 doi: 10.30948/afmi.2017.13.6.749
|
| [24] |
M. Sharma, D. Hazarika, Fuzzy bounded linear operator in fuzzy normed linear spaces and its fuzzy compactness, New Math. Nat. Comput., 16 (2020), 177–193. https://doi.org/10.1142/S1793005720500118 doi: 10.1142/S1793005720500118
|
| [25] |
J. Xiao, X. Zhu, Fuzzy normed linear space of operators and its completeness, Fuzzy Set. Syst., 133 (2003), 389–399. https://doi.org/10.1016/S0165-0114(02)00274-9 doi: 10.1016/S0165-0114(02)00274-9
|
Han Wang, Jianrong Wu. The norm of continuous linear operator between two fuzzy quasi-normed spaces[J]. AIMS Mathematics, 2022, 7(7): 11759-11771. doi: 10.3934/math.2022655
DownLoad: