Algebraic Results on Soft Normed Quasilinear Spaces

Abstract

As novel notions of soft normed quasilinear spaces, we define soft quasilinear dependence, soft quasilinear independence and soft quasi basis. One of the greatest obstacles to the improvement of soft normed quasilinear spaces is the presence of these properties. In this study, we will present the definitions of these significant concepts and give some illustrative examples. Additionally, we demonstrate that the proposed definitions agree with counterparts of similar results in soft linear spaces. Finally, in some soft normed quasilinear spaces, we have studied their singular and regular dimensions just as in quasilinear spaces.

Keywords

• Soft set
• Soft quasilinear space
• Soft normed quasilinear space
• Soft quasilinear dependence-independence
• Soft quasi base.

References

1. Aseev, S.M. Quasilinear operators and their application in the theory of multivalued mappings, Proceedings of the Steklov Institute of Mathematics, 2(1986), 23–52.
2. Bozkurt, H. Soft quasilinear spaces and soft normed quasilinear spaces, Adıyaman University Journal of Science, 10(2)(2020), 506–523.
3. Gönci, M.Ş ., Bozkurt, H., Soft inner product quasilinear spaces, Turkic Word Mathematical Society Journal of Applied and Engineering Mathematics, 14(1)(2024), 122–133.
4. Bozkurt, H., Soft quasilinear operator, Mathematical Sciences And Applications e-Notes, 10(2)(2022), 82–92.
5. Bozkurt, H., Soft interval spaces and soft interval sequence spaces, Filomat, 37(9)(2023), 2647–2658.
6. Çakan, S., Some New Results Related to Theory of Normed Quasilinear Spaces, Ph.D. Thesis, ˙In¨on¨u University, Malatya, 2016.
7. Das, S., Samanta, S.K., Soft real sets, soft real numbers and their properties, Journal of Fuzzy Mathematics and Informatics, 6(2)(2012), 551–576.
8. Das, S., Samanta, S.K., Soft metric, Annals of Fuzzy Mathematics and Informatics, 6(1)(2013), 77–94.

9. Das, S., Majumdar, P., Samanta, S. K., On soft linear spaces and soft normed linear spaces, Annals of Fuzzy Mathematics and Informatics, 9(1)(2015), 91–109.
10. Levent, H., Yılmaz, Y., Hahn- Banach extension theorem for interval-valued functions and existence of quasilinear functionals, New Trends in Mathematical Sciences, 6(2)(2018), 19–28.
11. Levent, H., Yılmaz, Y., Translation, modulation and dilation systems set-valued signal processing, Carpathian Mathematical Publications, 10(1)(2018), 143–164.
12. Levent, H., Yılmaz, Y., Inner-product quasilinear spaces with applications in signal processing, Advanced Studies: Euro-Tbilisi Mathematical Journal, 14(4)(2021), 125–146.
13. Levent, H., Yılmaz, Y., Analysis of signals with inexact data by using interval valued functions, The Journal of Analysis, 30(2022), 1635–1651.
14. Levent, H., Yılmaz, Y., Complex interval matrix and its some properties, Turkish Journal of Mathematics and Computer Science, 15(1)(2023), 20–26.
15. Maji, P.K., Biswas, R., Ropy, A.R., Soft set theory, Computation Mathematical Applications, 45(2003), 555–562.
16. Molodtsov, D., Soft set-theory first results, Computational and Applied Mathematics, 37(1999), 19–31.
17. Yazar, M.I., Bilgin, T., Bayramov, S., G¨und¨uz, C¸ ., A new view on soft normed spaces, International Mathematical Forum, 9(24) (2014), 1149–1159.
18. Yazar, M.I., Aras, C¸ .G., Bayramov, S., Results on Hilbert spaces, TWMS Journal of Applications Engineering Mathematics, 9(1)(2019), 159–164.
19. Yılmaz, Y., Bozkurt, H., Levent, H., C¸ etinkaya, U¨ ., Inner product fuzzy quasilinear spaces and some fuzzy sequence spaces, Journal of Mathematics, 2022(2022), 2466817.