Hahn-Banach type theorems and the separation of convex sets for fuzzy quasi-normed spaces

Ruini Li; Jianrong Wu; Ruini Li; Jianrong Wu

doi:10.3934/math.2022183

AIMS Mathematics

2022, Volume 7, Issue 3: 3290-3302. doi: 10.3934/math.2022183

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Research article

Hahn-Banach type theorems and the separation of convex sets for fuzzy quasi-normed spaces

Ruini Li ,
Jianrong Wu ^,

College of Mathematics Science, Suzhou University of Science and Technology, Suzhou, Jiangsu 215009, P. R. China

Received: 08 October 2021 Accepted: 25 November 2021 Published: 29 November 2021
MSC : 46S40, 46B10

In this paper, we first study continuous linear functionals on a fuzzy quasi-normed space, obtain a characterization of continuous linear functionals, and point out that the set of all continuous linear functionals forms a convex cone and can be equipped with a weak fuzzy quasi-norm. Next, we prove a theorem of Hahn-Banach type and two separation theorems for convex subsets of fuzzy quasinormed spaces.
- fuzzy quasi-normed space,
- continuous linear functional,
- Hahn-Banach extension theorem,
- separation theorem
Citation: Ruini Li, Jianrong Wu. Hahn-Banach type theorems and the separation of convex sets for fuzzy quasi-normed spaces[J]. AIMS Mathematics, 2022, 7(3): 3290-3302. doi: 10.3934/math.2022183

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Abstract

In this paper, we first study continuous linear functionals on a fuzzy quasi-normed space, obtain a characterization of continuous linear functionals, and point out that the set of all continuous linear functionals forms a convex cone and can be equipped with a weak fuzzy quasi-norm. Next, we prove a theorem of Hahn-Banach type and two separation theorems for convex subsets of fuzzy quasinormed spaces.

References

[1]	C. Alegre, S. Romaguera, On paratopological vector spaces, Acta Math. Hungar., 101 (2003), 237–261. doi: 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 Sets Syst., 161 (2010), 2181–2192. doi: 10.1016/j.fss.2010.04.002. doi: 10.1016/j.fss.2010.04.002
[3]	C. Alegre, S. Romaguera, The Hahn-Banach extension theorem for fuzzy normed spaces revisited, Abstr. Appl. Anal., 2014 (2014), 1–8. doi: 10.1155/2014/151472. doi: 10.1155/2014/151472
[4]	C. Alegre, S. Romaguera, On the uniform boundedness theorem in fuzzy quasi-normed spaces, Fuzzy Sets Syst., 282 (2016), 143–153. doi: 10.1016/j.fss.2015.02.009. doi: 10.1016/j.fss.2015.02.009
[5]	T. Bag, S. K. Samanta, Finite dimensional fuzzy normed linear spaces, Fuzzy Math., 6 (2003), 687–705.
[6]	S. Cobzas, Functional Analysis in Asymmetric Normed Spaces, Springer Basel, 2013. doi: 10.1007/978-3-0348-0478-3.
[7]	R. Gao, X. Li, J. Wu, The decomposition theorem for a fuzzy quasinorm, J. Math., 2020 (2020), 1–7. doi: 10.1155/2020/8845283. doi: 10.1155/2020/8845283
[8]	A. George, P. Veeramani, On some results in fuzzy metric space, Fuzzy Sets Syst., 64 (1994), 395–399. doi: 10.1016/0165-0114(94)90162-7. doi: 10.1016/0165-0114(94)90162-7
[9]	B.Y. Hussein, F.K. Al-Basri, On the completion of quasi-fuzzy normed algebra over fuzzy field, J. Interdiplinary Math., 25 (2020), 1–13.
[10]	B. Schweizer, A. Sklar, Statistical metric spaces, Pac. J. Math., 10 (1960), 314–334. doi: 10.2140/pjm.1960.10.313. doi: 10.2140/pjm.1960.10.313

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