On Paranormed and Locally Convex Quasilinear Spaces

Abstract

The concept of a normed quasilinear space was dened by Aseev in [1]. Based on this concept, topological quasilinear spaces are given in [5] and their properties are analysed. Denition of seminormed quasilinear spaces based on these denitions was given in [8] and their topological structure is analysed in this reference. In this paper we will introduce paranormed and locally convex quasilinear spaces and will give some examples. We observe that every normed quasilinear space is a paranormed quasilinear space. Finall we will prove that the topology of a locally convex quasilinear space comes from a family of seminorms which are dened on the quasilinear space.

Keywords

• Quasilinear space
• Paranormed quasilinear space
• Locally convex quasilinear space

References

1. [1] Aseev, S. M., Quasilinear operators and their application in the theory of multivalued mappings, Proceedings of the Steklov Institute of Mathematics, Issue 2(1986), 23-52.
2. [2] Yılmaz, Y. and Levent, H., Inner-product quasilinear spaces with applications in signal processing, Advanced Studies: Euro-Tbilisi Mathematical Journal, 14 (4) (2021).
3. [3] Yılmaz, Y., Bozkurt, H. and C¸ akan, S., On ortonormal sets in inner product quasilinear spaces, Creat. Math. Inform., 25 (2016), 229-239.
4. [4] Levent, H. and Yılmaz, Y., Translation, modulation and dilation systems in set-valued signal processing, Carpathian Math. Publ. 10 (2018), 10-31.
5. [5] Yılmaz, Y., C¸ akan, S. and Aytekin, S., Topological quasilinear spaces, Abstr. Appl. Anal., Article ID 951374(2012), 10 pages.
6. [6] Bozkurt, H. and Yılmaz, Y., Further results in inner product quasilinear spaces, International Journal of Advances in Mathematics, (2019), 34-44 .
7. [7] Bozkurt, H., ˙Ic¸ c¸arpım quasilineer uzayları ve bazı genelles¸tirmeleri, Fen Bilimleri Enstit¨us¨u, Doctorate thesis, 2016.
8. [8] C¸ akan, S. and Yılmaz, Y., A generalization of the Hahn-Banach theorem in seminormed quasilinear spaces, Journal of Mathematics and Applications, Vol. 42 (2019).

9. [9] Banazılı, H. K., On quasilinear operators between quasilinear spaces, Master Thesis, ˙In¨on¨u University, 2014.
10. [10] Maddox, I. J., Elements of functional analysis, Camb. Univ. Press, New York, 2. Edit, 1988.
11. [11] Kreyszig, E., Introductory functional analysis with applications, John Wiley & Sons, 1991.
12. [12] Wilansky, A., Modern methods in topological vector spaces, McGraw-Hill, New York, 1978.
13. [13] Levent, H. and Yılmaz, Y., Analysis of signals with inexact data by using interval-valued functions, The Journal of Analysis, Volume 30 (2022), 1635–1651.
14. [14] Khan, V. A. and Tuba, U., On paranormed ideal convergent sequence spaces defined by Jordantotient function, Journal of Inequalities and Applications, 96 (2021).
15. [15] Khan, V. A., Yasmeen, Fatima, H. and Altaf, H., A new type of paranorm intuitionistic fuzzy zweier I- convergentdouble sequence spaces, Filomat, 33(5) (2019), 1279-1286.
16. [16] Khan, V. A., Shafiq, M. and Lafuerza-Guillen, B., On paranorm I-convergent sequence spaces defined by a compact operator, Africa Matematika, Vol. 26, Issue 7-8 (2015), 1387-1398.
17. [17] Khan, V. A., Alshlool, K. M., Abdullah, S. A. A., Fatima, H. and Ahmad, A., Some new classes of paranorm ideal convergent double sequences of s- bounded variation over n-normed spaces, Cogent Mathematics, 5 (2018).